CHAPTER II.

The Structure of Linear Domains.

1. Preliminary Concepts and Definitions.

57. The totality of all points belonging to a rectilinear segment bounded by the points \(A\) and \(B\) we shall call a linear interval and denote by \(L\); the length of this interval we shall call \(l\); the endpoints of the interval will be its boundaries; its other points are called interior points of the interval; all the remaining points of the unbounded straight line \(OX\) will be the exterior points of \(L\).

This interval is called closed if its boundaries are included among the points of the interval, and open in the contrary case.

If \(AB\) is an open interval, then we may regard it as the limit of interior closed intervals \(a_i b_i\), whose boundaries lie inside \(AB\) and, according to some law, approach indefinitely: \(a_i\) to \(A\) and \(b_i\) to \(B\).

In exactly the same way, a closed interval may be represented as the difference between some segment \(OX\) and the limit of exterior closed intervals \(A_i B_i\), whose boundaries—the exterior points \(A_i\) and \(B_i\)—approach indefinitely, respectively, \(A\) and \(B\). In the latter case the interval \(L\) may be arbitrarily small and may consist of a single point, so that then \(l=0\).

Suppose we have on a straight line two intervals \(L_1\) and \(L_2\), lying outside one another. They are called adjacent if they have a common boundary \(B\); this boundary may belong to both intervals or to only one of them, so that the other will then be open, or to neither the one nor the other, in which case both intervals will be open at this point.

If the intervals \(L_1\) and \(L_2\) do not have a common boundary, so that between them there are points not belonging to them, \(L_1\) and \(L_2\) will be non-adjacent.

58. A linear point domain is the aggregate of a finite or infinitely large number of precisely determined values, selected according to some strictly definite law from the linear continuum; the latter we shall represent as rectilinear. Each element of the domain is a point.

If a system of points \(x\) constitutes a domain \(P\), we shall use the notation \(\{x\}\equiv P\).

C     A     X     B     D
──────•─────•─────•─────•──
          r     r

If, from the point \(x\), on both sides along the straight line, we lay off segments equal to \(r\), then the domain of points entering into the composition of the whole segment \(AB\) we shall call the neighborhood of radius \(r\) of the point \(x\).

If two domains \(P_1\) and \(P_2\) have no common points, then the domain \(P\), made up of all points entering into \(P_1\) and \(P_2\), we, following Cantor’s first notation¹, shall call their sum and shall put

\[ \tag{1} P_1 + P_2 \equiv P . \]

If the composition of \(P_1\) and \(P_2\) also includes common points, then the aggregate \(P\) of all possible distinct points entering into \(P_1\) and \(P_2\) we shall call their least multiple and denote

\[ \tag{2} M(P_1, P_2)\equiv P . \]

If we have to determine all possible distinct points entering into an infinite series of domains

\[ \tag{3} P_1, P_2, P_3, \ldots, P_n, \ldots, \]

we shall use the notation

\[ \overset{\infty}{\underset{1}{M}}(P_i)\equiv P . \]

If the points of \(P\) enter wholly into the composition of \(Q\), then \(P\) is called a divisor of \(Q\), and Cantor denotes this fact by

\[ \tag{4} P\equiv D(Q). \]

If we wish to denote the domain of common points entering into all domains of the series (3), we write

\[ \overset{\infty}{\underset{1}{D}}(P_i)\equiv P . \]

The domain of points common to two domains \(P\) and \(Q\) is denoted by \(D(P,Q)\). The domain \(R\) of the points of \(Q\) not entering into the composition of its part \(P\) is called the difference of \(Q\) and \(P\),

\[ R\equiv Q-P . \]

If, by chance, not a single point occurs in the composition of the domain \(P\), we shall write \(P \equiv 0\).

We constantly use these notations of Cantor’s; other notations were also proposed by Dedekind¹, but they did not take root in science because of their artificiality.

59. If a domain \(P\), situated on some interval \(L\), does not comprise all its points, then the totality of those points of \(L\) which do not enter into the composition of \(P\) constitutes a domain \(\Pi\), complementary to \(P\) with respect to \(L\). We shall always denote a principal domain and its complementary domain by the same capital Latin and Greek letters. Since \(P\) and \(\Pi\), having no common points, complete one another up to \(L\), it follows, according to (1), that

\[ P+\Pi \equiv L. \tag{5} \]

60. If \(P\) does not include all points of \(L\), and \(x_0\) is one of the points of \(L\), then one of the following three mutually exclusive alternatives holds for it:

a. It is possible to find for \(x_0\) such a sufficiently small neighborhood \(r\) that all its points will enter into the composition of \(P\); a point \(x_0\) satisfying this condition is called an interior point of \(P\).

b. It is possible to determine the neighborhood \(r\) so that all its points, and consequently also \(x_0\), will be points of \(\Pi\); then \(x_0\) will be an exterior point of \(P\).

c. Finally, it is possible that, however small \(r\) may be, every neighborhood \(r\) of the point \(x_0\) contains both points of \(P\) and points of \(\Pi\); then a point \(x_0\) possessing this property is called a boundary point of the domain \(P\).

The point \(x_0\) will also be a boundary point in the case when all points of a neighborhood, except \(x_0\), belong to \(P\), while \(x_0\) itself enters into the composition of \(\Pi\), or conversely.

The totality \(C\) of all boundary points of the domain \(P\) we shall call the contour of the domain, borrowing this concept from plane geometry. The points of the contour \(C\) may enter either into the composition of \(P\) or into the composition of \(\Pi\), and may also belong partly to \(P\), partly to \(\Pi\); in the first case or in the second, i.e. when

\[ C = D(P), \quad \text{or} \quad C = D(\Pi), \]

the domain \(\Pi\), or \(P\), is called open, and the other—closed.

Thus, a closed domain contains in itself all boundary points².

61. To determine the position of a point \(x\) on a straight line, we may choose on it, arbitrarily, an origin of coordinates, from which we shall measure the distances of points; the origin will be assumed chosen so that the distances of the points under investigation from the origin are always positive, or, in the limiting case, equal to zero.

By \(x\) we shall henceforth understand either a definite point of the straight line, or the distance of this point from the origin; in other words, we shall regard \(x\) as given geometrically or analytically; clearly, such a dual point of view can give rise to no misunderstanding.

If the coordinates \(x\) of the points of a domain \(P\) satisfy the condition

\[ a < x < A, \]

where \(a\) and \(A\) are finite positive numbers, the domain is called bounded (borné). For such a domain we can define\(^1\) the lower and upper bounds of the values of \(x\)

\[ \mathrm{uGr}\,x = a_0, \qquad \mathrm{oGr}\,x = A_0. \]

The introduction of such notation for the upper and lower bounds is very desirable, and it was proposed by us\(^2\) as early as 1898. According to the adopted convention, \(a_0\) may be positive or equal to \(0\).

If we take the interval \((a_0,A_0)\), then inside it there are points of \(P\); they may also coincide with its endpoints, but outside \((a_0,A_0)\) there are certainly no points of \(P\). Such an interval we shall call the fundamental interval, and it is precisely this interval that will henceforth be denoted by \(L\).

It is evident that \(a_0\) and \(A_0\), according to their definition, belong to the set of contour points.

62. Up to now we have considered the points \(x\) of a domain \(P\) in relation to the points of the interval occupied by them, or, in a narrower sense, of the fundamental interval. We shall now pass to the mutual arrangement of the points of \(P\), without taking into account their relation to the points \(P\).

Let \(x_0\) be one of the points of the domain \(P\). If we take for it some neighborhood, then in this neighborhood

a) there may be no point of \(P\);

b) there may be a finite number of them;

c) there may be infinitely many points of \(P\);

d) all points may turn out to be points of \(P\).

\(^1\) Bolzano, Rein analytischer Beweis, § 12.
\(^2\) Diary of the X Congress of Naturalists and Physicians, p. 427.

It is obvious that the second case reduces to the first, since, taking \(r_0\) smaller than the least of the distances from \(x_0\) to all points lying within the neighborhood \(r\), we obtain a neighborhood \(r_0\) satisfying condition a); thus here we can have to do with only three possibilities.

In the first case the point \(x_0\) bears the name of an isolated point of the domain \(P\); consequently, for such a point it is always possible to construct a neighborhood with finite \(r\), free from the other points of \(P\), except \(x_0\).

It is obvious that every isolated point is a point of the contour, since in an arbitrary neighborhood of it there are always points of \(P\) and of \(\Pi\), namely \(x_0\) is a point of \(P\) and all the rest, except it, are points of \(\Pi\).

In the third case c), if, however small \(r\) may be, in the neighborhood of \(x_0\) other points of \(P\) are always found, \(x_0\) is called a limiting point of the domain \(P\). Since in an arbitrary such neighborhood there may also be points of \(\Pi\) together with them, in this latter case \(x_0\) will also be a limiting point for the domain \(\Pi\). A common limiting point of \(P\) and \(\Pi\) must belong to the contour of \(P\).

In the last case d) the point \(x_0\) will also be a limiting point, but at the same time it will be an interior point of \(P\).

Thus all points of a domain, as regards their mutual disposition, will be either limiting points or isolated points.

All interior points of the domain \(P\) will eo ipso be its limiting points; points of the contour, however, may fall into either the one or the other category, i.e. be either limiting or isolated points of \(P\).

If, in an arbitrary neighborhood of the point \(x_0\), the points of \(P\) are always situated only on one side of \(x_0\), the point \(x_0\) will be called a one-sided limiting point; in the contrary case it will be two-sided.

63. If all points of the domain \(P\) are its limiting points, \(P\) is called a dense-in-itself domain (in sich dicht).

If, on the contrary, only isolated points enter into its composition, the domain is called isolated. But it does not follow from this that, if a domain is isolated, then it has no limiting points; these points, generally speaking, exist, but they simply do not enter into the composition of the domain.

If both isolated and limiting points enter into the composition of a domain, but in such a way that it is impossible to separate off from \(P\) any part consisting exclusively of limiting points, such a domain bears the name separate (separirt); in this domain the limiting and isolated points are, so to speak, in organic connection with one another.

Calling \(P_i\) the set of isolated points in \(P\), and \(P_g\) the set of limit points, we shall have

\[ P = P_i + P_g . \]

Let us note, however, that, considering \(P_g\) as an independent domain, we can by no means assert that it will be dense; indeed: \(x_0\), belonging to \(P_g\), may serve as a limit point for the points \(x_i\) that have entered into \(P_i\).

For an isolated domain,

\[ P_g = 0,\quad \text{and consequently}\quad P = P_i; \]

and for a dense one,

\[ P_i = 0,\quad \text{and consequently}\quad P = P_g . \]

64. On the basis of the well-known theorem of Weierstrass—for every domain consisting of an infinite number of points, there is, at least, one limit point.

The set of all limit points for the domain \(P\) constitutes a new domain \(P'\), which—according to Cantor—is called the first derivative.

The domain of limit points for the points \(P'\) will be the second derivative of \(P\); it is denoted by \(P''\); and so on; in general we shall have \(P^{(n)}\). The finding of successive derivatives \(P^{(n)}\) is possible as long as each of the domains \(P^{(n-1)}\) contains infinitely many points.

According as each of the \(P^{(n)}\) consists of an infinite number of points, or—on the contrary—there is one of the \(P^{(n)}\) that contains only a finite number of them, domains are divided into two kinds: the latter are called domains of the first kind, and the former—domains of the second kind.

If for a domain of the first kind \(P^{(n+1)} = 0\), then the domain is assigned to the \(n\)-th species.

If each \(P^{(n)}\) consists of an infinitely large number of points, then there exist points entering into all \(P^{(n)}\); the domain of such points will be

\[ P^{(\omega)} = \underset{1}{\overset{\omega}{D}}\{P^{(i)}\}. \]

If \(P^{(\omega)}\) does not consist of a finite number of points, the domains

\[ P^{(\omega+1)},\ P^{(\omega+2)},\ldots,\ P^{(\alpha)},\ldots; \]

if, no matter how far the process is continued, it is always possible to find a domain

\[ \mathrm{P}^{(\Omega)}=\underset{1}{\overset{\Omega}{D}}\{\mathrm{P}^{(i)}\} \]

of points belonging to all the \(\mathrm{P}^{(\alpha)}\), where \(\alpha\) is any of the numbers of the 1st or 2nd class, then this domain \(\mathrm{P}^{(\Omega)}\) will be perfect and closed.

65. Using the concept of the first derived domain \(\mathrm{P}'\), we may modify the definitions of 60°, introduced by Riemann, and take them in the form of Jordan¹, a form somewhat more complicated in appearance, but sometimes more convenient.

Let \(\xi\) be a point of the contour \(C\). If it is an isolated point of \(\mathrm{P}\), then it belongs to \(\mathrm{P}\) and to \(\Pi'\); if it is a limiting point and belongs to the composition of \(\mathrm{P}\), then it will belong to \(\mathrm{P}\) and to \(\Pi'\); if it is a limiting point but does not belong to \(\mathrm{P}\), then it must belong to \(\Pi\) and \(\mathrm{P}'\). Thus \(\xi\) must always belong to one of the domains \(\mathrm{P}\) or \(\Pi\) and to the derived domain of the other domain.

On the other hand, the interior points of \(\mathrm{P}\) belong only to \(\mathrm{P}'\), and the exterior ones only to \(\Pi'\).

Thus the distinction between points of the three categories may be expressed as follows:

a) the interior points of the domain \(\mathrm{P}\) are those points which belong to \(\mathrm{P}\) and do not belong to \(\Pi'\).

b) the exterior points are those which belong to \(\Pi\) and do not belong to \(\mathrm{P}'\).

c) the boundary points are those which belong simultaneously to \(\mathrm{P}\) and \(\Pi'\), or to \(\Pi\) and \(\mathrm{P}'\).

It is obvious that the two definitions, those of Riemann and of Jordan, coincide completely in substance, and we shall use whichever of them is more advantageous in the given case.

Using Jordan’s definition, it is easy to prove that for every domain, if it does not embrace the entire continuum, there exist boundary points; in other words—every domain has a contour.

66. We pass next to the classification of domains according to their composition.

A. The simplest domains will be those in which all points are isolated. For such a domain \(\mathrm{P}\) all limiting points do not enter into the composition of \(\mathrm{P}\), so that \(\mathrm{P}\) and \(\mathrm{P}'\) have no common elements, i.e.

\[ \mathrm{D}(\mathrm{P},\mathrm{P}')=0, \tag{6} \]

although in this case, generally speaking, \(\mathrm{P}'\ne 0\).

Equality (6) serves as the definition of an isolated domain.

B. If all points of \(P\) are limit points, but along with them there also exist limit points not included in \(P\), the domain is called condensed. For it, \(P\) enters as a component part into \(P'\), but does not coincide with \(P'\), so that

\[ P = D(P'), \]

or

\[ D(P, P') = P. \]

C. If all points of \(P\) are limit points and, besides them, \(P\) has no other limit points, so that every limit point belongs to \(P\), then the domain \(P\) is called perfect; for it \(P'\) completely coincides with \(P\)

\[ P = P'. \]

D. Suppose further that \(P\) includes isolated points and along with them all limit points of \(P\); then the entire contour of \(P\) belongs to \(P\), and—consequently—this domain will be closed. For it, \(P'\) is wholly included in \(P\), so that

\[ P = P_i + P', \qquad P' = D(P) \]

or

\[ D(P, P') = P'. \]

We see that the definition of a closed domain in §60 turns out to be identical with Cantor’s usual definition.

E. Finally, we obtain a domain \(P\) of general type if it includes both isolated points and limit points, but not in its entire composition.

According to §63, for it we have

\[ P = P_i + P_g \]

or

\[ P = P_i + D(P'), \]

where \(D(P')\) denotes some component part of \(P'\).

          (P = P_i)  ------>  (P = P_g)
              |                  |
              v                  v
          (P_i + P_g) ----->  (P = P')
                \              /
                 \            /
                  v          v
                    (P_i + P')

The interrelation of all the above-mentioned categories of domains may be represented by the following scheme:

Here the directions of the arrows show the addition of points which transfers domains from one category to another: the addition to \(P_i\) of some, but not all, limit points gives a domain of general type \(P_i + P_g\); the further addition of all the remaining limit points transfers \(P_i + P_g\) into the category of closed-

regions \(P_i + P'\). The addition to \(P_g\) of the remaining limit points, in the absence of isolated points, transforms \(P_g\) into the perfect region \(P'\); the addition to \(P'\) of isolated points, which admit no limits other than the points of \(P'\), gives \(P_i + P'\). Likewise, the addition to \(P_g\) first of the isolated points, and then of the remaining limit points, carries \(P_g\) into \(P_i + P_g\) and then into \(P_i + P'\). These will be the transformations of regions most often encountered; in the diagram they are indicated by solid arrows. Besides these, from \(P_i\) it is sometimes possible to obtain \(P_g\) and from \(P_i + P_g\), \(P'\); these accidental transformations are indicated by dotted arrows.

67. We shall next give a series of consequences of the definitions of §66.

A. From every region \(P\) one obtains a closed region if one takes

\[ \mathrm{M}(P, P'). \]

B. The derived region \(P'\) is always closed:

\[ P'' \equiv D(P'). \]

It follows further from this that

C. Every derived set \(P^{(n)}\) consists of the points of \(P'\):

\[ P^{(n)} \equiv D(P'). \]

D. The derivative of the sum of two regions \(P_1\) and \(P_2\) is the least multiple of their derivatives.

Let

\[ P \equiv P_1 + P_2. \]

The limit points of the region \(P\) may be divided into 3 categories:

\[ P' = Q_1 + Q_2 + Q_{12}, \]

where \(Q_1\) and \(Q_2\) are those limit points of \(P\) in whose derived neighborhoods there lie only points of \(P_1\), or only points of \(P_2\); while those points in whose derived neighborhoods there lie both points of \(P_1\) and points of \(P_2\) are included in \(Q_{12}\).

Then

\[ P_1' = Q_1 + Q_{12}, \qquad P_2' \equiv Q_2 + Q_{12}; \]

hence\(^1\)

\[ \mathrm{M}(P_1', P_2') = Q_1 + Q_2 + Q_{12} \equiv P'. \]

\(^1\) Cf. Schoenflies, Bericht, pp. 61–62.

Similarly one may prove that

E. The derivative of the sum of a finite number of regions without common points is the least multiple of their derivatives

\[ \left\{ \sum_{1}^{n} P_\nu \right\}' = M_{1}^{n}\{P_\nu'\}. \]

It is evident that, if the number of regions \(P_\nu\) is infinitely large, the theorem loses its force; then one can only assert that

F. For an infinite series of regions

\[ M_{1}^{\infty}\{P_\nu'\} = D\left\{ \sum_{1}^{\infty} P_\nu \right\}'. \]

Indeed: besides the limit points entering into the composition of \(P_\nu'\), there may still appear such points in every neighborhood of which there lies only a finite number of points of each of the regions \(P_\nu\).

G. For a dense region \(P\) its derivative \(P'\) is perfect.

H. A dense and closed region is perfect.

Theorems A, B, C belong to Cantor, G—to Cantor and Bendixson simultaneously; theorems D, E, F are implicitly contained in Cantor and were first introduced by Schoenflies.

68. Having a finite or infinite group of precisely distinguishable objects, and abstracting from their individual properties and order, we obtain a certain concept which Cantor calls the power (Mächtigkeit) of the group; for a finite group this will be the number of objects.

If between two groups it is possible to establish a one-to-one correspondence of the objects, the groups have one and the same power; such groups are called equipotent.

The simplest powers with which one has to deal are the power of the sequence of natural numbers and the power of the continuum.

If it is possible to establish a one-to-one correspondence with the sequence of natural numbers—in other words, to renumber the points of a region, i.e. to assign to each a definite number from the sequence of integers

\[ 1,\ 2,\ 3,\ \ldots,\ n,\ \ldots, \]

then the region is called countable (abzählbar); if such an enumeration is impossible, it is uncountable, since in what follows we

we shall be dealing only with cardinality of the continuum, then uncountability must always be understood precisely in this sense.

It is further known that a countable sequence of countable sequences will also be a countable sequence.

Applying these notions to sets, we have a series of theorems:

A. An isolated set is always countable.

B. A perfect set always has cardinality of the continuum.

C. Sets of the general type, closed and condensed, may be either countable or uncountable.

D. If \(P'\) is countable, then \(P\) is countable.

E. If \(P'\) is uncountable, then \(P\) may be countable or uncountable.

F. If \(P'\) is uncountable, then

\[ P' = R + S, \]

where \(R\) is countable, and \(S\) is perfect.

69. Let the interval \(l\) be divided into partial intervals, and let the number of division points increase without bound.

The question arises whether the number of these intervals, which do not overlap one another and fill the whole interval, will be countable, i.e., whether it is possible to assign to each a certain definite number?

To clarify this question, take some sufficiently small number \(\varepsilon\) and form the sequence

\[ \varepsilon,\ \frac{\varepsilon}{2},\ \frac{\varepsilon}{2^2},\ldots,\frac{\varepsilon}{2^n-1},\frac{\varepsilon}{2^n},\ldots; \tag{7} \]

select from the set of intervals those intervals whose length exceeds \(\varepsilon\); since \(\varepsilon\) is a finite number, and the length of the whole interval \(l\) is also finite, there can be only a limited number \(m\) of intervals exceeding \(\varepsilon\) in length. Let us call them, in order of their magnitudes,

\[ \lambda_1,\lambda_2,\lambda_3,\ldots,\lambda_m \quad \text{for } \lambda_i>\varepsilon. \tag{8} \]

Next select those intervals whose length is \(\leqq \varepsilon\), but \(> \dfrac{\varepsilon}{2}\); their number must also be limited,

\[ \lambda'_1,\lambda'_2,\lambda'_3,\ldots,\lambda'_{m_1} \quad \text{for } \varepsilon \geqq \lambda'_i > \frac{\varepsilon}{2}; \tag{9} \]

and so on; take those intervals whose length is \(\leqq \dfrac{\varepsilon}{2^{n-1}}\), but \(> \dfrac{\varepsilon}{2^n}\),

call them

\[ (10)\qquad \lambda^{(n)}_1,\ \lambda^{(n)}_2,\ \lambda^{(n)}_3,\ldots,\lambda^{(n)}_{m_n} \quad \text{when} \quad \frac{\varepsilon}{2^{n}-1} \ge \lambda^{(n)}_i > \frac{\varepsilon}{2^n}; \]

their number again must be finite, since \(\frac{\varepsilon}{2^n}\), for every finite \(n\), is a finite quantity. In this way we shall distribute all intervals of the given domain of intervals into the denumerable sequence of rows (8), (9), …, (10), …, in each of which there will be a bounded number of intervals. It is obvious that for every interval of the domain, however small it may be, provided only that it is not equal to \(0\), one can find two values \(\nu\), i.e. two values \(\frac{\varepsilon}{2^{\nu-1}}\) and \(\frac{\varepsilon}{2^\nu}\) in the sequence (7), between which it lies in magnitude; consequently—every interval will enter one of the groups (10). Thus all the intervals of the domain will be arranged in a denumerable sequence of groups, each with a finite number of terms; and in such a case the given domain of intervals will be denumerable; therefore henceforth, instead of “domain of intervals,” we shall use the term sequence of intervals.

This theorem is of fundamental importance; our further reasoning will rely upon it at every step. From this theorem it follows directly that

A. “The domain of one-sided limiting points is always denumerable or finite,” since it is equivalent to a sequence of intervals; and from this it follows that

B. “Every non-denumerable domain contains two-sided limiting points.”

70. Having set forth all these preliminary facts, before passing to the study of the structure of various kinds of domains, we must establish a few more facts.

For this purpose we shall first of all determine the arrangement of intervals on a certain linear segment.

Let the finite segment \(AB\) be divided by \(n-1\) points \(M_1, M_2,\ldots,M_{n-1}\) into a finite number of segments \(l_1,l_2,\ldots,l_n\). In this case the whole aggregate of points of the linear segment \(L\), determined by the segment \(AB\), breaks up into two domains: \(G\)—the domain of the interior points of the parts \(l_i\), and \(Q\)—the domain of the boundaries of these parts.

The fundamental interval for the domain \(G\) will be the part \(AB\), with the points \(A\) and \(B\) either able to be included in the composition of the domain \(G\)

or may not belong to it; the principal interval of the set \(Q\) will be either \(L\), if \(A\) and \(B\) enter into this set, or \(M_1M_{n-1}\) in the contrary case.

Each of the constituent parts of the set \(G\) is an open segment; if we wished to make one of these segments closed, i.e. to include one of the points \(M_i\) in the composition, for example, of \(l_i\), then the two intervals \(l_i\) and \(l_{i+1}\) would coincide into one and, consequently, the number of intervals would be reduced by one.

It is obvious that the set \(Q\) will be isolated; there are no limiting points for it, since the number of its points is finite; the set \(G\), however, will be everywhere scattered and condensed, but not closed.

71. If the number of subintervals into which the interval \(AB\) is decomposed is infinitely large, then—on the basis of 69°—it must be countable; thus we shall always have either a finite or a countable number of points of division.

Here the segment \(L\), determined by the segment \(AB\), is again decomposed into sets: \(G\)—the interior points of a countable sequence of segments \(l_1,l_2,\ldots,l_n,\ldots\), and \(Q\)—the set of the boundaries of these segments; the question now is whether there will be some further new category of points.

Since the set \(Q=\{x\}\) consists of an infinite number of points, there must exist for it—on the basis of 64°—limiting points, which we shall denote by \(\{x'\}\); depending on the chosen law of division, the number of these points \(x'\) may vary from one to an infinite multitude.

Let \(x'_0\) be one of the limiting points, and let

\[ x_{\nu_1},\ x_{\nu_2},\ x_{\nu_3},\ldots,\ x_{\nu_n},\ldots \tag{11} \]

be the sequence of boundaries determining it.

Since in an arbitrarily small neighborhood of \(x'_0\) there are infinitely many points (11), in this neighborhood there must also be infinitely many corresponding intervals \(l_{\nu_i}\); and if this is so, the intervals \(l_{\nu_i}\) must decrease indefinitely in the neighborhood of the point \(x'_0\), and we shall not be able to indicate, in the direction of these decreasing intervals, a segment for which \(x'_0\) would serve as a boundary—in the present case the right-hand one—and within which there would be no longer any points \(Q\); at most, if such a segment

will lie on the other side of \(x'_0\), i.e. it will be the left boundary of some interval not included among the intervals \(\{l_{\nu_i}\}\).

Thus, if the point \(x'_0\) is also the boundary of some interval \(l_{n_1}\), then for \(l_{n_1}\) there does not exist an adjacent interval in the direction of the intervals \(\{l_{\nu_i}\}\). If, however, the point \(x'_0\) turns out to be a two-sided limit, then we shall have two sequences of adjacent intervals \(\{l_{\nu_i}\}\) and \(\{l_{n_i}\}\), which will not be adjacent to one another; in other words, there is no last interval of the first sequence immediately adjoining the first interval of the second. The common limiting point of the two sequences of intervals will therefore not be the boundary of any interval, and it does not lie inside any one of them; consequently, the two preceding categories of points, in the case of the existence of two-sided limits, do not comprise all possible points of the portion \(L\). These new points will be exterior points of the indicated intervals.

We have given above one case in which exterior points appear; but this case, as we shall see below, is not the only one. For us it is now important only to establish the possibility of the appearance of exterior points.

Next, what is important for us is the fact that the existence of every limiting point, one-sided or two-sided, eo ipso entails the appearance of non-adjacent intervals.

Therefore a sequence of all adjacent intervals is possible only when \(l\) is divided into a finite number of parts¹.

Let us also note that, if for an interval \(l_0\) there is no adjacent interval on either side, so that both its boundaries \(x_0\) and \(\xi_0\) serve as limits for two sequences of points \(\{x_i\}\) and \(\{\xi_i\}\),

\[ \lim x_i = x_0, \qquad \lim \xi_i = \xi_0, \]

then we shall call such an interval isolated.

2. Nowhere dense sets.

72. Suppose that, according to some law, we divide the fundamental interval successively into ever smaller and smaller parts. If, under an arbitrary law and with an arbitrarily continuing process of division, in every partial interval there are points of the given set \(P\), then this set is called everywhere dense in the interval (überall dicht); for such a set, however small a part of the fundamental interval is taken, points of \(P\) are always found on it.

If, on the contrary, in every arbitrary part of the fundamental interval there can be found a partial interval of finite length free of points of the set, it is called a nowhere dense set (niergends dicht).

According to the definition adopted above—“the derivative of an everywhere dense set coincides with the fundamental interval.”

This latter property may be adopted, as Cantor already indicated, as the definition of an everywhere dense set, and then its first definition becomes a consequence. Thus the two definitions turn out to be equivalent, and we shall use one or the other definition depending on convenience.

73. Let \(P\) be a nowhere dense set, concerning which there is no further information. The question is, what can then be said about it? In other words—what properties do all nowhere dense sets possess?

Since the set \(P\) is nowhere dense, in an arbitrary part of the segment \((x_{\bar 0}, x_0)\) an interval free of points \(P\) can be indicated; we shall enlarge this interval to the right until the boundary of the interval becomes either a point \(P\), or a point \(P'\); let us call such a point \(x_1\); to the left we likewise enlarge the interval to the point \(x_{\bar 1}\), which will again belong either to \(P\), or to \(P'\); we shall call the obtained free interval \(l_1\).

Next let us take the parts of the segment \((x_{\bar 0}, x_{\bar 1})\) and \((x_1, x_0)\); in each of them there can be found, by the method indicated above, intervals \(l_2\) and \(l_3\), free of points \(P\), with their boundaries \(x_{\bar 2}, x_2, x_{\bar 3}, x_3\) being points of \(P\) or \(P'\); and so on, we shall construct such intervals indefinitely; their number must necessarily be countable, and the number of intervals exceeding in length a given small finite quantity will be finite, because the sum of all the intervals cannot exceed \(l\).

Every point not belonging to \(P\) will be either an interior point of one of the intervals \(l_i\), or one of the endpoints of such an interval, or—possibly—a two-sided limiting point of these endpoints, i.e. an exterior point of all the intervals.

Therefore the series of intervals \(\{l_i\}\) distributes the points \(L\) into three domains \(G, Q\), and \(E\), of which \(G\)—the domain of interior points of the intervals—always enters wholly into \(\Pi\): not one of its points will be a point of \(P\)

\[ G \equiv D(\Pi). \]

The domain \(E\)—the domain of exterior points—enters wholly into \(P'\); indeed: the points \(E\) serve as two-sided limits of the endpoints of free intervals, and these endpoints will be either points of \(P'\) or of \(P\); consequently—in any case the points \(E\) belong to \(P'\)

\[ E \equiv D(P'), \]

but they may be both points of \(P\) and points of \(\Pi\); let us suppose that, in the general case, \(E\) consists of two parts

\[ E \equiv E_1 + E_2, \]

of which

\[ E_1 \equiv D(\Pi), \quad E_2 \equiv D(P). \]

Finally the domain of endpoints of the intervals \(Q \equiv \{x\}\) may consist both of points of \(P\) and of points of \(\Pi\); let us also call

\[ Q_1 \equiv D(\Pi), \quad Q_2 \equiv D(P). \]

In this most general case the domain \(P\) consists of the part of the endpoints of the intervals and the part of the exterior points

\[ P \equiv E_2 + Q_2, \]

or, if we call

\[ E + Q \equiv T, \]

then

\[ P \equiv D(T). \]

Thus, if we impose no restrictions on a nowhere dense domain, then concerning it we can state only the following theorem:

For every nowhere dense domain \(P\) there can be constructed a series of intervals whose boundaries and interiors include all points of \(P\).

Such a series of intervals defines a closed domain \(E+Q\); therefore we may say that “for every given nowhere dense domain \(P\) there can be constructed a closed domain \(T\), containing the given domain \(P\) as a part.”

It is obvious that in two such domains all isolated points will be one and the same, so that the domain \(T-P\) consists only of one- or two-sided limiting points of the domain \(P\).

74. We saw above, in 71°, that, unless the series \(\{l_i\}\) consists of a finite number of intervals—and this we shall, of course, assume in what follows—it must contain non-adjacent intervals. Consequently, with respect to the series \(\{l_i\}\) only two suppositions are possible: either \(\{l_i\}\) contains no adjacent intervals at all, or else intervals of both kinds occur in the series \(\{l_i\}\).

These two mutually exclusive suppositions make it possible to divide all nowhere dense point domains into two categories: domains determined by series \(\{l_i\}\) without adjacent intervals, and domains with adjacent intervals. We shall still have occasion to deal with the latter domains later; in view of their diversity, we cannot at present undertake their further investigation. As for domains without adjacent intervals, i.e.—in other words—domains determined by a series of isolated intervals, they are rather simple, and we shall say a few words about them now.

Since the series of intervals \(\{l_i\}\)—on the basis of 69°—is always denumerable, and the domains \(\{x'_i\}\) and \(\{x_i\}\) of the left and right endpoints of the intervals are equinumerous with the series of intervals, \(\{x'_i\}\) and \(\{x_i\}\) will both be denumerable; and in that case the domain \(\{x'_i, x_i\}\) is also denumerable.

Since adjacent intervals are entirely absent from \(\{l_i\}\), and consequently all intervals \(l_i\) will be isolated, in an arbitrary neighborhood of each of their endpoints there are infinitely many boundaries of other intervals; therefore each of these boundaries will be a limiting point of the domain of boundaries \(Q\), so that this domain will be perfect

\[ Q \equiv D(Q'). \tag{12} \]

Each of the exterior points \(E\) of the interval domain \(\{l_i\}\) will be the limit of some sequence of boundaries, or the limit of their limit points; therefore each of the points \(E\) will enter into the composition of \(Q'\)

\[ E \equiv D(Q'). \]

Then not one of the interior points of \(\{l_i\}\) can be a limit point of \(Q\); consequently \(Q'\) can consist only of the points \(Q\) and \(E\)

\[ Q' \equiv Q + E; \]

and we know, on the basis of G 67°, that—in virtue of (12)—the domain \(Q\) will be perfect. Thus,

“If, for a nowhere dense domain, all the free intervals are non-adjacent, the points \(P\) belong to the composition of some perfect domain.”

75. In addition, incidentally, from the arguments of the preceding paragraph there follows a very important theorem of Cantor:

“The domain of non-adjacent intervals determines, by its boundaries and exterior points, a perfect domain.”

Conversely: “for every perfect domain there can be constructed a domain of non-adjacent intervals, free of the points of the domain”¹.

Among the nowhere dense domains that are determined by non-adjacent intervals, let us note three domains obtained in three special cases:

A. Suppose that \(P \equiv Q + E\), i.e. all non-interior points of the interval domain enter into the composition of \(P\); then \(P\) will be a perfect domain, and—consequently—will be uncountable.

B. If \(P \equiv Q\), i.e. the nowhere dense domain consists only of the boundaries of the non-adjacent intervals, it will be countable and dense.

C. Finally, if \(P \equiv E\), i.e. \(P\) consists only of the exterior points of the interval domain \(\{l_i\}\), \(P\) will be dense and uncountable; indeed: if \(P \equiv E\) were countable, then, by the countability of \(E\), \(Q + E\) would also be countable, which is not true, since \(Q + E\) is perfect.

3. Principal and derived types of arrangement.

76. We have already had above a division of the segment \(L\) into a finite number of parts; now we must pass to further investigations, assuming that the number of division points may be infinite.

In view of the infinite variety of the laws according to which an interval \(L\) may be divided into a finite or countable number of parts \(l_i\), it is of interest to determine what principal types of divisions are possible in this case.

Among these types the simplest will be the division of the interval \(L\) into a finite number \(n\) of parts: we shall call it a placement of intervals of type \(n\). Under this notation, type 1 means that the interval is not subjected to any division.

Next take the interval \(L\) and, from one end, namely from the left, lay off the segment \(l_1\); from the end \(x_1\) of this segment lay off the segment \(l_2\), and so on, constructing according to some definite law a countable sequence of segments \(l_i\) in such a way that they fill the whole interval \(L\)

\[ l=\sum_{1}^{\infty} l_i, \]

and so that—consequently—the boundary \(x_0\) of the interval \(L\) is a limit point of the sequence \(\{x_i\}\)

\[ x_0=\lim x_i; \]

then on \(L\) we obtain a countable domain of points \(\{x_i\}\)—the boundaries of the intervals \(\{l_i\}\), whose basic interval will be \(L\).

All points of the interval \(L\) will split into three domains: \(G\)—the domain of interior points of the intervals \(\{l_i\}\), \(Q\)—the domain of the boundaries of the intervals, and \(E\)—the domain consisting of the single point \(x_0\), serving as the limit of \(\{x_i\}\); this will be an external point of the domain of intervals \(\{l_i\}\).

It is obvious that

\[ Q' \equiv \{x_0\}=E,\qquad Q''=0; \]

\[ P'\equiv \{Q+E\}'=E,\qquad P''=0. \]

Such an elementary distribution of partial intervals over a given interval is typical in the construction of domains, and this simplest type of placement, by analogy with the designation of Cantor, we shall denote by the letter \(\omega\)—when the limit point is placed as in the drawing, \({}^{*}\omega\)—in the reverse placement, i.e. when the limit point is the beginning of the interval.

77. Suppose next that on the segment \(L\) an interval \(l_0\) is taken, neither of whose endpoints \(x'_0\) and \(x_0\) coincides with the endpoints of \(L\); to the right and to the left of \(l_0\) we construct intervals \(\{l_i\}\) and \(\{l'_i\}\) in such a way that

they are adjacent and fill the whole segment \(L\), so that

\[ l=\sum_{1}^{\infty} l'_i+l_0+\sum_{1}^{\infty} l_i, \]

\[ \lim x'_i=x'_0,\qquad \lim x_i=x_0. \]

With such a construction of the intervals \(l'_i\) and \(l_i\) we divide the points of the segment \(L\) into three regions: \(G\)—the region of interior points of the intervals \(l'_i\) and \(l_i\), \(Q\)—the region of the boundaries \(\{x'_i,x_i\}\), and \(E\)—the region consisting of the two points \(\{x'_0,x_0\}\), serving as limits of \(x'_i\) and \(x_i\); this will be the region of points external with respect to the intervals \(l'_i\) and \(l_i\); the regions \(G\), \(Q\), and \(E\) possess the same properties as above.

It is obvious that

\[ Q'\equiv \{x'_0,x_0\}\equiv E,\qquad Q''\equiv 0; \]

\[ P'\equiv (Q+E)'\equiv Q',\qquad P''\equiv 0. \]

This type of distribution of intervals over the segment \(L\) we shall denote by \({}^{*}\omega+\omega\), since it represents a combination of the two types \({}^{*}\omega\) and \(\omega\) of the preceding type.

It is clear that, if for the initial interval we had taken not \(l_0\) but some other one, the essence of the matter would not have changed thereby: to the right and to the left of it there would still be a countable sequence of intervals, and—consequently—all the reasoning would retain its force.

78. Let us next take on the segment \(L\) an interval \(l_1\) with endpoints \(x'_1\) and \(x_1\), neither of which coincides with the endpoints of \(L\); on the segments \((x'_0,x'_1)\), \((x_1,x_0)\) we arrange intervals \(l_2\) and \(l_3\) again in such a way that,

so that none of the boundaries \(x_2\) and \(x_2\) of the segment \(l_2\), and \(x_3\) and \(x_3\) of the segment \(l_3\), coincide respectively with \(x_0, x_1; x_1, x_0\).

The same process we shall continue further as well: in each free part of the fundamental interval we shall place new intervals, only not making them adjacent to those constructed earlier; continuing such a process indefinitely, we shall construct a domain of isolated intervals and shall again divide the points \(L\) into three domains: \(G\)—the domain of the interior points of the intervals \(l_i\), \(Q\)—the domain of the boundaries of these intervals, and \(E\)—the domain of points exterior with respect to them.

In the two preceding paragraphs the intervals \(l_i\) were chosen so that they filled up the segment \(L\), i.e. their sum was always equal to \(l\); as regards the present case, here one cannot impose such a condition, and we shall see below that, depending on the choice of the lengths of the intervals \(l_i\), \(\sum l_i\) may be either equal to, or less than, the length \(l\) of the fundamental interval \(L\).

As we saw in 74°, the domains \(Q\) and \(E\) will be dense; of them \(Q\) is countable, while \(E\) is not countable, and \(P = Q + E\) is perfect.

It is obvious that

\[ Q' = Q + E = P,\qquad Q'' = P' = P. \]

It is clear that, if as the initial interval one takes not \(l_1\), but any other one whatsoever, everything remains as before: to the right and to the left of it there would still be placed, in a countable sequence, non-adjacent intervals; only the numbering of the intervals would change, and this numbering is not essential for the construction of the sequence of intervals.

This type of arrangement of partial intervals on the segment \(L\) we shall call the perfect type of partition or type \(\tilde{\omega}\).

79. The four types we have examined:

a) type \(n\)—for a finite number of points,
b) type \(\omega\) or \({}^{*}\omega\),
c) type \({}^{*}\omega + \omega\),
d) perfect type \(\tilde{\omega}\)

embrace all the elements on the basis of which any domain may be constructed, however complicated in its structure.

composition; therefore we shall call them the basic types of disposition of intervals.

We shall analyze further the other most important types, which are obtained if we apply, a finite or an infinite number of times, the basic types of disposition and combine them with one another in every possible way. Let us note here that the actual disposition of the intervals within each of the types may be infinitely varied, depending on the chosen law of their succession.

80. Let there be a segment \(L\); divide it into a finite number \(n\) of parts and place on each of them a domain of type \(\omega\). Denote by \(\{x_i\} = Q^{(1)}\) the domain of the boundaries of the first division, and for the second division of the domain of points, internal to \(l_i\), call

\[ \{x_{ij}\} = Q_i^{(2)}, \quad \sum_{i=1}^{n} Q_i^{(2)} = Q^{(2)}; \quad Q = Q^{(1)} + Q^{(2)} + \{x_0\}; \]

the domain \(Q\) is the domain of the boundaries of the final disposition.

On each of the \(n\) segments \(l_i\) there will be placed, in a countable series, intervals \(l_{ij}\), which in the aggregate—on the basis of 69°—will again give a countable series. Each limiting point here is a one-sided limiting point; the number of such points will be finite and equal to \(n\). Each limiting point of an arbitrary series of boundaries on one of the intervals \(l_i\), being, except for \(x_0\), in its turn the boundary of an interval on \(l_{i+1}\), will enter into the domain of boundaries \(Q\), and consequently all the points of \(L\) are divided into three domains: the domain \(Q\) of the boundaries of the final disposition, the domain \(G\) of internal points, and, finally, \(E\)—the domain of external points, which will consist of one single point \(x_0\), the boundary of the whole segment.

The domain \(Q\), always countable, will necessarily contain in its composition all its limiting points, except the last; the boundaries of adjacent intervals will be isolated points for it, and the presence of isolated points for \(Q\) is obligatory. The domain

\[ Q+E=Q+\{x_0\}=P \]

will be countable and closed, and it is obvious that

\[ P'=Q^{(1)}+\{x_0\},\qquad P''=0. \]

Taking the arrangement \({}^{*}\omega\), we would obtain analogous results.

The type of arrangement obtained by combining the division of \(L\) into \(n\) parts with a type \(\omega\) or \({}^{*}\omega\), we shall denote by \(n\omega\) or \(n{}^{*}\omega\), depicting the derived type as the product of the component types.

If into the intervals of an arrangement \(\omega\) or \({}^{*}\omega\) we place intervals of type \(n\), i.e.—in other words—divide each interval into \(n\) parts according to some law, then obviously the type \(\omega n\) or \({}^{*}\omega n\) will not differ from \(\omega\) or \({}^{*}\omega\); we must therefore put

\[ \omega n=\omega,\qquad {}^{*}\omega n={}^{*}\omega. \]

81. Let us now divide the segment \(L\) into intervals of type \(\omega\), and in each of them place a series of intervals of the same type; then the segment \(L\) will be decomposed into a countable series of countable series of intervals \(\{l_{ij}\}\), which, taken together, form—by virtue of 69°—again a countable series. Denote the points of division by

\[ \{x_0,x_i\}=Q^{(0)},\qquad \{x_i\}=Q^{(1)},\qquad \{x_{ij}\}=Q_i^{(2)},\qquad \sum Q_i^{(2)}=Q^{(2)}. \]

All points of \(L\) will be decomposed into the \(G\)-domain of interior points of the intervals \(\{l_{ij}\}\), the \(Q\)-domain of their boundaries, and the \(E\)-domain of the exterior points of the intervals of the final arrangement.

The domain \(G\) is of no particular interest; the domain

\[ Q=\{x_0\}+Q^{(1)}+Q^{(2)} \]

will consist of a countable series of countable series of points \(\{x_{ij}\}\), a countable series \(Q^{(1)}\), and the point \(x_0\). The set \(Q\) contains all its limiting points, except for the single \(x_0\), since all of them, except \(x_0\), are boundar-

by the ends of the intervals \(l_{ij}\); the set \(E\) thus consists of the single point \(x_0\)

\[ E=\{x_0\}. \]

If we set

\[ P=Q+E=\{x_0\}+Q^{(1)}+Q^{(2)}+\{x_0\}=\sum_0^2 Q^{(i)}, \]

then it is obvious that

\[ P'=Q^{(1)}+\{x_0\}=Q',\qquad P''=\{x_0\},\qquad P'''=0, \]

i.e. \(P\) will be a set of the second kind; it is countable and closed. The set \(Q\), which consists of the boundaries of all intervals \(l_{ij}\) lying inside \(l_i\), will be dense.

Since, in constructing the present set, we have applied the arrangement \(\omega\) successively twice, we may denote the present type of arrangement by \(\omega^2\).

82. In exactly the same way we could obtain the type \({}^{*}\omega^2\), which presents no special feature in comparison with \(\omega^2\), and the types \(\omega{}^{*}\omega\) and \({}^{*}\omega\,\omega\).

In the arrangement of intervals of type \(\omega{}^{*}\omega\) we take a set of intervals of type \(\omega\) and in each of them place a set of type \({}^{*}\omega\); then we shall have

\[ Q^{(0)}=\{x_0',x_0\},\qquad Q^{(1)}=\{x_i\},\qquad Q^{(2)}=\{x_{ij}\}, \]

\[ Q=\{x_i\}+\{x_{ij}\}=Q^{(1)}+Q^{(2)},\qquad E=\{x_0',x_0\}=Q^{(0)}; \]

since neither \(x_0'\) nor \(x_0\) will already here be boundaries of intervals of the final arrangement; hence

\[ P=Q+E=\sum_0^2 Q^{(i)},\qquad P'=Q^{(0)}+Q^{(1)}=Q',\qquad P''=\{x_0\},\qquad P'''=0. \]

In the type \({}^{*}\omega\,\omega\) we first take the arrangement \({}^{*}\omega\) and in each of its intervals place a set \(\omega\); for this case the sets \(Q\), \(E\), and \(P\) have the same composition as in the preceding one.

Placing further, in the intervals of type \(\omega^2\), intervals of type \(\omega\), we obtain a new type \(\omega^3\), for which

\[ P'''=\{x_0\},\qquad P^{(\mathrm{IV})}=0; \]

and so on. Obviously, if, first taking the arrangement \(\omega\), and then \(\omega^2\), we obtain the same arrangement \(\omega^3\); thus

\[ \omega\cdot\omega^2=\omega^3=\omega^2\cdot\omega . \]

Combining successively \(\omega\) and \(*\omega\), we can obtain domains \(Q\) that are always countable, of the first kind and of the \(n\)-th species, if the number of operations \(\omega\) and \(*\omega\) is \(n\); in particular, for a domain of type \(\omega^n\) or \(*\omega^n\), the domain \(P^{(n)}\) consists of the single point \(x_0\) or \(x'_0\).

Thus, from a given type

\[ \omega^\alpha *\omega^\beta \omega^\gamma \ldots *\omega^\nu \tag{1} \]

we can always reconstruct the structure of the domain and directly indicate its species \(n=\alpha+\beta+\gamma+\ldots+\nu\).

It goes without saying that the notation (1) gives only the type of the domain; in order to obtain some domain of this type, one must specify each of the constituent fundamental arrangements.

In accordance with the notation we have adopted, the domains of Smith\(^1\) will be \(P_1\) of type \(*\omega\), \(P_2\) of type \(*\omega^2\), \(P_m\) of type \(*\omega^m\); to these same types belong the domains of Mittag-Leffler\(^2\), for which \(P^{n+1}\equiv 0\).

Let us further note that in all domains of type \(\omega^n\) and \(*\omega^n\) the limiting points will always be one-sided: in the first case—right-hand, and in the second—left-hand; as for type (1), there both kinds will occur; moreover, for \(n>2\), two-sided limiting points must also enter.

83. Let us now turn to domains arising from the fundamental type \(*\omega+\omega\), and first take a division of the interval \(L\) into a finite number \(n\) of parts; the domains of points of the various systems of division we shall call

\[ Q^{(0)} \equiv \{x_0, x'_0\},\quad Q^{(1)} \equiv \{x_1, x_2,\ldots,x_{n-1}\},\quad Q^{(2)} \equiv \{x_{ij}\}. \]

The domain of intervals of the final arrangement will have as its boundaries the domain \(Q\equiv Q^{(2)}\), whereas not one of the points of \(Q^{(0)}\) and \(Q^{(1)}\) will enter into \(Q\). The type of arrangement obtained in this way will be denoted, in agreement with the preceding, by \(n(*\omega+\omega)\).

With such a distribution of the segments \(\{l_i\}\) on \(L\), we shall have a finite number of countable rows of intervals; each of the points—

\(^1\) See A° 5.

\(^2\) See 18°.

of the first division \(Q^{(1)}\), and also the boundaries of the interval \(L\), will be limit points.

The points of the segment \(L\) will here be distributed into three domains: \(G\)—the domain of interior points of the segments \(\{l_{ij}\}\), \(Q\)—the domain of their boundaries, and finally \(E\)—the domain of limit points of the boundaries of the segments that are exterior points of the intervals of the final arrangement.

The domain \(G\)—the domain of interior points of the segments—does not present anything new in comparison with what was set forth above; the domain \(Q\) of boundaries will be countable and isolated, since none of the limit points \(\{x_{ij}\}\) enters into the composition of the boundaries of the intervals \(\{l_{ij}\}\); the domain \(E\) will consist of the division points \(Q^{(1)}\) and of the boundaries of the fundamental interval \(x_{\bar 0}\) and \(x_0\)

\[ E = Q^{(0)} + Q^{(1)}. \]

It is obvious that here

\[ \tag{2} Q' = E,\qquad Q''' = 0, \]

whereas all limit points of the domain \(Q\), except for the points \(Q^{(0)}\), will be two-sided. The domain

\[ P = Q + E = \sum_{0}^{2} Q^{(i)} \]

will give, by virtue of (2),

\[ P' = Q' = E,\qquad P''' = 0. \]

It is obvious that

\[ ({}^{*}\omega + \omega)n = {}^{*}\omega + \omega \ne n({}^{*}\omega + \omega). \]

83. Suppose further that we have first taken the type \({}^{*}\omega + \omega\), and in each of its intervals have again placed a domain of type \({}^{*}\omega + \omega\); then the boundaries of the segment and the domains of points of the first and second divisions will be

\[ Q^{(0)} = \{x_{\bar 0}, x_0\},\quad Q^{(1)} = \{x_j\},\quad Q_i^{(2)} = \{x_{ij}\},\quad \sum Q_i^{(2)} = Q^{(2)}, \]

[Diagram of the arrangement of intervals and points, with labels \(x_{\bar 0}, x_0, x_{\bar j}, x_j, x_{ij}\) and intervals \(l_{\cdot}\), illustrating the above construction.]

where \(i\) and \(j\) assume all integral positive and negative values; moreover, \(i\) may be equal to \(0\). The type of arrangement of intervals obtained in this way must be denoted by

\[ ({}^{*}\omega + \omega)^2. \]

Under such a law of distribution of the intervals on the segment \(L\), we divide all its points into three regions: \(G\)—the region of interior points of the intervals \(\{l_{ij}\}\) of the final placement, \(Q\)—the region of their endpoints, and \(E\)—the region of points exterior to \(\{l_{ij}\}\). It is clear that \(E\) is composed of all points \(Q^{(1)}\) together with the endpoints of the fundamental interval \(L\), while \(Q\) coincides with \(Q^{(2)}\)

\[ E \equiv Q^{(0)} + Q^{(1)}, \qquad Q \equiv Q^{(2)}; \]

the region \(Q\) will be countable and dense, since all its limiting points enter into \(E\); therefore

\[ Q' \equiv Q^{(0)} + Q^{(1)}, \qquad Q'' \equiv Q^{(0)}, \qquad Q''' \equiv 0. \]

The combination of the regions \(Q\) and \(E\) gives a countable closed region

\[ P \equiv Q + E, \]

for which

\[ P' \equiv Q', \qquad P''' \equiv 0. \]

Let us note that all limiting points occurring here, except—of course—the endpoints of the fundamental interval, will be two-sided.

84. Placing, in the intervals of a placement of type \((*\omega+\omega)^2\), intervals of type \(*\omega+\omega\), or conversely, we obtain a new type of placement \((*\omega+\omega)^3\), for which we shall have

\[ Q''' \equiv (x_0, x_0), \qquad Q^{(\mathrm{IV})} \equiv 0. \]

Repeating the operation \(*\omega+\omega\) a finite number \(n\) of times, we shall have the type \((*\omega+\omega)^n\), determining a dense region \(Q\) and a closed \(P\) of the first genus and of the \(n\)-th species, for which the endpoints of the fundamental interval will serve as the \(n\)-th derivative and for which, apart from these endpoints, all the remaining limiting points will be two-sided.

85. We pass next to the combination of two types of placement \(\omega\) or \(*\omega\) and \(*\omega+\omega\).

A. Let us first take a placement \(\omega\), for which

\[ Q^{(0)} \equiv \{x_0, x_0\}, \qquad Q^{(1)} \equiv \{x_i\}, \]

and in each interval let us place a region of type \(*\omega+\omega\); the type of placement thereby obtained will be denoted by \(\omega(*\omega+\omega)\); let us call

\[ Q_i^{(2)} \equiv \{x_{ij}\}_{j=\pm 1}^{\pm\infty}, \qquad \sum_{1}^{\infty} Q_i^{(2)} \equiv Q^{(2)}. \]

Intervals of this type distribute the points \(L\) into three regions: \(G\)—the region of interior points of the intervals \(\{l_{ij}\}\) of the final arrangement, \(Q\)—the region of the endpoints of these intervals, and \(E\)—the region of their exterior points.

[[diagram]]

To the last category here will belong all points \(\{x_i\}\) of the first division \(Q^{(1)}\), and, besides these, the points \(x_{-0}\) and \(x_0\); thus

\[ E \equiv Q^{(1)} + \{x_{-0}, x_0\} \equiv Q^{(0)} + Q^{(1)}. \]

This will be a countable closed region, for which \(E' = \{x_0\}\). The region of endpoints \(Q = Q^{(2)}\) will be isolated, since all its limiting points enter into \(E\), so that

\[ Q' = E, \qquad Q'' = \{x_0\}, \qquad Q''' = 0. \]

It is further evident that \(P = Q + E\) will be countable and closed, and that all its limiting points, except \(x_{-0}\) and \(x_0\), will be two-sided; for it

\[ P' = Q^{(1)} + Q^{(0)} = E, \qquad P''' = 0. \]

B. If for the first arrangement one takes \(*\omega+\omega\), and in its intervals places the type \(\omega\), then a new type \((*\omega+\omega)\omega\) will be obtained.

Let us call, as above, the endpoints of the segment and the region of points of the first and second divisions respectively

\[ Q^{(0)} = \{x_{-0}, x_0\}, \qquad Q^{(1)} = \{x_i\}_{i=\pm 1}^{\pm\infty}, \qquad Q_i^{(2)} = \{x_{ij}\}_{j=1}^{\infty}, \qquad Q^{(2)} = \sum_{0}^{\pm\infty} Q_i^{(2)}. \]

[[diagram]]

The region \(G\) here is the same as above; the region \(Q\) consists, first, of the points \(Q^{(2)}\), which is self-evident, and, second, of the points of the first division; indeed—each of the points \(Q^{(1)}\) will be

left-hand boundary of one of the intervals \(l_{ij}\), so that it will enter into the domain of boundaries of the final arrangement; thus the domain

\[ Q = Q^{(1)} + Q^{(2)} \]

will be countable and of general form, since not all its limit points will enter into its composition; the domain \(E\) consists only of the boundaries of the principal interval

\[ E \equiv \{x'_0, x_0\}; \]

for the domain \(Q\) we have

\[ Q' \equiv Q^{(1)} + Q^{(0)}, \quad Q'' \equiv Q^{(0)}, \quad Q''' = 0. \]

The domain

\[ P = Q + E \]

will here again be countable and closed; all its limit points are one-sided.

Comparing the types \(\omega(*\omega+\omega)\) and \((*\omega+\omega)\omega\), we see that they are to a considerable degree different, so that here it follows that

\[ \omega(*\omega+\omega) \ne (*\omega+\omega)\omega . \]

It is evident that the types \(*\omega(\omega+\omega)\) and \((*\omega+\omega)*\omega\) will give us nothing special in comparison with the types we have investigated; likewise,

\[ *\omega(*\omega+\omega) \ne (*\omega+\omega)*\omega . \]

86. Let us consider still another combination of the types \(\omega\), \(*\omega\), and \(*\omega+\omega\): first place on \(L\) intervals of type \(\omega*\omega\), and then in each of the intervals arrange domains of type \(\omega+\omega\); the type of arrangement obtained in this way will be

\[ \omega*\omega(*\omega+\omega). \]

Let us denote the boundaries of the interval and of the domains of the first, second, and third divisions respectively by

\[ Q^{(0)} = \{x'_0, x_0\}, \quad Q^{(1)} = \sum_{i=1}^{\infty}\{x_i\}, \quad Q_i^{(2)} = \sum_{j=1}^{\infty}\{x_{ij}\}, \quad Q_{ij}^{(3)} = \sum_{k=\pm 1}^{\pm \infty}\{x_{ijk}\}, \]

\[ \sum Q_i^{(2)} = Q^{(2)}, \quad \sum\sum Q_{ij}^{(3)} = Q^{(3)}. \]

The domain of the boundaries of the intervals of the final arrangement \(Q\) consists here only of points of the third division,

\[ Q \equiv Q^{(3)}; \]

the outer ...

points form a set \(E\), which consists of all points of the second and first deletions and of the endpoints of the fundamental interval

\[ E=Q^{(0)}+Q^{(1)}+Q; \]

the set \(Q\) will be isolated and countable; for it

\[ Q'=Q^{(2)}+Q^{(1)}+Q^{(0)}=E,\quad Q''=Q^{(1)}+Q^{(0)},\quad Q'''=Q^{(0)},\quad Q^{(\mathrm{IV})}=0. \]

We thus see that the set \(Q\) is a set of genus 1 and of the 3rd kind; the set \(E\) will be closed and likewise a countable set of the 2nd kind; the set \(P=Q+E\) gives a countable closed set, for which \(P'=E\).

Obviously—by combining the types \(\omega\), \({}^{*}\omega\), and \({}^{*}\omega+\omega\) in various ways, we shall obtain increasingly complex sets; the order of the last derivative not reducing to \(0\) will always be equal to the number of operations indicated by the number of factors of the type of arrangement. All sets obtained by a finite number of combinations of these fundamental types of arrangement have a type consisting of a product of types

\[ \omega^{\alpha},\quad {}^{*}\omega^{\beta},\quad ({}^{*}\omega+\omega)^{\gamma}; \]

for them \(P\) will always be countable.

Let us note, incidentally, that if the last factor of the type is \({}^{*}\omega+\omega\), then the set of external points of the intervals of the final arrangement will always be closed and will consist of the points of all but the last deletions.

87. We proceed further to the third fundamental type \(\tilde{\omega}\). Taking an arrangement of intervals of this type, let us place in each of them a new set of intervals of the same type; the type thereby obtained must be denoted by \(\tilde{\omega}^{2}\).

Let us call the endpoints of the segment, and the points of the first and second deletions, respectively,

\[ Q^{(0)}=\{x_0',x_0\},\quad Q^{(1)}=\left\{x_i\right\}_{\pm 1}^{\pm\infty},\quad Q_i^{(2)}=\left\{x_{ij}\right\}_{j=\pm 1}^{\pm\infty},\quad \sum Q_i^{(2)}=Q^{(2)}; \]

we shall further call \(E^{(1)}\) the set of external points of the first deletion, of which \(Q^{(0)}\) is a part; analogously, \(E_i^{(2)}\) shall denote the set

of the outer points of the perfect set situated on \(l_i\), while again \(\{x_i,x'_i\}=D\left(E_i^{(2)}\right)\). The set \(Q\) of boundaries of the intervals of the final arrangement will consist exclusively of the points of the second division \(Q=Q^{(2)}\), whereas the boundaries of the intervals of the first division will now all pass into the category of outer points. The set \(Q\) is countable, since it is equipotent to the set of intervals, and is dense, since each of its points \(x_{ij}\), being the boundary of an interval of the perfect set on \(l_i\), is a limiting point. The set \(E\) of outer points of the final arrangement will include all outer points \(E^{(1)}\) of the first division, among which are also \(x_1\) and \(x_0\), all the boundaries of this division \(\{x_i\}\), i.e. the whole set \(Q^{(1)}\), and finally all outer points \(E_i^{(2)}\) of each of the perfect sets situated on the intervals \(l_i\), except for the already counted boundaries \(x_i\) and \(x'_i\); thus

\[ E=E^{(1)}+Q^{(1)}+\sum\left[E_i^{(2)}-(x_i,x'_i)\right]=E^{(1)}+\sum E_i^{(2)}. \]

Since—according to B 68°—the sets \(E^{(1)}\) and \(E_i^{(2)}\) will be countable, the set \(E\) will also prove countable.

The intervals \(\{l_{ij}\}\), situated on \(l_i\), are all isolated; the double series of such non-adjacent intervals on all \(\{l_i\}\) can be transformed into a simple series of non-adjacent intervals, and the latter, as we know, determine on \(L\) the perfect set of non-interior points \(Q+E=P\).

Thus it turns out that the type \(\tilde{\omega}^{2}\) determines the same perfect set \(P\) as the type \(\tilde{\omega}\); and in this case, after a twofold application of the operation \(\tilde{\omega}\), the type of the arrangement does not change: for the perfect set \(\tilde{\omega}^{2}\), on the basis of the converse of Theorem 75°, we can construct free intervals which must coincide with \(\{l_{ij}\}\), and we shall therefore obtain a set of type \(\tilde{\omega}\).

After all the preceding we arrive at the following important result: applying successively twice the arrangement of intervals of type \(\tilde{\omega}\), we again obtain the same arrangement \(\tilde{\omega}\); we must therefore set

\[ \tilde{\omega}^{2}=\tilde{\omega}. \]

It is also evident that repeated, but finite, application of such an arrangement gives us the same result, so that for every finite \(n\)

\[ \tilde{\omega}\, n = \tilde{\omega}, \]

and we can obtain no new types from this.

We consider it not superfluous to note that a distribution of type \(\tilde{\omega}\) distributes the points of an interval into a countable region \(Q\) and two uncountable regions \(G\) and \(E\).

88. Combining the type \(\tilde{\omega}\) with the preceding types, we obtain a number of interesting regions, some of which have already appeared in the literature. Let us first take the type \(n\) and, in its intervals, place intervals of type \(\tilde{\omega}\); let us denote

\[ Q^{(0)}=\{x_0,x_n\},\quad Q^{(1)}=\{x_i\}_{1}^{\,n-1},\quad Q_i^{(2)}=\{x_{ij}\}_{\pm1}^{\pm\infty},\quad \sum_{1}^{n} Q_i^{(2)}=Q^{(2)},\quad \mathop{\mathrm{M}}_{1}^{n} E_i^{(2)}=E, \]

where the points \(x_{-i}\) and \(x_i\) belong to \(E_i^{(2)}\).

[Diagram of nested intervals and marked points \(x_0,\ldots,x_n\), with subintervals \(l_i,l_{ij}\) and points \(x_{ij}\).]

It is obvious that \(Q=Q^{(2)}\) will be condensed; consequently \(Q'\) is perfect, and

\[ Q'=Q^{(2)}+E=Q+E; \]

thus the final distribution of the intervals turns out to be such that the region \(P=Q+E\) will be perfect. Hence it follows that

\[ n\tilde{\omega}=\tilde{\omega} \]

for every finite \(n\).

89. On the contrary, it is easy to show that

\[ \tilde{\omega}\,n\ne\tilde{\omega}. \]

Let us place in the intervals of a perfect region a finite number \(n\) of new intervals; let us denote, as usual,

\[ Q^{(0)}=\{x_0,x_0'\},\quad Q^{(1)}=\{x_i\}_{\pm1}^{\pm\infty},\quad Q_i^{(2)}=\{x_{ij}\}_{j=1}^{\,n-1},\quad \sum_{1}^{\infty} Q_i^{(2)}=Q^{(2)},\quad Q^{(1)}+E^{(1)}=P^{(1)}, \]

where \(P^{(1)}\) is perfect, and each of the \(Q_i^{(2)}\), and consequently \(Q^{(2)}\), is isolated.

The domain of the boundaries of the intervals of the final arrangement \(Q\) will consist of the new points of division \(Q^{(2)}\) and of \(Q^{(1)}\), whereas the domain of exterior points \(E\) coincides with \(E^{(1)}\)

\[ Q = Q^{(2)} + Q^{(1)},\quad E = E^{(1)}. \]

In view of the fact that the points \(x_{ij}\), situated inside the intervals \(l_i\), will be isolated, while even the endpoints of the interval \(x_{-i}\) and \(x_i\) cannot be their limit points, since the number of points of division \(x_{ij}\) on \(l_i\) is finite, \(Q'\) fully coincides with \(Q^{(1)'}\), i.e. it will be

\[ Q' = Q^{(1)'} = Q^{(1)} + E^{(1)} = P^{(1)}; \]

then, if we call

\[ P = Q + E = Q^{(2)} + Q^{(1)} + E^{(1)} = P^{(1)} + Q^{(2)}, \tag{3} \]

then \(P' = P^{(1)}\); hence it follows that \(P\) is closed; moreover it is non-denumerable, since it contains in its composition the non-denumerable part \(P^{(1)}\).

The expression (3), in accordance with the Cantor–Bendixson theorem, gives the decomposition of the non-denumerable closed domain \(P\) into the perfect part \(P^{(1)}\) and the denumerable \(Q^{(2)}\), for which \(Q^{(2)'} = 0\).

We thus see that the type \(\tilde{\omega}n\) differs from \(\tilde{\omega} = n\tilde{\omega}\).

90. Let us now take the type \(\omega\), and in each of its intervals place domains of type \(\tilde{\omega}\); the final arrangement must be denoted by the symbol \(\omega\tilde{\omega}\).

Let us call, as before,

\[ Q^{(0)} = \{x_{-0},x_0\},\quad Q^{(1)}_{\omega} = \{x_i\},\quad E^{(1)} = \{x_0\},\quad Q_i^{(2)} = \{x_{ij}\}_{j=\pm 1}^{\pm\infty},\quad \sum_1^\infty Q_i^{(2)} = Q^{(2)}. \]

let also \(E_i^{(2)}\) be the domain of the exterior points on each of the intervals \(l_i\), where the points \(x_{i-1}\) and \(x_i\) enter into \(E_i^{(2)}\); therefore the sum

\[ \sum \left[E_i^{(2)} - \{x_{i-1}, x_i\}\right] \]

will give all such exterior points of the perfect domains on \(l_i\) which lie inside \(l_i\). If we take the sum \(\sum E_i^{(2)}\), then in it each of the points \(x_i\) of the domain \(Q^{(1)}\) will occur twice, and the point \(x_0\) once; therefore, in order to obtain all exterior points of all the domains, we must exclude the points of \(Q^{(1)}\) that have been included a second time in \(\sum E_i^{(2)}\); the domain thereby obtained will be

\[ \tag{4} \sum_1^\infty E_i^{(2)} - Q^{(1)}. \]

The domain \(Q\) of the endpoints of the intervals of the final arrangement consists only of the points \(Q^{(2)}\); the points \(x_i\) will cease to be endpoints of intervals and will pass into the category of exterior points; thus \(Q = Q^{(2)}\); the domain \(E\) will be composed, first of all, of all the exterior points of the intervals \(l_{ij}\), i.e. of the points of the domain (4), which also include the points of \(Q^{(1)}\); in addition, the point \(x_0\), which does not enter into any of the \(E_i^{(2)}\), will be an exterior point; thus

\[ E = \{x_0\} + \sum E_i^{(2)} - Q^{(1)}; \]

the domain \(Q\) is countable and dense, so that \(Q'\) will be perfect; \(E\) is dense and uncountable, since each of its constituent parts \(E_i^{(2)}\) is uncountable. Since the domain

\[ Q' = Q^{(2)} + E = Q + E = P \]

is perfect, and it is determined by the intervals \(l_{ij}\), the type of arrangement of these intervals will be \(\tilde{\omega}\), so that again

\[ \omega \tilde{\omega} = \tilde{\omega}. \]

The domain \(*\omega \tilde{\omega}\) gives the same result and therefore it is not of interest to examine it in detail; for it also

\[ *\omega \tilde{\omega} = \tilde{\omega}, \]

91. We shall obtain an entirely different result by considering a domain of type \(\tilde{\omega}\omega\), i.e. by first taking a perfect domain \(\tilde{\omega}\) and placing in its intervals domains of type \(\omega\).

Let us call, as before,

\[ Q^{(0)}=\{x_0',x_0''\},\quad Q^{(1)}=\{x_i\}_{\pm 1}^{\pm\infty},\quad Q_i^{(2)}=\{x_{ij}\}_{j=1}^{\infty},\quad \sum_{1}^{\infty} Q_i^{(2)}=Q^{(2)}. \]

\(E^{(1)}\) is the domain of exterior points of the first division, which also includes \(x_0'\) and \(x_0''\); the domain

\[ P^{(1)}=Q^{(1)}+E^{(1)} \]

is perfect by assumption; let us call, as always, \(Q\) the domain of the boundaries of the intervals of the final placement \(\tilde{\omega}\omega\), and \(E\) their exterior points.

Obviously, \(Q\) will contain all points of the second division \(\{x_{ij}\}\), i.e. the whole domain \(Q^{(2)}\), and, in addition, the left endpoints \(\{x_i\}\) of the intervals \(l_i\):

\[ Q=Q^{(2)}+\{x_i\}; \tag{5} \]

the exterior points of the final placement will be, first of all, the exterior points of the domain \(\tilde{\omega}\), i.e. \(E^{(1)}\), and, in addition, all right boundaries of the intervals of the first division, so that

\[ E=E^{(1)}+\{x_i\}; \tag{6} \]

the domain \(Q\) will be countable and of the general type, the points \(x_i\) being its limiting points; the domain \(E\) is non-countable, since the \(E^{(1)}\) entering into it is non-countable, and is not closed, since \(\{x_i\}\) do not enter into it; for \(E\) and \(Q\) we have

\[ E'=P^{(1)},\quad Q'=Q^{(1)}+E^{(1)}=P^{(1)}; \]

finally, for the domain

\[ P=Q+E \tag{7} \]

we shall have

\[ P'=M(Q',E')=P^{(1)}; \]

and since from (7)—in view of (5) and (6)—it follows that

\[ P=Q^{(2)}+E^{(1)}+\{x_i\}_{\pm 1}^{\pm\infty} =Q^{(2)}+P^{(1)}, \]

then \(P\) turns out to be a closed domain. We see here that the closed uncountable domain \(P\) consists of two parts: the perfect domain \(P^{(1)}\) and the countable domain \(Q^{(2)}\), for which \(Q^{(2)\prime}\equiv P^{(1)}\); this is the result predicted by the Cantor–Bendixson theorem.

Here we have

\[ \tilde{\omega}\not\equiv\tilde{\omega}; \]

the domain \(\tilde{\omega}*\omega\) will be of the same kind; for it likewise

\[ \tilde{\omega}*\omega\not\equiv\tilde{\omega}. \]

92. Let us next examine a domain of type \((*\omega+\omega)\tilde{\omega}\), i.e. in each of the intervals of type \(*\omega+\omega\) let us place a domain \(\tilde{\omega}\).

As before, for the points of the first division let us call

\[ Q^{(1)}:=\{x_i\}_{\pm1}^{\pm\infty},\qquad E^{(1)}:=\{x_0,x'_0\}:=Q^{(0)}. \]

[Figure: a line diagram of nested intervals \(l_i\) and points \(x_i, x_{ij}\), with endpoints \(x_0\) and \(x'_0\).]

the boundaries of the intervals of the perfect domain on \(l_i\) we shall denote by

\[ Q_i^{(2)}:=\{x_{ij}\}_{i=\pm1}^{\pm\infty},\qquad \sum Q_i^{(2)}:=Q^{(2)}; \]

the domain of the corresponding exterior points we shall call \(E_i^{(2)}\), where this domain contains the points \((x_{i+1},x_i)\), \((x_{-1},x_1)\), \((x_i,x_{i+1})\), according as \(i<0,=0,>0\); finally let \(Q\) and \(E\) be the domains of the boundaries and exterior points of the intervals of the final subdivision. Turning to the investigation of the domain \(Q\), we see that it will consist exclusively of the points \(Q^{(2)}\): not a single point of the first division will remain the boundary of an interval, so that \(Q=Q^{(2)}\), since each of its points is a point of one of the perfect domains

\[ Q_i^{(2)}+E_i^{(2)}:=P_i^{(2)}, \]

situated on the interval \(l_i\); hence \(Q\) will be a condensed domain, and consequently \(Q'\) is perfect.

In the composition of \(Q'\), besides the points \(Q\), there will also enter all exterior points \(E\), and not a single interior point of the intervals of the final-

of the final arrangement, so that \(Q' = Q + E = P\), and this domain is perfect; it thus turns out that the intervals of the arrangement \((*\omega+\omega)\tilde{\omega}\) give the same distribution of intervals as \(\tilde{\omega}\); consequently we must put

\[ (*\omega+\omega)\tilde{\omega}=\tilde{\omega}. \]

93. Finally, let us take a domain of type \(\tilde{\omega}(*\omega+\omega)\), i.e. in the intervals of a perfect domain let us place a domain of intervals of type \(*\omega+\omega\); let us denote the points of the first division and the exterior points of the intervals of this division respectively by

\[ Q^{(1)}=\{x_i\}_{\pm 1}^{\pm\infty},\quad E^{(1)}, \]

where the number of points \(E^{(1)}\) also includes \(\{x_{-0},x_0\}=Q^{(0)}\); let us further call

\[ Q_i^{(2)}=\{x_{ij}\}_{j=\pm 1}^{\pm\infty},\quad \sum Q_i^{(2)}=Q^{(2)}; \]

[diagram of nested intervals and points on a line, with labels \(X_0\), \(x_0\), \(I_1,\ldots,I_7\), \(I_{10}, I_{11}, I_{12}\), etc.]

then for the intervals of the final arrangement we obtain

\[ Q=Q^{(2)},\quad E=E^{(1)}+Q^{(1)}=P^{(1)}. \]

Since the intervals of the first division give the perfect domain \(P^{(1)}\), it follows that the domain of exterior points \(E\) will be perfect. As for \(Q\), it will be countable and isolated. The domain

\[ P=Q+E=Q^{(2)}+P^{(1)} \]

is a closed and non-countable domain, which—in accordance with the Cantor–Bendixson theorem—consists of the perfect part \(P^{(1)}\) and the countable part \(Q^{(2)}\), and for which, moreover, \(Q^{(2)\prime}=P^{(1)}\). Here again

\[ \tilde{\omega}(*\omega+\omega)\ne\tilde{\omega}. \]

94. We see from the preceding that the application of the operation \(\tilde{\omega}\) after the operations \(\omega\), \(*\omega\), and \(*\omega+\omega\) is equivalent to only one operation \(\tilde{\omega}\).

Let the various \(\omega_1,\omega_2,\omega_3,\ldots,\omega_n\) belong to the number of types

\[ \omega,\quad *\omega,\quad *\omega+\omega, \tag{8} \]

where for the moment we suppose the number \(n\) to be finite. Performing, on some domain that is a combination of the types (8),

\[ \prod_{1}^{n}\omega_i \]

of operations \(\tilde{\omega}\) transforms a domain into a domain of type \(\tilde{\omega}\); indeed,

\[ \prod_{1}^{n}\omega_i\cdot\tilde{\omega} = \prod_{1}^{\,n-1}\omega_i\cdot\omega_n\tilde{\omega} = \prod_{1}^{\,n-1}\omega_i\cdot\tilde{\omega} = \prod_{1}^{\,n-2}\omega_i\cdot\tilde{\omega} =\ldots=\tilde{\omega}; \]

thus the operation \(\tilde{\omega}\) prevails over all the preceding ones, which, so to speak, are finally drowned in it.

Conversely: if some combination of the operations (8) is performed on a domain of type \(\tilde{\omega}\), the result will be a closed uncountable domain, and the type

\[ \tilde{\omega}\cdot\prod_{1}^{n}\omega_i \ne \tilde{\omega}. \]

It is also obvious that again

\[ \tilde{\omega}\cdot\prod_{1}^{n}\omega_i\cdot\tilde{\omega}=\tilde{\omega}, \]

since the type \(\tilde{\omega}\prod\tilde{\omega}\) can be obtained by raising, within the interval \(\tilde{\omega}\), a domain \(\prod\tilde{\omega}\), which has type \(\tilde{\omega}\).

Since the application of the operation \(\tilde{\omega}\) destroys all previous operations, converting them into the operation \(\tilde{\omega}\), we may reduce all possible derived types to such combinations

\[ \tag{9} \tilde{\omega},\quad \tilde{\omega}\cdot\prod_{1}^{n}\omega_i,\quad \prod_{1}^{n}\omega_i; \]

the composition of the domains corresponding to them is quite clear by virtue of the symbols (9).

It would be possible, with further investigation, to establish still some further results concerning repeated operations of various types, but we shall not dwell on them.

Let us also note that the Bendixson domain indicated in 14 belongs to the type \(\tilde{\omega}(*\omega+\omega)^2\).

4. Complex types of arrangement.

95. Having considered domains of intervals obtained after a finite number of combinations of the fundamental types studied, we now pass to such types where the number of fundamental operations is infinitely large. Let us first consider the operation \(\eta\).

Let us take on the segment \(L\) an arrangement \(n\); in each of its intervals let us place an arrangement of the same type; and so on, continuing this process indefinitely; the arrangement thus obtained we shall denote by the symbol \(n^\infty\).

In order to clarify the character of regions of this kind, let us examine several examples.

A. Divide \(l\) into four equal parts \(\{l_i\}\); then each \(l_i\) again into four, \(l_i'\) again into four, and so on; the lengths of these intervals will in this case be

\[ l_i'=\frac{1}{4}l,\quad l_i''=\frac{1}{4^2}l,\ldots,\quad l_i^{(n)}=\frac{1}{4^n}l,\ldots . \]

In view of the circumstance that the points of each new system of divisions are distributed uniformly over the intervals of the preceding system, and also because, when the process of division is continued indefinitely, the intervals \(l_i^{(n)}\) become arbitrarily small, the region of all points of division \(Q\) will be everywhere dense in the interval \(L\); since it is everywhere dense, there will be no free intervals at all in the final arrangement.

B. Let us take another example: take on \(L\) three points \(x_i'\), in such a way that

\[ \begin{aligned} &l_1'=\frac{1}{2}l,\qquad l_2'=\frac{1}{2^2}l,\qquad l_3'=\frac{2}{3}\cdot\frac{1}{2^2}l,\quad l_4'=\frac{1}{3}\cdot\frac{1}{2^2}l;\\ &l_1''=\frac{1}{2}\cdot l_1'=\frac{1}{2^2}l,\qquad l_2''=\frac{1}{2^2}\cdot\frac{1}{2}l=\frac{1}{2^3}l,\quad l_3''=\frac{2}{3}\cdot\frac{1}{2^3}l,\quad l_4''=\frac{1}{3}\cdot\frac{1}{2^3}l,\\ &l_5''=\frac{1}{2}l_2'=\frac{1}{2}\cdot\frac{1}{2^2}l=\frac{1}{2^3}l,\quad l_6''=\frac{1}{2^4}l,\qquad l_7''=\frac{2}{3}\cdot\frac{1}{2^4}l,\quad l_8''=\frac{1}{3}\cdot\frac{1}{2^4}l, \end{aligned} \]

and so on; and in this case, despite the fact that the points cluster toward the right ends of the intervals, the region \(Q\) will be everywhere dense; to convince oneself of this, it is enough to show that the left boundary of an arbitrary interval will be a limiting point of the region \(Q\) of the boundaries of the final arrangement; in particular—if this is established for \(x_0\), then the same will be true also for the other analogous points. As for \(x_0\), in view of…

that every \(x_1^{(i)}\) divides the segment \(l_1^{(i-1)}\) in half; it is clear that
\[ x_0=\lim x_1^{(i)}. \]

C. Let us finally take the following law of division: divide the interval \(L\) into two parts \(l'_1\) and \(l'_2\) in such a way that the ratio of \(l'_1\) to \(l'_2\) is equal to the ratio of \(2^1+1\) to \(1\); then
\[ l'_1=\frac{3}{4}l,\quad l'_2=\frac{1}{4}l; \]
divide each of the \(l'_i\) into two parts in the ratio of \(2^2+1\) to \(1\);

in this case
\[ l''_1=\frac{5}{6}\cdot\frac{3}{4}l=\frac{5}{8}l,\quad l''_2=\frac{1}{8}l,\quad l''_3=\frac{5}{6}\cdot\frac{1}{4}l=\frac{5}{24}l,\quad l''_4=\frac{1}{24}l; \]
divide each of the \(l''_i\) in the ratio \(2^3+1\) to \(1\), and so on; each of the \(l_i^{(n-1)}\) in the ratio \(2^n+1\) to \(1\). Then in each of the intervals the first interval of the next division will constitute, respectively,
\[ \frac{3}{4},\ \frac{5}{6},\ \frac{9}{10},\ldots,\frac{2^n+1}{2^n+2},\ldots \]
of the whole preceding interval; here the points of the first three systems of division correspond to the following fractions of the length of the segment \(l\):
\[ 0,\qquad \frac{3}{4},\qquad 1, \]
\[ 0,\qquad \frac{5}{8},\qquad \frac{3}{4},\qquad \frac{23}{24},\qquad 1, \]
\[ 0,\qquad \frac{9}{16},\qquad \frac{5}{8},\qquad \frac{59}{80},\qquad \frac{3}{4},\qquad \frac{15}{16},\qquad \frac{23}{24},\qquad \frac{239}{240},\qquad 1. \]

It is clear that, under such a law for constructing the domain \(R\) of points of division, to the right of all the left endpoints of each of the intervals there will be free intervals of the final arrangement; for example, the right endpoints of the first intervals \(l_1^{(i)}\) of each system, expressed in fractions of \(l\), will be

\[ x'_1=\frac{3}{4}l=\left(\frac{1}{2}+\frac{1}{2^2}\right)l, \]

\[ x''_1=\frac{5}{6}\cdot\frac{3}{4}l =\frac{2^2+1}{2^2+2}\left(\frac{1}{2}+\frac{1}{2^2}\right)l =\frac{2^2+1}{2\cdot 2^2}l =\left(\frac{1}{2}+\frac{1}{2^3}\right)l, \]

\[ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ , \]

\[ x^{(n)}_1=\frac{2^n+1}{2^n+2}\left(\frac{1}{2}+\frac{1}{2^n}\right)l =\frac{2^n+1}{2\cdot 2^n}l =\left(\frac{1}{2}+\frac{1}{2^{n+1}}\right)l, \]

so that

\[ \xi'_1=\lim x^{(n)}_1=\frac{1}{2}; \]

consequently, the interval \(\left(0,\frac{1}{2}\right)\) will be free of points of \(Q\); this free interval constitutes \(\frac{2}{3}\) of the interval \(l'_1\).

On the interval \(l'_2\) the division points are situated in the same ratios to the whole interval as the points on \(l'_1\); therefore on \(l'_2\) as well, to the right of the point \(x'_1\), there will be a free interval equal to \(\frac{2}{3}\) of the interval \(l'_2\), i.e. \(\frac{2}{3}\cdot\frac{1}{4}l=\frac{1}{6}l\), and moreover
\[ \xi'_2=\frac{3}{4}l+\frac{1}{6}l=\frac{11}{12}l. \]

The odd intervals of the second division constitute \(\frac{5}{6}\) in relation to the preceding intervals; the first interval of the second division \(l''_1\), equal to \(\frac{5}{6}\) of \(\frac{3}{4}l\), will be equal to \(\frac{5}{8}l\). For it the free interval, extending from \(x''_0\) to \(\xi'_1\), constitutes
\[ \frac{1}{2}:\frac{5}{8}=\frac{4}{5} \]
of \(l''_1\); in view of the similar arrangement of the points on all intervals of the second division, on each of \(l''_2,l''_3,l''_4\) the free part will constitute \(\frac{4}{5}\) of the corresponding intervals; and in this case, after the second division, we shall obtain two new free intervals \((x''_1,\xi''_2)\), \((x''_3,\xi''_4)\), and all the free intervals taken together will give

\[ \frac{4}{5}(l''_1+l''_2+l''_3+l''_4)=\frac{4}{5}l, \]

i.e. \(\frac{4}{5}\) of the length of the whole segment \(L\).

The odd intervals of the third division constitute \(\frac{9}{10}\) of the corresponding preceding intervals; the first interval of this division will be
\[ \frac{9}{10}\cdot\frac{5}{8}l=\frac{9}{16}l, \]
and its free part, equal to \(\frac{1}{2}l\), will constitute
\[ \frac{1}{2}:\frac{9}{16}=\frac{8}{9} \]
of the whole interval; on each of the intervals \(l'''_i\) of the third division the free parts will likewise constitute \(\frac{8}{9}l'''_i\); there will be four new free intervals, and all together they will give \(\frac{8}{9}l\).

Continuing such a process of division, after \(n\) divisions we shall always have \(2^n\) free intervals, the sum of which is equal to \(\dfrac{2^n}{2^n+1}l\); with an indefinitely continuing process the sum of the free intervals will evidently tend to \(l\).

We see from this that in the present example there can no longer be any question of a frequently dispersed domain of points of division \(R\); it is not only rarely dispersed, but even the sum of the free intervals is arbitrarily close to \(l\).

It is further evident that each of the left endpoints of the intervals of an arbitrary division, except for the point \(x_0\), will here be a right-hand unilateral limit of the right endpoints of successive intervals; the limits of the left endpoints will be points \(\xi_j^{(i)}\) not belonging to the domain of endpoints \(R\); these limits will be left-hand.

Since all endpoints, except \(x_0\), serve as limiting points of \(R\), we have
\(R-\{x_0\}=D\{R'\}\), so that \(R-\{x_0\}\) will be a condensed domain, and in that case \(R'\) is perfect; let us see of what further points \(R'\) consists. It will contain, first of all, all the endpoints \(\{x_j^{(i)}\}\), except \(x_0\); secondly, all the points \(\{\xi_j^{(i)}\}\); finally, the limits of these and of other points, distinct from \(\{\xi_j^{(i)}\}\).

Denote the perfect domain \(R'\) by \(P\), and by \(G,Q,E\) the corresponding parts of \(L\). First of all it is clear that the fundamental interval for the domain \(P\) will be the interval \(\{\xi_1',x_0\}\). It is further evident that the free intervals of the domain \(P\) will be the free intervals of the domain \(R\) lying to the right of all the points of \(R\) up to the corresponding points \(\{\xi_j^{(i)}\}\); consequently
\[ Q=\{x_j^{(i)},\xi_j^{(i)}\}. \]

We now arrive at the following conclusion: under the indicated construction of the domain \(R\), all its points, except \(x_0\), turn out to be left endpoints of free intervals of type \(\omega\); this domain \(R\) will be countable, non-closed, since the points \(\{\xi_j^{(i)}\}\) and their limits are absent from it, non-condensed, since the point \(x_0\) is included in its composition, and, finally, rarely dispersed, and even in such a way that the sum of the free intervals is arbitrarily close to \(l\). We see, finally, that the domain \(R\) has assumed a completely distinctive character, very far from the type of arrangement \(n\) or \(n'\).

96. Returning now to the beginning of §44, denote by
\[ Q^{(1)},\ Q^{(2)},\ \ldots,\ Q^{(n)},\ \ldots \]
the endpoints of the first, second, etc., division, and by \(Q^{(0)}\) the endpoints of the segment; let us also call

\[ Q^{(0)}+Q^{(1)}=Q_1,\quad Q^{(0)}+Q^{(1)}+Q^{(2)}=Q_2,\ldots,\sum_0^m Q_i=Q_m; \]

with the endlessly continuing process of applying intervals of type \(n\) to the preceding intervals, i.e., in other words, of dividing these intervals according to a certain law into \(n\) parts, we shall obtain the domain of points

\[ Q_\omega=\sum_0^\omega Q_i. \]

It is first of all evident that the domain \(Q_\omega\) will be countable; indeed, each \(Q_m\) consists of a finite number of points; a countable series of such finite \(Q_m\) again gives a countable series of points of division. Since each of the \(Q^{(m)}\) consists of a finite number of points, \(Q_m\) will always be finite. Therefore for no finite value of \(m\) do limiting points exist; consequently all points \(L\) will be either interior points or boundaries of the intervals \(Q_m\), and there are no exterior points for \(Q_m\) at all.

From the preceding examples we have seen that the domain \(Q_\omega\) may be densely or sparsely scattered over the segment \(L\); in the latter case, as follows from 95° and the general theorem 73°, for the sparsely scattered domain \(Q_\omega\) there may be constructed a domain of intervals, whose boundaries and exterior points comprise all the points \(Q_\omega\).

The points \(Q_m\) will be boundaries of the intervals of division \(n^m\); for the domain \(Q_\omega\), however, some of these boundaries may remain boundaries of the intervals of the final arrangement, but besides them new boundaries will also appear, while the former ones thereby become exterior points of the free intervals. Thus the character of the points \(Q_m\) may change completely in \(Q_\omega\), and the arrangement \(n^\omega\) represents a wholly peculiar type, far removed from those we have studied before.

In this arrangement we encounter for the first time densely scattered domains, which are a consequence of one of the basic arrangements—the arrangement \(n\).

Comparing the composite type \(n^\omega\), when \(P\) is sparsely scattered, with all the preceding ones, we find the following characteristic feature of it: the free intervals of this type are not, as before, intervals of earlier divisions. Thus the intervals \(\{l_i\}\) of the final arrangement do not belong to the number of intervals \(\{l_i^{(n)}\}\).

of different systems of division; we shall find this feature also in all the other complex types.

97. If, instead of successively dividing intervals into \(n\) parts, we divide \(l\) first into \(n_1\) parts, then the intervals thus obtained into \(n_2\), into \(n_3\), and so on, we obtain the type \(\prod\limits_1^\omega n_i\). The boundaries \(P\) of the intervals of the various systems of division of this type will likewise be countable, more or less frequently scattered, depending on the law of division.

In the investigation of the properties of the domain \(P\) in the preceding section, the number of divisions \(n\) played no role whatever; no assumptions were based upon it; therefore, from the point of view of the arrangement of the intervals \(\{l_i^{(n)}\}\) along \(l\) and of the rarefaction of the points \(P\) associated with it, the two types \(n^\omega\) and \(\prod\limits_1^\omega n_i\) appear identical; this does not mean that the domains obtained are one and the same; for different numbers \(n\), for different \(n_i\), and consequently for different laws of distribution of the intervals of each system, the domains \(P\) will of course be different, but their general character, their type, will be the one investigated in §§ 95–96. Obviously, if we distribute the points of division uniformly over the intervals \(l_i^{(n)}\) and take

\[ n = 10,\quad x_{0} = 0,\quad x_{\omega} = 1, \]

then we obtain, as the domain \(P\), all proper rational decimal fractions.

The points \(\{x_i^{(n)}\}\) in the type \(\prod\limits_1^\omega n_i\) constitute a generalized expression of the fractions found in Brodén.

98. We proceed next to the type \(\omega^\omega\). We obtain such an arrangement by placing on \(l\) the intervals \(\{l_i\}\) of type \(\omega\); in each of these intervals we place a countable sequence of intervals \(\{l_{ij}\}\) of type \(\omega\); these again give a countable sequence of intervals \(\{l_i''\}\) of type \(\omega^2\); in each of them we construct intervals \(\omega\), thereby obtaining the type \(\omega^3\); and so on.

The question arises: what feature will such an arrangement of intervals on \(l\) have, what will be the domain of the boundaries of the intervals of successive divisions, and in what relation will it stand to the domain of free intervals of the final arrangement?

Let us call \((x_1^0,\ x_2^0)\) the boundaries of the fundamental portion \(L\), and

\[ Q^{(0)}=\{x_1^0,\ x_2^0\}. \]

Take on \(l\) a countable series of intervals \(\{l_i'\}\), and let us call the domain of the points \(\{x_i'\}\) determining them, which are interior with respect to \(l\),

\[ Q^{(1)}=\{x_i'\}_1^\omega,\qquad Q_1=Q^{(0)}+Q^{(1)}-\{x_2^0\}, \]

where \(Q_1\) is the domain of boundaries after the first placement. The point \(x_2^0\) will be the only exterior point of this placement: \(E^{(1)}=\{x_2^0\}\).

[Figure: a line segment from \(x_1^0\) to \(x_2^0\), subdivided by points \(x_i'\) and \(x_{ij}'\), with nested intervals \(l_i'\) and \(l_{ij}'\).]

Let us now place on the intervals \(l_i'\) of the first division the intervals of the second division \(l_{ij}'\); the division points thereby put on \(l_i'\), interior for \(l_i'\), will be \(\{x_{ij}'\}\). Let us call the domains of these points

\[ Q_i^{(2)}=\{x_{ij}'\}_{j=1}^\omega,\qquad \sum_1^\omega Q_i^{(2)}=Q^{(2)},\qquad Q_2=\sum_0^2 Q^{(i)}-\{x_2^0\}. \]

The intervals \(\{l_{ij}'\}\) form a double series, i.e. a countable series of countable series of intervals; as is known, by establishing a certain law of succession we can transform it into a simple series \(l_i''\), so that then

\[ \{l_{ij}'\}_{i=1,\ j=1}^{\omega,\ \omega}=\{l_i''\}_{i=1}^{\omega}; \]

this identity must be understood to mean that all intervals of the first domain enter into the second, and conversely.

Let us call the boundaries of the intervals \(\{l_i''\}\) by \(\{x_k''\}\); evidently,

\[ \{x_k''\}=\{x_{ij}'\}+\{x_i'\}+\{x_1^0\}=Q^{(0)}+Q^{(1)}+Q^{(2)}-\{x_2^0\}, \]

since the points \(x''_{ij}\), together with \(x'_1\) and \(x^0_1\), are the same as \(x''_k\), only renumbered differently; thus \(\{x''_k\}\equiv Q_2\) is the set of boundaries after the second placement.

Next, on each of the intervals \(l''_i\) we construct intervals \(l''_{ij}\), defining them by the points \(\{x''_{ij}\}\), and moreover

\[ Q^{(3)}_i \equiv \{x''_{ij}\}_{j=1}^{\infty}, \qquad \sum_{1}^{\infty} Q^{(3)}_i \equiv Q^{(3)}; \]

then we again transform the double sequence \(\{l''_{ij}\}\) into a simple one \(\{l'''_i\}\), calling \(\{x'''_k\}\) the boundaries of the intervals of this sequence. Then it is again evident that

\[ \{x'''_k\}\equiv \sum_{0}^{3} Q^{(i)}-\{x^0_2\}\equiv Q_3, \]

and so on; after the \(n\)-th division we obtain intervals and the inner points of division determining them,

\[ \{l^{(n-i)}_{ij}\}\equiv \{l^{(n)}_i\},\qquad \{x^{(n-1)}_{ij}\}\equiv Q^{(n)},\qquad \{x^{(n)}_k\}\equiv \sum_{0}^{n} Q^{(i)}-\{x^0_2\}\equiv Q_n . \]

99. Let us take the successive sets

\[ Q_1=Q^{(0)}+Q^{(1)}-\{x^0_2\},\qquad Q_2=Q^{(0)}+Q^{(1)}+Q^{(2)}-\{x^0_2\},\ldots,\qquad Q_n=\sum_{0}^{n}Q^{(i)}-\{x^0_2\}. \]

If we take the set of non-inner points of all the intervals, then we shall have to add only the single point \(x^0_2\), which lies outside these intervals; the sets thus obtained we denote by

\[ P_n \equiv Q_n+\{x^0_2\}\equiv \sum_{0}^{n} Q^{(i)}; \]

we also denote

\[ P_0\equiv Q^{(0)}\equiv \{x^0_1,x^0_2\}; \]

thus the set \(P_n\) contains all the points of division, beginning with the boundaries of the segment \(l\) and ending with the points determining the intervals of the \(n\)-th division.

It is not hard to see that, determining for \(Q_i\) derivatives of various orders, we obtain the following table:

\[ Q'_0 = 0, \]

\[ Q'_1 = P_0 - x'_1,\quad Q''_1 = 0, \]

\[ Q'_2 = P_1 - x^0_1,\quad Q''_2 = P_0 - x^0_1,\quad Q'''_2 = 0, \]

\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots , \]

\[ Q'_n = P_{n-1}-x^0_1,\quad Q''_n = P_{n-2}-x^0_1,\quad Q'''_n = P_{n-3}-x^0_1,\ldots,\quad Q^{(n)}_n = P_0-x^0_1,\quad Q^{(n+1)}_n=0. \]

Looking at the table, we see that the derivative of any of the domains \(Q_n\) contains all points of the preceding domain \(P_{n-1}\), and consequently all points \(Q_{n-1}\), except for the initial point \(x^0_1\). Each of the \(Q_n\) turns out to be a domain of the first rank and of the \(n\)-th kind, and its \(n\)-th derivative always consists of the single point \(x^0_2\); all the domains \(P_n - x^0_1\) will be closed domains.

The question now is what will happen to the domain \(Q_n\) if we let \(n\) increase without bound? Let us call the domain obtained in this way \(Q_\omega\),

\[ Q_\omega=\lim_{n\ldots\infty} Q_n \quad\text{or}\quad Q_\omega=\sum_{0}^{\omega} Q^{(i)}-\{x^0_2\}; \]

this will be a domain of type \(\omega^\omega\).

The domain \(Q_\omega\) is composed of the points belonging to the countable row of domains

\[ Q^{(0)}-\{x^0_2\},\quad Q^{(1)},\quad Q^{(2)},\ldots,\quad Q^{(n)},\ldots, \]

each of which is countable, and which have no common points with one another; every point of \(Q_\omega\), if it is not \(x'_1\), necessarily belongs to one \(Q^{(i)}\); \(x'_1\), however, belongs to \(Q^{(0)}-\{x^0_2\}\).

All these domains \(Q^{(n)}\) are rarely scattered; in this case we can, in an arbitrary part \((\alpha_0,\beta_0)\) of the fundamental interval \(l\), find an interval \((\alpha_1,\beta_1)\) within which there are no points of \(Q^{(1)}\); the endpoints \(\alpha_1\) and \(\beta_1\) of this interval we can choose, moreover, so that they are interior points of some interval \(l_i\). Within \((\alpha_1,\beta_1)\) we can choose an interval \((\alpha_2,\beta_2)\), inside which there will be no points \(x'_i\), and whose endpoints will be interior points of one

from the intervals \(l_i''\), and so on; we shall thus obtain two sequences of numbers

\[ \alpha_1,\ \alpha_2,\ \alpha_3,\ldots,\alpha_n,\ldots;\quad \beta_1,\ \beta_2,\ \beta_3,\ldots,\beta_n,\ldots, \]

for which

\[ \lim \alpha_n=\alpha \leq \beta=\lim \beta_n . \]

The points \(\alpha\) and \(\beta\) will not be points of \(Q_\omega\), since they do not enter into any one of the \(Q^{(i)}\), whatever the value of \(i\).

Thus on \(l\) there certainly exist points not belonging to \(Q_\omega\).

100. Let \(\xi\) be one of the points of \(Q_\omega\), distinct from \(x_1^0\); if \(\xi\) is a point of \(Q_\omega\), it must enter into one of the \(Q^{(i)}\)—for instance into \(Q^{(n)}\); it therefore enters into \(Q_n\) and into \(P_n\); if it enters into \(P_n\) and is distinct from \(x_1^0\), then it serves as a limiting point for the domain \(Q_{n+1}\), and, consequently, for \(Q_\omega\), since all the points of \(Q_{n+1}\) enter into \(Q_\omega\); thus every point of \(Q_\omega\), except \(x_1^0\), will be its limiting point.

Conversely: let \(\xi\) be a limiting point of \(Q_\omega\). With respect to it one may make two assumptions: either in an arbitrary neighborhood of it there are infinitely many points of at least one of the \(Q^{(i)}\), for instance \(Q^{(n)}\), or this is not so. Under the first assumption \(\xi\) will be a limiting point for \(Q^{(n)}\) and consequently for \(Q_n\), into which \(Q^{(n)}\) enters in its entirety; and if \(\xi\) is a limiting point of \(Q_n\), then it must enter into the composition \(P_{n-1}-x_1^0\); and in such a case it will be either a point of \(Q_{n-1}\) or the point \(\{x_2^0\}\). In the first case \(\xi\) will enter into the composition of \(Q_\omega\); in the second it lies outside \(Q_\omega\).

Let us now examine the second assumption: let \(\xi\) be a limiting point, and in its neighborhood there are not infinitely many points of any one of the \(Q^{(i)}\); then it is necessary that in an arbitrary neighborhood of it there should lie infinitely many points of distinct \(Q^{(i)}\), but only finitely many from each.

In such a case, around \(\xi\), for every given \(Q^{(n)}\), there may be determined such a neighborhood \(r\) that inside it there will be no points of \(Q^{(n)}\); consequently \(\xi\) will not be a limiting point for any \(Q_n\).

Since this is so, since the point \(\xi\) will not be a limiting point for any \(Q_n\), whatever finite \(n\) may be, it must not enter into the domain \(P_{n-1}-x_1^0\), and consequently it will not enter into \(Q_{n-1}\), which differs from \(P_{n-1}\) only by the point \(x_2^0\); and if it does not enter into any \(Q_{n-1}\) for any value of \(n\), it cannot

enter into \(Q_\omega\). Thus, under the second supposition, the limit point \(\xi\) will not be a point of \(Q_\omega\). That such points exist we shall now see from the example given below.

Thus it turns out that every point of \(Q_\omega\), except \(x_1^0\), is its limit point, but not conversely; therefore \(Q_\omega - x_1^0\) will be a condensed domain, and consequently \(Q_\omega'\) will be perfect.

101. The point \(x_1^0\) may be a limit point of \(Q_\omega\), but this is not necessary.

A. It is easy to convince oneself that, for a certain choice of the displacements, it will be an isolated point of \(Q_\omega\).

Indeed: take \(l=1\); make the first interval of the first division equal to \(\frac12\), the first interval of the second division equal to \(\frac38\), of the third division—\(\frac5{16}\), of the fourth division—\(\frac9{32}\), etc.; then the first points of the successive systems of division will be, respectively,

\[ \frac12=\frac{2^0+1}{2^2},\quad \frac38=\frac{2+1}{2^3},\quad \frac5{16}=\frac{2^2+1}{2^4},\ldots,\quad \frac{2^{\,n-2}+1}{2^n},\ldots, \]

so that, as \(n\) increases without bound,

\[ \lim_{i\ldots\infty} x_{\mu_i}^{(i)}=\frac14,\quad \text{but } \ne 0, \]

where \(x_{\mu_i}^{(i)}\) are the endpoints of the intervals indicated above; thus, in the present case, around the point \(x_1^0\) there must necessarily be an interval of length \(\frac14\) free of points of \(Q_\omega\). It is clear that such free intervals are possible to the right of each of the division points.

Points like \(\frac14\) will be precisely those which do not enter into \(Q_\omega\), but serve as limit points of \(Q_\omega\), and of which we spoke in the preceding paragraph.

B. Take another example: let, for the same segment \((0,1)\), the first point of the first division be \(\frac12\), the first point of the second division \(\frac1{2^2}\), the first point of the third division \(\frac1{2^3}\), and so on; then

\[ \lim_{i\ldots\infty} x_{\mu_i}^{(i)}=0, \]

so that \(0\) will also be a limit point of \(Q_\omega\).

102. We thus arrive at the conclusion that, first, the domain \(Q_{\omega}-x_1^0\) will be condensed, and sometimes \(Q_{\omega}\) itself will be condensed; and, second, in an arbitrary part \((\alpha_0,\beta_0)\) of the interval \(l\) there may be found points \(\alpha \leqq \beta\) which do not belong to \(Q_{\omega}\).

Depending on whether in every part \((\alpha_0,\beta_0)\) of the interval \(l\) it is possible to find points \(\alpha=\beta\) or \(\alpha<\beta\) not belonging to \(Q_{\omega}\), or whether in separate parts of \(l\) now one, now the other contingency occurs, in the first case we shall have a domain \(Q_{\omega}\) that is everywhere dense on the interval \(l\), in the second—nowhere dense on the interval \(l\), and in the third \(Q_{\omega}\) will prove to be a mixed domain; that the second case is possible we have seen from example A 101°.

103. We shall now show that, with a suitable choice of the law of placement of the intervals, the domain \(Q_{\omega}\) may be everywhere dense.

To reveal this, let us take the following example: divide the interval \(l\) into four equal parts, and take the first three for the intervals \(l'_1,l'_2,l'_3\), each of which will therefore be equal to \(\frac14 l\); divide the fourth part again into four, and call the first three \(l'_4,l'_5,l'_6\), so that

\[ l'_4=l'_5=l'_6=\frac{1}{4^2}l; \]

divide the last \(\frac{1}{16}l\) again into four parts, and the first three will be \(l'_7,l'_8,l'_9\), where

\[ l'_7=l'_8=l'_9=\frac{1}{4^3}l; \]

and so on. Thus the intervals \(l'_{3n+1}, l'_{3n+2}, l'_{3n+3}\) will be equal to one another and will constitute a \(\frac{1}{4^{\,n+1}}\) part of the interval \(l\). Then on each of the intervals \(l'_i\) we take the same distribution of intervals; and so on.

It is obvious that the points \(x_i^{(n)}\) will be placed along \(l\) more and more densely, and, as \(n\) increases without bound, the domain \(\{x\}\) becomes everywhere dense, since it will be the domain of finite quaternary fractions, if \(l\) is taken as the unit.

104. In the case when \(Q_{\omega}\) is nowhere dense, its derived set, coinciding with the derived set of the condensed domain \(Q_{\omega}-x_0\), will be perfect; if the domain \(Q_{\omega}\) is everywhere dense, its derived set will be identical with the continuum \(L\).

Thus we obtain here the domain \(Q_\omega\), which is countable and may be either sparsely or densely scattered over \(L\), i.e. we arrive at the same type as we had in 95°–96°.

Let \(Q_\omega\) be sparsely scattered; let us call \(P\) the perfect domain \(Q'_\omega\); its free intervals will be \(\{\lambda_i\}\), the domain of their boundaries and exterior points—respectively \(Q\) and \(E\); then \(P=Q+E\). Since the points \(Q_\omega-x^0_1\) can here serve only as left boundaries of free intervals, while they cannot be right boundaries or exterior points, it follows that \(Q_\omega-x^0_1=D(Q)\), i.e. the domain of the boundaries of the intervals of all systems of placement of type \(\omega^\omega\), except for \(x^0_1\), enters into the composition of the boundaries of some perfect domain of intervals.

Thus domains of intervals of type \(n^\omega\) and \(\omega^\omega\) give us analogous domains. The difference between them is that for \(\omega^\omega\) all points of \(Q_\omega\) will certainly be right limiting points of the boundaries of free intervals of the perfect domain. It is clear that the domain \(*\omega^\omega\) will be of the same \(\omega^\omega\) type, with the only difference that in \(*\omega^\omega\) the role of the point \(x'_1\) will be played by the point \(x^0_2\).

105. Let us further investigate the domain of type \((*\omega+\omega)^\omega\), which we obtain if in each of the intervals of type \(*\omega+\omega\) we place new intervals \(*\omega+\omega\), in these again intervals of the same type, and so on.

Analogously to the preceding, we shall denote by \(x'_1\) and \(x''_1\) the boundaries of the segment \(L\), and shall call the domain consisting of these two points

\[ Q^{(0)}=\{x'_1,x''_1\}; \]

this will be the domain of zero division.

Let us further call \(Q^{(1)}\) the domain of points of the first division, internal with respect to \(L\),

\[ Q^{(1)}=\left\{x'_i\right\}_{\pm 1}^{\pm\omega}; \]

as for the exterior points, we have \(E^{(1)}=\{x''_i\}=Q^{(0)}\). The points of the second division, internal with respect to \(l'_i\), we shall call \(\{x'_{ij}\}\); they constitute—

take the domains

\[ Q_i^{(2)}=\{x'_{ij}\}_{j=\pm1}^{\pm\omega},\qquad \sum_{0}^{\pm\omega} Q_i^{(2)}=Q^{(2)}; \]

for the exterior points of the intervals of the second division

\[ E^{(2)}=\{x'_i\}_{1}^{2}+\{x'_i\}_{\pm1}^{\pm\omega}=Q^{(0)}+Q^{(1)}. \]

The intervals of the second division \(\{l'_{ij}\}\) constitute a double and even, if desired, a quadruple sequence; they may be brought into the form of a simple sequence \(\{l''_i\}\); it is evident that then \(\{l'_{ij}\}=\{l''_i\}\), since for each \(l'_{ij}\) there is a place in the sequence \(l''_i\), and conversely. The boundaries of the intervals \(\{l''_i\}\), which will also be countable, we renumber in some order; let them be \(\{x''_k\}\). It is clear that then \(\{x'_{ij}\}=\{x''_k\}\), since no other points, except the points of the second division, are boundaries of the intervals \(l''_i\); hence \(\{x''_k\}=Q^{(2)}\).

In the intervals \(l''_i\) we again place intervals \(\{l''_{ij}\}\), bounded by the points \(x''_{ij}\), for which

\[ Q_i^{(3)}=\{x''_{ij}\}_{j=\pm1}^{\pm\omega},\qquad \sum_{0}^{\pm\omega} Q_i^{(3)}=Q^{(3)}, \]

whereas for the exterior points of \(l''_{ij}\) we shall have

\[ E^{(3)}=\{x'_i\}_{1}^{2}+\{x'_i\}_{\pm1}^{\pm\omega}+\{x''_k\}_{1}^{\omega} =\sum_{0}^{2} Q^{(i)}. \]

Having brought the intervals \(\{l''_{ij}\}\) into the sequence \(\{l'''_i\}\), and denoting their boundaries in some order by \(\{x'''_k\}\), we shall have \(\{x'''_k\}=Q^{(3)}\), and so on; we shall successively obtain

\[ Q_i^{(n)}=\{x_{ij}^{(n-1)}\}_{j=\pm1}^{\pm\omega},\qquad \sum_{0}^{\pm\omega} Q_i^{(n)}=Q^{(n)}=\{x_k^{(n)}\},\qquad E^{(n)}=\sum_{0}^{\,n-1} Q^{(i)}. \]

Let us now see what the domains \(Q^{(n)}\), \(E^{(n)}\), and

\[ P_n = Q^{(n)} + E^{(n)} = \sum_0^n Q^{(i)} \]

will represent.

The domain of boundaries of the intervals of the \(n\)-th arrangement consists exclusively of the points of the \(n\)-th division; all the preceding division points are external points of the intervals \((*\omega+\omega)^n\); the domain \(P_n\) comprises the boundaries of all divisions from the zeroth to the \(n\)-th inclusive.

Let us further clarify what will result in the case where we increase \(n\) without bound; let us call

\[ P_\omega = \lim P_n = \sum_0^\omega Q^{(i)}. \]

It is obvious that also

\[ \lim_{n\ldots\infty} E^{(n)} = \lim \sum_0^{n-1} Q^{(i)} = \lim \sum_0^n Q^{(i)} = P_\omega. \]

The domain \(P_\omega\) will contain all points belonging to all domains of the denumerable sequence

\[ Q^{(0)},\ Q^{(1)},\ Q^{(2)},\ldots,\ Q^{(n)},\ldots; \]

this will be the domain of the boundaries of the intervals of all divisions.

If we now, analogously to 99°, find the derived domains of the domains \(P_n\), then we obtain the following table:

\[ P'_0 = 0, \]

\[ P'_1 = P_0,\quad P''_1 = 0, \]

\[ P'_2 = P_1,\quad P''_2 = P_0,\quad P'''_2 = 0, \]

\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots, \]

\[ P'_n = P_{n-1},\quad P''_n = P_{n-2},\quad P'''_n = P_{n-3},\quad P^{\mathrm{IV}}_n = P_{n-4},\ldots,\ P^{(n)}_n = P_0,\quad P^{(n+1)}_n = 0. \]

We see from this that each domain \(P_n\) will be closed, since all its limiting points enter into the composition of the domain

\(P_{n-1}\), contained in \(P_n\); each of the \(P_n\) will be a region of the first order and of the \(n\)-th kind, as in 99°.

Each of the regions \(Q^{(i)}\) is nowhere dense; therefore—according to 99°—in any part \((\alpha_0,\beta_0)\) of the interval \(L\) one can find points \(\alpha<\beta\) not belonging to \(P\), since they will not belong to any of the \(Q^{(i)}\).

106. Before setting forth the further study of the properties of the region \(P_n\), we shall examine several examples which will make clear to us what we may expect from this study.

A. Let us divide the interval \(l\) into four equal parts, and take the two middle ones as the intervals \(l'_0\) and \(l''_0\); then divide the two outer parts

[Diagram of successive subdivisions of an interval, with labels such as \(X^0_1\), \(X'_1\), \(X_4\), \(X_3\), \(X_2\), \(X_1\), \(X_{\bar{0}1}\), \(X_{\bar{0}0}\), \(X_{01}\), \(X'_0\), \(X_{01}\), \(X_{00}\), \(X_{01}\), \(X_1\), \(X_2\), \(X_3\), \(X_4\), \(X^0_1\), and intervals \(l'_1,l'_2,l'_3,\ldots,l'_0,\ldots\).]

again into 4 equal parts and denote by \(l'_3,l'_2,l'_{-1}\) and \(l'_1,l'_2,l'_3\) the parts adjacent three at a time to \(l'_0\) and \(l''_0\) on the two sides; divide the two outermost parts again into 4, and so on. Then the countable sequence of intervals \(\{l'_i\}\) will consist of two intervals of length \(\frac14\,l\), six intervals of length \(\frac{1}{4^2}\,l\), six intervals of length \(\frac{1}{4^3}\,l\), and so on, where the intervals \(l'_{3n+3}, l'_{3n+2}, l'_{3n+1}, l'_{3n+1}, l'_{3n+2}, l'_{3n+3}\) have length equal to \(\frac{1}{4^{n+2}}\,l\). On each of the intervals \(\{l'_i\}\) we arrange in exactly the same way the intervals \(\{l'_{ij}\}\); and so on.

Obviously, into the region \(P_\omega\) of the boundaries of the intervals of all divisions will enter all points of the infinitely repeated division of the interval \(l\) by 4; these points will be everywhere dense in the interval.

If \(l\) is taken as unity, the points \(\{x_i^{(n)}\}\) can be expressed by proper finite quaternary fractions \(0,a_1a_2a_3\ldots a_n\), where the numbers \(a_i\) take only the values \(0,1,2,3\).

Each of the points of \(P_\omega\) will be a limiting point, but not all infinite fractions in the quaternary system will enter among the points of \(P_\omega\).

Thus the region \(P_\omega\) will be everywhere dense, countable, dense-in-itself, but not closed; for it \(P'_\omega \equiv L\).

B. Let us now take another law of division of the segment \(l\). Let the first interval \(l'_0\) be situated symmetrically on \(l\) and constitute its half; to the right and to the left of it are taken the intervals \(l'_{-1}\) and \(l'_1\), equal to the halves of the segments \((x^0_1, x'_{-1})\) and \((x'_1, x^0_1)\); further—the intervals \(l'_{-2}\) and \(l'_2\), again equal to the halves of \((x^0_1, x'_{-2})\) and \((x'_2, x^0_1)\), etc.; such will be the intervals of the first division.

For the intervals of the second division let us take a different ratio of the middle interval to the corresponding interval of the first division; for the intervals of the third division let us again change this ratio, etc., while for the subsequent intervals we shall leave the same law as above, i.e. we take each following interval equal to half of the segment from the end of the preceding interval to the end of the whole interval of the preceding division. We shall take these ratios in the following manner.

The endpoints for the interval \(l'_0\) are the points

\[ \frac{1}{4}l=\left(\frac{3}{8}-\frac{1}{2^3}\right)l,\qquad \frac{3}{4}l=\left(\frac{5}{8}+\frac{1}{2^3}\right)l; \]

on this \(l'_0\), for the endpoints of the middle interval of the second division, let us take the points

\[ \frac{5}{16}l=\left(\frac{3}{8}-\frac{1}{2^4}\right)l,\qquad \frac{11}{16}l=\left(\frac{5}{8}+\frac{1}{2^4}\right)l; \]

for the endpoints of that middle interval of the third division which is situated on the interval \(\left(\frac{5}{16}l,\frac{11}{16}l\right)\), let us take

\[ \frac{11}{32}l=\left(\frac{3}{8}-\frac{1}{2^5}\right)l,\qquad \frac{21}{32}l=\left(\frac{5}{8}+\frac{1}{2^5}\right)l; \]

and so on; for the \(n\)-th division the endpoints will be

\[ \left(\frac{3}{8}-\frac{1}{2^{n+2}}\right)l,\quad \left(\frac{5}{8}+\frac{1}{2^{n+2}}\right)l. \tag{1} \]

In the same ratio there must be taken, in order to obtain the intervals of the \(n\)-th division, the middle intervals for each interval of the \(n-1\)-st division.

Then the disposition of the boundaries of the intervals on all intervals of one system will be similar, i.e. plotted only on different scales. Therefore what happens on one interval of a known division will also happen on every other interval of the same system.

Let us now examine the middle interval of the whole segment \(l\). Since the limits of the ends of the middle interval \(l^{(n)}_0\)—on the basis of (1)—are the points \(\frac{3}{8}l\) and \(\frac{5}{8}l\), it follows, first, that these points do not enter into the composition of the set \(P_\omega\) of the boundaries of the intervals of all divisions \(\{l^{(n)}_i\}\); secondly, the interval from \(\frac{3}{8}l\) to \(\frac{5}{8}l\), of length \(\frac{1}{4}l\), will be free from points of \(P_\omega\).

This free interval is related to the original one as \(\frac{1}{4} : \frac{1}{2}\), i.e. it constitutes one half of it; consequently, on each interval of the first division we shall have a free interval equal to \(\frac{1}{2}\) of the corresponding interval.

Thus, after the first division there is already revealed a countable series of intervals free from points of \(P_\omega\); the total length of such intervals will be \(\frac{1}{2}l\).

In exactly the same way we can compute the new free intervals that appear after the second division, and indicate the limiting points \(\{\xi''_{ij}\}\) that must thereby be determined; and so on.

We see that in the present example each of the points of \(P_\omega\) will be a limiting point; but they by no means exhaust the entire abundance of limiting points, since there exist such points not entering into the composition of \(P_\omega\).

It follows from this that here the set \(P_\omega\) will be countable, condensed, but not closed, and spread everywhere densely; the derivative \(P'_\omega\) of this set will be perfect. If we construct, for \(P'_\omega\), the set of free intervals \(\{\lambda_i\}\), then not one of the points \(P\) will serve as a boundary of these intervals; indeed, each \(x\) will necessarily be a two-sided limiting point, whereas the boundaries of intervals of the type

\(\omega\) must necessarily be one-sided limit points. Consequently the set \(P_\omega\) will enter wholly into the composition of the external points of the set of intervals \(\{\lambda_i\}\). The points \(\xi\)—obviously—will, on the contrary, serve as the boundaries of the intervals \(\{\lambda_i\}\).

107. Having set forth these examples, we now return to the investigation of sets of type \((*\omega+\omega)^\omega\) in their general form. As we saw in 105°, each of the \(P_n\) is closed; all those limit points in neighborhoods of which there lie infinitely many points of one of the sets \(P_n\) belong to \(P_{n-1}\); but, when we pass from \(P_n\) to \(P_\omega\), the appearance of other limit points is possible as well, namely those which are limits of countable sequences of points taken from different \(Q^{(i)}\). In the examples of 106° we have already seen such points; these points, being limits of sequences of points \(Q^{(i)}\) for various \(i\), themselves do not belong to any one of the \(Q^{(i)}\) and consequently will not enter into \(P_\omega\) either. Thus all points of the set \(P_\omega\) are limit points, but there exist for it further limit points not belonging to it. Consequently \(P_\omega\) will be a condensed set, but not perfect, since it is not closed.

Once \(P_\omega\) is condensed, its derivative \(P_\omega'\), which we shall call \(P\), will be perfect; in it there will also appear the limit points previously missing.

The set \(P_\omega\), representing a countable sequence of countable sequences \(Q^{(i)}\), will itself be countable; \(P\), however, will be uncountable; consequently the set of limit points not entering into \(P_\omega\) must be uncountable.

All the sets \(P_n\) are of the first species, while the set \(P_\omega\), since \(P'\) is perfect, is a set of the second species.

Further—each of the sets \(P_n\) is nowhere dense; as for \(P_\omega\), it turns out that it may be both nowhere dense and everywhere dense; examples of both cases we have seen above. Therefore, in an arbitrary part of the segment \(l\), for the former there are free intervals, and for the latter—points belonging to \(P_\omega\). Each of the points of \(P_\omega\) is a two-sided limit point; therefore, if for a nowhere dense set \(P_\omega\) we construct the free intervals \(\{\lambda_i\}\) of the perfect set \(P=P_\omega'\), then not one of the points of \(P_\omega\) will be a boundary of such intervals: all of them will turn out to be their external points. If \(P_\omega\) is everywhere dense, then—obviously—\(P\) is identical with \(L\).

Thus we may state, for a nowhere dense set, the following assertion:

The set \(P_\omega\) of endpoints of intervals of all divisions of type \((*\omega+\omega)^\omega\) is contained in the set of exterior points of the intervals \(\{\lambda_i\}\) of the perfect set \(P'_\omega\). Hence it follows that the arrangement of intervals of type \((*\omega+\omega)^\omega\) gives results close to those of the arrangement \(\omega^\omega\), but not coinciding with them. As regards the arrangement of the free intervals \(\{\lambda_i\}\) themselves, \((*\omega+\omega)^\omega\) does not differ from the arrangement \(\tilde{\omega}\).

108. Let us take an arrangement of type \(\tilde{\omega}\); in its intervals let us again place intervals of the same type, and let us continue this operation indefinitely; denote the resulting type by \(\tilde{\omega}^{\omega}\). We shall call
\(Q^{(0)}=\{x''_1,x^0_1\}\), \(Q^{(1)}\) the endpoints of the intervals of the first arrangement

\[ Q^{(1)}=\{x'_i\}_{i=\pm1}^{\pm\omega} \]

and \(E^{(1)}\) the set of their exterior points, not including in it the exterior points \(x''_1\) and \(x^0_1\), and

\[ \overline{E}^{(1)}=Q^{(0)}+E^{(1)}; \]

then

\[ \tag{2} P^{(1)}=Q^{(1)}+E^{(1)} \]

is, by hypothesis, a perfect set.

In each of its free intervals \(l'_i\) there must be placed a perfect set of intervals \(\{l'_{ij}\}\); call

\[ Q_i^{(2)}=\{x'_{ij}\}_{j=\pm1}^{\pm\omega},\qquad \sum_{1}^{\omega} Q_i^{(2)}=Q^{(2)},\qquad \sum_{1}^{\omega} E_i^{(2)}=\overline{E}^{(2)}, \]

where \(E_i^{(2)}\) is the set of exterior points of the interval \(l_i\), not including \(x_{-i}\) and \(x_i\), and \(\overline{E}^{(2)}\) is the set of exterior points of the intervals \(\{l'_{ij}\}\), only again without the points \(\{x'_i\}=Q^{(1)}\). The set of all exterior points after the second division will be

\[ \tag{3} E^{(2)}=Q^{(0)}+\overline{E}^{(1)}+Q^{(1)}+\overline{E}^{(2)} =P^{(1)}+\overline{E}^{(2)}, \]

whereby—on the basis of §7—the set

\[ \mathrm{P}^{(2)} \equiv \mathrm{Q}^{(2)} + \mathrm{E}^{(2)} \tag{4} \]

will again be perfect. On the basis of (3) and (4), \(\mathrm{P}^{(1)}\) is contained in \(\mathrm{P}^{(2)}\):

\[ \mathrm{P}^{(1)} = \mathrm{D}\bigl(\mathrm{P}^{(2)}\bigr). \]

Let us note here—and this is very important—that all points of \(\mathrm{P}^{(2)}\), except the points of \(\mathrm{P}^{(1)}\), belong to the set \(\mathrm{G}^{(1)}\)—the set of interior points of the intervals \(\tilde{\omega}\) of the first division.

Before proceeding to the third division, we must transform the double sequence of intervals \(\{l'_{ij}\}\) into a simple sequence \(\{l''_i\}\), and perform such a transformation after each operation; under this, the numbering of the points \(\{x'_{ij}\}\) will by itself pass into the numbering \(\{x''_i\}\), since to each interval \(l''_i\) there corresponds a pair of points that do not serve as endpoints of other intervals; thus

\[ \left\{x''_i\right\}_{\pm 1}^{\pm \omega} \equiv \sum_{i=1}^{\omega}\left\{x'_{ij}\right\}_{j=\pm 1}^{\pm \omega} = \mathrm{Q}^{(2)} . \]

After the third division, with the new numbering of the intervals, we obtain

\[ \mathrm{Q}^{(3)}_i = \{x''_{ij}\}, \qquad \mathrm{Q}^{(3)} \equiv \sum_{1}^{\omega}\mathrm{Q}^{(3)}_i, \qquad \overline{\mathrm{E}}^{(3)} \equiv \sum_{1}^{\omega}\overline{\mathrm{E}}^{(3)}_i, \qquad \mathrm{E}^{(3)} = \mathrm{P}^{(2)} + \overline{\mathrm{E}}^{(3)}, \]

where again \(\overline{\mathrm{E}}^{(3)}_i\) consists of the exterior points of the intervals \(l''_{ij}\) lying on the interval \(l''_i\), not including its endpoints, and \(\overline{\mathrm{E}}^{(3)}\) is the set of all such exterior points, except the points \(\{x''_i\} = \mathrm{Q}^{(2)}\). Again the set

\[ \mathrm{P}^{(3)} \equiv \mathrm{Q}^{(3)} + \mathrm{E}^{(3)} \]

will be perfect, and \(\mathrm{P}^{(2)} = \mathrm{D}\bigl(\mathrm{P}^{(3)}\bigr)\); as also with regard to \(\mathrm{P}^{(2)}\), it must be noted here that the points of \(\mathrm{P}^{(3)}\), except the points of the set \(\mathrm{P}^{(2)}\), are all taken from the set \(\mathrm{G}^{(2)}\)—the set of interior points of the second division.

Thus we see that after two, three, and so on, a finite number of operations \(\tilde{\omega}\), we always obtain a perfect set

\[ \mathrm{P}^{(n)} \equiv \mathrm{Q}^{(n)} + \mathrm{E}^{(n)}, \]

which embraces the preceding domain \(\mathrm P^{(n-1)}\)

\[ \mathrm P^{(n-1)} = \mathrm D\bigl(\mathrm P^{(n)}\bigr), \]

and the points \(P_n\), apart from the points of this last domain, are all taken from among the interior points \(G^{(n-1)}\) of the intervals of the \((n-1)\)-st division. The question now is what form the domain

\[ \mathrm P_\omega = \lim_{n\ldots\infty} \mathrm P^{(n)} \]

will take under an infinite continuation of the process \(\delta\)?

However large \(n\) may be, every new planting of the domain \(\delta\) in the intervals of the preceding perfect domain binds the countable series of perfect domains on these intervals into one single perfect domain. Let us see whether the domain \(\mathrm P_\omega\) will remain perfect as \(n\) increases without bound, or, on the contrary, will lose this property.

In order to clarify this, one must ascertain whether every point of the domain \(\mathrm P_\omega\) will be a limiting point, and whether all limiting points are contained in \(\mathrm P_\omega\)? To the first question we shall obtain, as will be seen below, an affirmative answer, and to the second—a negative one.

109. The domain \(\mathrm P\) is composed successively of all possible distinct points of the perfect domains

\[ \mathrm P^{(1)},\ \mathrm P^{(2)},\ \mathrm P^{(3)},\ldots,\mathrm P^{(n)},\ldots, \]

with respect to which we have seen that each domain \(\mathrm P^{(n)}\) is contained wholly in the one following it, \(\mathrm P^{(n+1)}\).

Let us denote

\[ \mathrm P^{(1)} = Q^{(1)} + E^{(1)} = P_1, \]

\[ \mathrm P^{(2)} = Q^{(2)} + E^{(2)} = Q^{(2)} + \mathrm P^{(1)} + \overline{E}^{(2)} = P_1 + P_2, \]

\[ \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots, \]

\[ \mathrm P^{(n)} = Q^{(n)} + E^{(n)} = Q^{(n)} + \mathrm P^{(n-1)} + \overline{E}^{(n)} = \sum_{1}^{n} P_i, \]

\[ \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots, \]

where \(P_n = Q^{(n)} + \overline{E}^{(n)}\); then the domain \(\mathrm P_\omega = \lim \mathrm P^{(n)}\) will consist of all points of the domains

\[ P_1, P_2, P_3,\ldots, P_n,\ldots, \]

which no longer have common points with one another,

\[ \mathrm{P}_\omega=\sum_{1}^{\omega}\mathrm{P}_i; \]

if we substitute here the value of \(\mathrm{P}_i\), we obtain

\[ \mathrm{P}_\omega=\sum_{1}^{\omega}Q^{(i)}+\sum_{1}^{\omega}E^{(i)}, \]

whence it follows that \(\mathrm{P}\) is composed of the boundaries and exterior points of all the domains.

Let us see what the domains \(\mathrm{P}_i\) represent. The domain \(\mathrm{P}_1\) is perfect; \(\mathrm{P}_2\) differs from the perfect domain of the second derivation \(\mathrm{P}^{(2)}\) only by the absence of the points \(\mathrm{P}^{(1)}\), i.e. it will consist of a countable sequence of perfect domains, but on the open intervals \(l_i\); thus the boundaries of these intervals and their limit points will not enter into it. Each of the domains \(\mathrm{P}_n\) stands in the same relation to the perfect domain \(\mathrm{P}^{(n)}\), and—consequently—it will be similar to the domain \(\mathrm{P}_2\). All these domains will be dense, uncountable, and nowhere dense, while the missing limit points of each of them \(\mathrm{P}_n\) enter into the composition of the preceding domains \(\mathrm{P}_i\), since

\[ \sum_{1}^{n}\mathrm{P}_i=\mathrm{P}^{(n)} \]

is perfect.

Let us now examine the domain \(\mathrm{P}_\omega\); let \(x\) be one of its points; if this is so, then the point \(x\) must enter into one of the domains \(\mathrm{P}_i\), for example—into \(\mathrm{P}_n\); since \(\mathrm{P}_n\) is dense, \(x\) will be a limit point of \(\mathrm{P}_n\); therefore, in an arbitrary neighborhood of it there lie points of \(\mathrm{P}_n\), and consequently also points of \(\mathrm{P}_\omega\), since \(\mathrm{P}_n\) is contained in \(\mathrm{P}_\omega\) entirely, and \(x\) will thus be a limit point of \(\mathrm{P}_\omega\); hence every point of the domain \(\mathrm{P}_\omega\) will be its limit point. Now let, conversely, \(x\) be one of the limit points of \(\mathrm{P}_\omega\); then in an arbitrary neighborhood of it there lie points of \(\mathrm{P}_\omega\). With regard to these points one may make, as above, only two mutually exclusive assumptions: either in an arbitrary neighborhood of \(x\) there lie infinitely many points, at least of one of the domains \(\mathrm{P}_i\), or this is not the case.

In the first case, \(x\) will be a limiting point at least of one region—\(P_n\), and consequently it will enter the preceding regions \(P_i\); then it will also appear in the composition of \(P_\omega\).

Under the second supposition, if \(x\) is a limiting point of \(P_\omega\), but in its neighborhood there are not infinitely many points of any one of the \(P_n\), it is necessary that \(x\) be a limit for some sequence of points

\[ \tag{6} x_1, x_2, x_3, x_4, \ldots, x_n, \ldots, \]

of which only a finite number belongs to each \(P_i\). In this case we can easily prove, similarly to how we did above, that the point \(x\) will not enter \(P_\omega\).

Thus \(P_\omega\) will be a dense, but not closed, region, i.e. the operation \(\tilde{\omega}^{\omega}\) no longer gives the same result as \(\tilde{\omega}^{n}\), which reduced to \(\tilde{\omega}\); here something else was bound to result.

110. Let us now see what can be said concerning the scattering of the region \(P_\omega\) over the segment \(L\)?

For this purpose we shall first examine two examples.

A. Let on the segment \(L \equiv (0,1)\) the first interval \(l'_1\), equal to \(\frac{1}{2}\), be placed symmetrically; its boundaries will be the points \(\frac{1}{4}\) and \(\frac{3}{4}\); on the interval \(\left(0,\frac{1}{4}\right)\) let the second interval \(l'_2\), equal to half of this interval, i.e. \(\frac{1}{8}\), be placed symmetrically; its boundaries will be \(\frac{1}{16}\) and \(\frac{3}{16}\); on the interval \(\left(\frac{3}{4},1\right)\) we place in the same way the interval \(l'_3\) of length \(\frac{1}{8}\), with boundaries \(\frac{13}{16}\) and \(\frac{15}{16}\); on the intervals

\[ \left(0,\frac{1}{16}\right),\quad \left(\frac{3}{16},\frac{1}{4}\right),\quad \left(\frac{3}{4},\frac{13}{16}\right),\quad \left(\frac{15}{16},1\right), \]

we symmetrically place the intervals \(l'_4, l'_5, l'_6, l'_7\), each equal in length to half the preceding interval, i.e. equal to \(\frac{1}{32}\); and so on. Such will be—

the region of the first division; on each of the intervals \(l_i'\) of this division we place a region of intervals of the same structure; in its free intervals again such a region, and so on.

Since the boundaries of the greatest interval of each division always divide in half the distance between the midpoint of the preceding interval and its ends, the limit of this greatest interval is equal to \(0\), and consequently its boundaries have as their limit the common point \(\xi\)—the midpoint of the initial interval. Hence it follows that the region of the boundaries of the intervals will be everywhere dense; moreover it is countable; the region \(P_\omega\), however, will be uncountable, since each \(P_n\) contained in \(P_\omega\) is uncountable.

It is evident that in an arbitrary part of the segment \(L\) there can be found points, and indeed limiting points, which are not included in \(P_\omega\): these will be the points \(\xi\)—the midpoints of each interval first appearing in the process of division.

Thus \(P_\omega\) will be non-closed, although it is dense; \(P_\omega'\) here coincides with the segment \(L\) itself.

B. Let us take another law of distribution, differing slightly from the first.

On the segment \(L\), for which \(l=1\), as before we place symmetrically an interval equal in length to \(\frac{1}{2}\); its boundaries will be

\[ \frac{1}{4}=\frac{3}{2^3}-\frac{1}{2^3},\qquad \frac{3}{4}=\frac{5}{2^3}+\frac{1}{2^3}; \]

the remaining intervals of the first division we leave the same as they were taken in the first example. On the interval \(l_1'\) we take

![diagram]

again symmetrically an interval \(l''\), defining it by the points

\[ \frac{5}{16}=\frac{3}{2^3}-\frac{1}{2^4},\qquad \frac{11}{16}=\frac{5}{2^3}+\frac{1}{2^4}; \]

this interval, equal to \(\frac{3}{8}\), will constitute \(\frac{3}{8}:\frac{1}{2}=\frac{3}{4}\) of the interval \(l_1'\).

On the intervals \(\left(\frac14,\frac{5}{16}\right)\) and \(\left(\frac{11}{16},\frac34\right)\) we shall place the remaining intervals of the second division according to the same law as above, i.e. take intervals whose magnitudes are equal to one-half of the parts indicated above. We shall take the same law of division on all intervals \(l_i'\) of the first division, i.e. everywhere the middle symmetrically situated interval will be equal to \(\frac34\,l_i'\).

For the middle interval of the third division we take the interval

\[ \left(\frac{11}{32}=\frac{3}{2^3}-\frac{1}{2^5},\quad \frac{21}{32}=\frac{5}{2^3}+\frac{1}{2^5}\right), \]

which constitutes \(\frac{5}{16}:\frac{3}{8}=\frac{5}{6}\) of the interval \(l''\); and so on; in general, the middle interval of the \(n\)-th division will be

\[ \left(\frac{3}{2^3}-\frac{1}{2^{n+2}},\quad \frac{5}{2^3}+\frac{1}{2^{n+2}}\right). \]

By these middle intervals each of the systems of divisions is determined, since the remaining intervals in each system are taken according to one and the same law.

As \(n\) increases without bound, the points

\[ \frac{3}{8}-\frac{1}{2^3},\quad \frac{3}{8}-\frac{1}{2^4},\quad \frac{3}{8}-\frac{1}{2^5},\ldots,\quad \frac{3}{8}-\frac{1}{2^{n+2}},\ldots, \]

\[ \frac{5}{8}+\frac{1}{2^3},\quad \frac{5}{8}+\frac{1}{2^4},\quad \frac{5}{8}+\frac{1}{2^5},\ldots,\quad \frac{5}{8}+\frac{1}{2^{n+2}},\ldots \]

have, respectively, as limits the points \(\frac38\) and \(\frac58\), which will not be boundaries of intervals and which themselves determine a new interval \(\left(\frac38,\frac58\right)\), free of points \(P_\omega\). This interval constitutes \(\frac14:\frac12=\frac12\) of the interval \(l_i'\). Since on all intervals of the first division the arrangement of the intervals of the further systems of division will be similar to the arrangement on \(l_i'\), i.e. it will be the same, but on another scale, then on each of the intervals \(l_i'\) there will be an interval, free of points \(P_\omega\), equal in length to \(\frac12 l_i'\). An analogous thing will take place on the intervals of the second division.

Thus the law of division indicated above will give on \(L\) a non-closed, nowhere dense domain.

It is obvious that here \(P_\omega\) will be dense and consequently \(P_\omega'\) perfect.

111. Thus, considering the domain of intervals of type $\tilde{\omega}^{\omega}$, we arrive at the following conclusions:

The domain of the boundaries of the intervals, always countable, may be sparse and often scattered over the segment $L$, depending on the law of arrangement of the intervals of each of the component types $\tilde{\omega}$.

The domain $P_{\omega}$ is always uncountable and dense, but not closed; therefore $P'_{\omega}=P$ for an often scattered domain coincides with the segment $L$, while for a sparsely scattered one it will be perfect.

For a sparsely scattered domain $P_{\omega}$, each of the division points $\{x_i^n\}$, for arbitrary $n$, being an external point of the perfect domain $P^{(n+1)}$, proves to be a two-sided limiting point; consequently, not one of the points of $P_{\omega}$ can be the boundary of a free interval $\lambda_i$ of the perfect domain $P \equiv P'_{\omega}$; all these points thus enter into the composition of the external points $E$ of the domain of detached intervals $\{\lambda_i\}$; in this respect $\tilde{\omega}^{\omega}$ reminds us of $(*\omega+\omega)^{\omega}$.

The boundaries $\{\lambda_i\}$ are determined by limiting points of the domain $P_{\omega}$ which do not enter into $P_{\omega}$.

It is not difficult to see that, for a domain of intervals of type $\tilde{\omega}^{\omega}$, the domain

$P_{\omega}=\lim P^{(n)}$

coincides with the domain of external points of the intervals of the perfect domain $P'_{\omega}$. To be convinced of this, we shall argue as follows: every point of $P_{\omega}$ enters into the composition of the external points of the intervals $\{\lambda_i\}$; is there now possible any external point of $\{\lambda_i\}$ which would not be a point of $P_{\omega}$? Suppose this is so, and let $\xi$ be such a point; it, as a point of $P'_{\omega}\equiv P$, will be a limiting and, in addition, a two-sided limiting point of $P_{\omega}$. On the basis of 109°—this point will not be a point of $P_{\omega}$ if it serves as a limit for a series (6) of points of different divisions. But in that case these points will be the boundaries of a series of intervals

\[ l'_{p_1},\ l''_{p_2},\ l'''_{p_3},\ldots,\ l^{(n)}_{p_n},\ldots, \tag{7} \]

which are situated one within another; the boundaries of the intervals (7) determine either the point $\xi$—the boundary of a free interval of $P\equiv P'_{\omega}$, if $P_{\omega}$ is sparsely scattered; in this case $\xi$ will be a one-sided limiting point, which is contrary to the supposition; or else $\xi$ will prove to be a limiting

a point for two rows of boundaries of the intervals (7); then \(P_\omega\) will not be sparsely scattered, which again contradicts our condition. Thus \(\xi\) cannot fail to be a point of \(P_\omega\).

It follows from this that the region of intervals \(\{\lambda_i\}\), free of points of \(P_\omega\), coincides with the free intervals of the perfect region \(P'_\omega=P\), and only the boundaries of the intervals \(\{\lambda_i\}\) do not belong to \(P_\omega\).

112. Let us now compare all the types of disposition of intervals that have been examined,

\[ n^\omega,\quad \omega^\omega,\quad *\omega^\omega,\quad (*\omega+\omega)^\omega,\quad \widetilde{\omega}^{\,\omega} \]

and try to draw some general conclusions. For convenience let us call, for a moment,

\[ \omega^\omega=\Omega_1,\quad *\omega^\omega=\Omega_2,\quad (*\omega+\omega)^\omega=\Omega_3,\quad \widetilde{\omega}^{\,\omega}=\Omega_4 \]

and let us first compare the first three types, calling \(P_\omega\) the region of boundaries and exterior points of the intervals of all systems of division.

We saw above that the region \(P_\omega\) may be sparsely scattered over the interval \(L\) and may be densely scattered; in the latter case it is of no special interest, and therefore we shall pass to the first case.

For the first of the types \(\Omega_1\), all points of \(P_\omega\), except the left boundary of the whole interval, will be right-hand limiting points; for \(\Omega_2\) the same will hold for all points except the right boundary of the basic interval, but the points of \(P_\omega\) will be left-hand limits; and finally for \(\Omega_3\) all points without exception turn out to be limiting, and moreover two-sided, except—of course—the boundaries of the interval \(L\). We may say that in all three cases the region \(P_\omega\) will be condensed, except—possibly—for a single isolated point.

If the region \(P_\omega\) is condensed up to the point \(x_{\bar 0}\) or \(x_0\), and is sparsely scattered, then its derivative \(P'_\omega\) will be a sparsely scattered perfect region, and the isolated point \(x_{\bar 0}\) or \(x_0\) will no longer appear in \(P'_\omega\); with this disappearance of \(x_{\bar 0}\) or \(x_0\) there is connected a diminution of the interval \(L\), so that for \(P'_\omega\) the basic interval may already be different.

If we construct for \(P'_\omega=S\) the free intervals, then we shall see that among their boundaries \(Q\) there may occur some of the points of \(P_\omega\), but besides them there will certainly occur points not belonging to the set \(P_\omega\). Those points of \(P_\omega\) which do not enter into \(Q\) will be exterior points of the region of intervals \(P'_\omega\), i.e. will enter into \(E\). For

of the domains \(\Omega_1\) and \(\Omega_2\), one of the endpoints of the free intervals of the final arrangement necessarily belongs to \(P_\omega\); it will be the left endpoint for \(\Omega_1\) and the right endpoint for \(\Omega_2\). For the domain \(\Omega_3\), none of the points \(P_\omega\) will be an endpoint of a free interval of the domain \(Q\).

Thus we may now represent the structure of the domain \(P_\omega\) as follows: having constructed the domain of intervals \(P'_\omega\), we obtain the points \(P_\omega\) as the left endpoints, for \(\Omega_1\), or the right endpoints, for \(\Omega_2\), of the free intervals, and for \(\Omega_3\) as a countable sequence of exterior points of these intervals; for \(\Omega_1\) and \(\Omega_2\) we shall still have to adjoin to the fundamental interval of the domain \(S\), on the left or on the right, a free interval with a solitary point at its end.

In \(\Omega_1\) and \(\Omega_2\) all points, except \(x_{-0}\) or \(x_0\), will be one-sided limit points, and in \(\Omega_3\), on the contrary, two-sided limit points, except for the endpoints of the segment. The limit points of the domain \(P_\omega\) that are not contained in \(P_\omega\) are divided into two categories: first, the countable sequence of endpoints of the free intervals \(P'_\omega\), and second, an uncountable domain of limit points for these endpoints; that this latter domain is uncountable follows from the fact that \(P'_\omega\) is perfect. Thus the free intervals of the domain \(P'_\omega\) prove to be open intervals for \(P_\omega\), since at least one of the endpoints of each interval does not belong to \(P_\omega\).

A domain of type \(\Omega_4\) presents certain analogies with \(\Omega_3\): both are dense; in both, all points, except the endpoints of the fundamental interval, are two-sided limit points; in both, not one of the points of division will be an endpoint of a free interval of the final arrangement. But there is also a profound difference between them: the exterior points of each system of division \((*\omega+\omega)^n\) are always countable, whereas for \(\tilde{\omega}^{\,n}=\tilde{\omega}\) they are uncountable; furthermore, for \(\Omega_3\) the domain \(P_\omega\) embraces all the endpoints of all divisions, among which are also the exterior points of these divisions; whereas for \(\Omega_4\) within \(P_\omega\) one may distinguish, first, the points of all divisions, and second, the exterior points of these divisions; the domain \(P_\omega\) for \(\Omega_3\) will be countable, whereas for \(\Omega_4\) it is uncountable. Finally, the peculiarity of the domain \(P_\omega\) for \(\Omega_4\) is that it coincides with the domain of all exterior points of intervals \(P'_\omega\) of type \(\tilde{\omega}\).

Finally, with regard to type \(n^\omega\), if it gives a rarely scattered domain, we can say nothing beyond the general theorem \(73^\circ\), so long as the law of arrangement of the points of the successive divisions has not been given with sufficient definiteness.

All these considerations prove very essential for the construction of further domains of ever more complicated type.

113. Let now \(\omega_i\) denote one of the types

\[ \tag{8} \omega,\ *\omega,\ *\omega+\omega . \]

Consider a compound type of arrangement of the form

\[ \Omega=\prod_{1}^{\omega}\omega_i, \]

which is a generalization of the preceding types \(\Omega_1,\Omega_2\), and \(\Omega_3\).

The simplest of the types \(\Omega\) will be those in which an infinite number of factors of only one of the types (8) occurs, while the others enter only a finite number of times.

Let, in the first \(m\) factors, all three, or at least two, types occur, while each \(m+i\)-th factor belongs to one of the forms (8); then

\[ \Omega=\prod_{1}^{m}\omega_j\cdot\prod_{m+1}^{\omega}\omega_i =\prod_{1}^{m}\omega_j\cdot\omega_i^{\omega}; \]

if we denote

\[ \prod_{1}^{m}\omega_i=\omega_0, \]

then

\[ \Omega=\omega_0\cdot\omega_i^{\omega} \]

represents an arrangement of one of the types \(\Omega_1,\Omega_2,\Omega_3\) in the free intervals of the type \(\omega_0\)

\[ \Omega=\omega_0\cdot\Omega_i. \]

We shall show first of all that in \(\Omega\) there can be no isolated points.

In the arrangement \(\omega_0\) some points may be limiting and others isolated; those points which were limiting in \(\omega_0\) will remain limiting also for \(\Omega\). Let now \(x\) be one of the isolated points; first of all, it cannot be the boundary of the whole segment \(L\). Indeed: if this were so, \(\omega_0\) would have to be of type \(\omega^n\) or \({}^*\omega^n\), which is excluded by the hypothesis; therefore \(x\) is one of the interior points of \(L\); consequently it will be the common boundary of two adjacent intervals \(l_\mu\) and \(l_\nu\).

Since intervals of type \(\omega_i^{\circ}\) are placed in the intervals \(\omega_0\), \(x\) will be a limiting point of the domain \(\omega_i^{\circ}\) on \(l_{\mu}\), if \(\omega_i=\omega\), or on \(l_{\nu}\), if \(\omega_i={}^{*}\omega\), or, finally, both on \(l_{\mu}\) and on \(l_{\nu}\) in the case where \(\omega_i={}^{*}\omega+\omega\). Thus—in every case it will be a limiting point of the final arrangement.

Finally, in a type \(\omega_i^{\circ}\) the isolated point may be one of the endpoints of the fundamental interval; here, however, these endpoints are precisely the endpoints of the intervals \(\omega_0\), which, as we have just seen, cannot be isolated.

We thus arrive at the conclusion that, under the indicated assumption, the type \(\Omega\) gives a condensed domain \(P\) of endpoints of the intervals of all division systems; consequently \(P'\) will be perfect.

Since the free intervals of \(P'\) will, together with the free intervals of \(\Omega\), and all endpoints and outer points of \(\omega_0\), pass into the composition of the endpoints and outer points of the perfect domain \(P'\), all the conclusions of 112° continue to hold here.

If we now take the other assumption, i.e. suppose that at least two types (8) occur in \(\Omega\) an infinite number of times, then we see immediately that still more in the domain \(P\) obtained in this way there can be no isolated points, so that \(P\) will be condensed and \(P'\) perfect.

To these domains, again, everything that pertains to the preceding one applies.

Thus every composite domain will be condensed, except, perhaps, for one point for \(\omega^{\circ}\) and \({}^{*}\omega^{\circ}\); the endpoints \(P\) of all systems of its divisions enter into the composition of the endpoints and outer points of the perfect domain \(P'\), if \(P\) is nowhere dense, which we shall henceforth assume, since the other assumption is of little interest.

114. If in the composition of the type

\[ \Omega=\prod_{1}^{\omega}\omega_i \]

under the form of one of the \(\omega_i\) there could be hidden an arrangement \(\tilde{\omega}\), then—according to 94°—all the preceding types standing before \(\omega_n=\tilde{\omega}\) would be dissolved in \(\tilde{\omega}\), i.e. we would have

\[ \Omega=\prod_{1}^{n}\omega_i\cdot\prod_{n+1}^{\omega}\omega_i =\tilde{\omega}\cdot\prod_{n+1}^{\omega}\omega_i =\tilde{\omega}\Omega_0; \]

in other words, the whole matter would reduce to placing a certain complicated set \(\Omega_0\) in the intervals \(\tilde{\omega}\); we shall return to such a problem below.

We shall obtain the same thing if the composition of \(\Omega\) contains several factors equal to \(\tilde{\omega}\). If there are infinitely many such factors, then it is not hard to see that one must have

\[ \Omega=\tilde{\omega}^{\omega}. \]

115. In the preceding considerations of 113° we did not at all take into account the type \(n\); indeed, if it entered into some combination \(\omega_{i-1} n \omega_i\), then by virtue of 80° and 83° this combination would give

\[ \omega_{i-1} n \omega_i=\omega_{i-1}\omega_i, \]

so that \(n\) in the final result would have to disappear; finally, the combination \(n\omega_1\) could not be reduced to any one of the types \(\omega_i\), but then we could divide the segment into \(n\) parts, and in each of them one of the types \(\Omega\) would be obtained.

Thus, in what follows, we may completely leave aside the type \(n\); below we shall return to this placement from a somewhat more general point of view.

116. In determining the position of a countable set \(P\) of type \(\Omega_1, \Omega_2\), and \(\Omega_3\) with respect to a perfect set \(P'\), we encounter a very important circumstance and arrive, by an entirely different route compared with Baire, at an extension of the notion of point sets.

When we construct the complementary intervals of the perfect set \(P'\), the points of \(P\) are taken—first—from certain endpoints of the intervals, and—second—from the exterior points of the set \(P'\).

The set \(P'\) is perfect and, consequently, uncountable, whereas the set \(P\) is countable; hence the points of the countable set must be chosen, except—perhaps—for one point, from among the points of the uncountable set \(P'\); in other words, the set \(P\), with respect to the perfect set \(P'\), plays exactly the same role as any point set with respect to the continuum.

Thus, alongside sets with respect to the continuum, we must also consider sets with respect to given perfect sets. Without entering here into the full theory of such sets, we can say only that, as Baire also pointed out,\(^1\)

\(^1\) Annali di Matematica, (3) 7.

to such domains one may apply all the notions established by Cantor for domains with respect to the continuum; in particular, a domain \(P\), referred to the perfect domain \(P'\), will be\(^1\) everywhere dense in \(P'\).

It follows from this that, for the domain \(P\), the free intervals will be only the intervals \(P'\), since outside them the points of \(P\) are situated with respect to \(P'\) in the same way as the rational points are situated with respect to the continuum.

117. If, in the intervals of some domain

\[ \omega_0=\omega_1\omega_2\omega_3\ldots\omega_n, \]

with the notation (8), we place one of the domains of intervals \(\Omega\), we obtain nothing new, since adjoining a finite number of factors to the beginning of the countable series \(\Omega\) again gives a countable series.

The matter will be different if we place \(\Omega\) in the intervals \(\tilde{\omega}\), obtaining thereby the type of arrangement \(\tilde{\omega}\Omega\).

Situated in each of the free intervals \(l_i\) of type \(\tilde{\omega}\), the domain \(P_i\) will have the character indicated above in §§112–113; it will be condensed, except—possibly—for one point, which, as the boundary of the interval \(l_i\), will already be a limiting point of \(\tilde{\omega}\); therefore the whole domain \(P\) of the boundaries of intervals of all divisions will be condensed.

Each of the \(P_i\) is everywhere dense in the perfect domain \(P'_i\), situated on \(L_i\). These latter, taken together, give a perfect domain on \(L\), and the domain \(P\) will stand, with respect to this latter perfect domain, in exactly the same relation as do the domains \(P_i\) with respect to \(P'_i\).

Thus the character of a domain of type \(\tilde{\omega}\Omega\) proves to be similar to the general character of the composite types.

118. We now pass to the arrangement of some domain \(\omega_0\) in the free intervals \(\Omega\); the type of arrangement thereby obtained will be denoted by \(\Omega\omega_0\). Since—on the basis of §116—the free intervals of a domain \(P\) of type \(\Omega\) will be only the intervals of the perfect domain \(P'\), while in the remaining part of the segment \(L\) the points \(P\) will be everywhere dense in the domain of external points \(P'\), the domain \(\omega_0\) must be situated exclusively in the intervals of the perfect domain \(P'\).

\(^1\) See §48.

Thus, with respect to the arrangement of intervals, the type \(\Omega\omega_0\) differs in no way from the type \(\tilde{\omega}\omega_0\), but the difference will be essential if we take the domain of boundaries and exterior points of the intervals of all divisions: for \(\tilde{\omega}\omega_0\) this domain contains 1) the boundaries and exterior points of \(\omega_0\), 2) the boundaries of the free intervals \(\tilde{\omega}\), and 3) all exterior points of \(\tilde{\omega}\); whereas \(\Omega\omega_0\) consists 1) of the same boundaries and exterior points of \(\omega_0\), 2) perhaps of some boundaries of \(\tilde{\omega}\), 3) of a countable sequence of exterior points of the intervals \(\tilde{\omega}\). Thus for \(\tilde{\omega}\omega_0\) the domain \(P\) will be uncountable, whereas for \(\Omega\omega_0\) it is countable.

It is obvious that the arrangement \(\Omega\omega_0\) defines, generally speaking, a domain of general form; here its isolated points will be, besides the single possible point \(\Omega\), the isolated points of the domain \(\omega_0\).

The arrangement of the type

\[ \Omega\omega_0=\prod_{1}^{\omega}\omega_i\cdot\prod_{1}^{n}\omega_i =\prod_{1}^{\omega+n}\omega_i \]

gives occasion to speak of transfinite types of arrangement.

119. Let us next place in the intervals \(\Omega\) perfect domains \(\tilde{\omega}\) and see by what features such an arrangement \(\Omega\tilde{\omega}\) is distinguished.

Since the free intervals of \(\Omega\) are the free intervals of the perfect domain \(P'\), and since in each of them a perfect domain \(\tilde{\omega}\) is placed, the final arrangement of the intervals will be the same as for the type \(\tilde{\omega}\cdot\tilde{\omega}\), which coincides with \(\tilde{\omega}\). But if we turn to the boundaries and exterior points of the intervals \(\tilde{\omega}\) and compare them with the whole domain of boundaries and exterior points of all systems of divisions \(\Omega\tilde{\omega}\), then we shall arrive at different conclusions.

The boundaries of the intervals of the final arrangement \(\Omega\tilde{\omega}\) are exclusively the boundaries of the intervals \(\tilde{\omega}\), since those points of \(\Omega\) which served as boundaries of intervals in \(\Omega\) will now become exterior points of the domain of type \(\tilde{\omega}\); the exterior points of the intervals \(\Omega\tilde{\omega}\) will remain, in addition to the exterior points of the countable sequence of domains \(\tilde{\omega}\), of which the boundaries \(\Omega\) will be included, also all exterior points of \(\Omega\).

120. Further possible is a new transfinite type of arrangement

\[ \Omega^{(1)}\cdot\Omega^{(2)} =\prod_{1}^{\omega}\omega_i\cdot\prod_{1}^{\omega}\omega_i =\prod_{1}^{2\omega}\omega_i, \]

which, in particular, if all \(\omega_i\) are identical with one another, will be one of the dispositions

\[ \omega^{2\omega},\quad {}^{*}\omega^{2\omega},\quad ({}^{*}\omega+\omega)^{2\omega},\quad \tilde{\omega}^{2\omega}. \]

This type will be obtained if we arrange, in the free intervals \(\Omega^{(1)}\), domains of the compound type \(\Omega^{(2)}\).

Again—if the result of such a disposition gives a set that is often scattered, this set can arouse no interest; this interest will be preserved in the case when the set \(\prod_{1}^{2\omega}\omega_i\) proves to be rarely scattered.

Since the free intervals \(\Omega^{(1)}\) and \(\Omega^{(2)}\) will be free intervals of a certain perfect set, then, inasmuch as we are dealing exclusively with free intervals, the disposition \(\Omega^{(1)}\Omega^{(2)}\) does not differ from the disposition \(\tilde{\omega}\cdot\tilde{\omega}\), i.e. from \(\tilde{\omega}\); in other words—the free intervals of type \(\Omega^{(1)}\Omega^{(2)}\) will be the free intervals of a certain new perfect set; the boundaries of the final disposition will be exclusively the boundaries of the intervals \(\Omega^{(2)}\). If, however, we pass to the boundaries and to the exterior points of all systems of division, the matter assumes quite a different form.

Among the boundaries of all the divisions \(P^{(1)}\), determined by the disposition \(\Omega^{(1)}\), there belong 1) some of the boundaries of the isolated intervals \(l_i\) of the set \(P^{(1)'}\), and 2) a countable series of exterior points, often scattered over that same perfect set; the set \(P_i^{(2)}\), placed inside each free interval \(l_i\), likewise consists of 1) certain boundaries and 2) a countable series of exterior points; the boundaries of \(l_i\) will be included among the exterior points of the intervals \(P_i^{(2)}\).

Thus the points of the final set \(P\), comprising all the points of all the divisions, will be chosen 1) from some of the boundaries of the intervals \(\{l_i\}\) of the final disposition \(\Omega^{(1)}\Omega^{(2)}\), and 2) from a countable series of exterior points of these intervals, often scattered over \(P'\). The composition of the set of points determined by the type \(\Omega^{(1)}\Omega^{(2)}\) turns out to be similar to those we have seen for other transfinite sets.

121. It is obvious that, with respect to \(\prod_{1}^{2\omega}\omega_i\), we may continue the process of disposing new sets and obtain sets

\[ \prod_{1}^{2\omega+1}\omega_i,\quad \prod_{1}^{2\omega+n}\omega_i,\quad \prod_{1}^{2\omega}\omega_i,\ldots,\quad \prod_{1}^{\alpha}\omega_i, \]

where \(\alpha\) is any of the Cantor numbers; in particular, we would have the types

\[ \omega^{\alpha},\ *\omega^{\alpha},\ (*\omega+\omega)^{\alpha},\ \tilde{\omega}^{\alpha}. \]

Comparing all that has preceded, we see that the general character of all transfinite types of arrangements remains one and the same: a sequence of free intervals always turns out to be a sequence of type \(\tilde{\omega}\); the domain of the points \(P\) is often scattered over the perfect domain of boundary and exterior points of the intervals \(\tilde{\omega}\), but the selection of a countable sequence of points \(P\) from the uncountable sequence \(P'\) gives rise to an infinite variety of domains \(P\).

122. Besides the complex types investigated above, obtained by infinitely repeating one definite one of the basic types, we could obtain an infinite set of complex types by taking, a) as initial any one of the derived types

\[ \omega_0=\prod_{1}^{n}\omega_i \]

and, considering it as basic, construct the types

\[ \tag{9} \omega_0^n,\ \omega_0^\omega,\ \omega_0^\alpha; \]

then b) we could obtain a complex type

\[ \tag{10} \Omega^{(n)},\ \Omega^{(\omega)},\ \Omega^{\alpha}, \]

by repeating, a finite or infinite number of times, one of the complex types; or again c) take an infinite combination of complex types in a definite sequence

\[ \tag{11} \prod_{1}^{\omega}\Omega_i; \]

or d) take a finite or infinite combination of the complex types (9), (10), and (11) in the corresponding sequence; and so on.

Thus before us there opens an infinite variety of complex types, the simplest examples of which we have considered above. But this by no means exhausts all the wealth of point domains.

5. Mixed types of arrangement.

123. All the basic types of arrangement investigated by us above and their combinations we shall henceforth call pure types, while assigning to this term the following meaning:

By taking, for the construction of series of intervals and the point domains connected with them, some segment \(L\), we place on it new intervals of one or another fundamental type.

Thus the whole segment was subjected, in its entirety, to one and the same operation. Further, in repeated operations, in all intervals of the given division we placed domains of one and the same type; for example—in a domain of type \(\tilde{\omega}(*\omega+\omega)\), in all intervals of the initial perfect domain \(\tilde{\omega}\), we place a domain of type \(*\omega+\omega\).

It is clear that the possibility of other processes must also be admitted; we have the right, for different intervals \(l_i^{(n)}\) at each given \(n\)-th division, to subject them in the \((n+1)\)-st division to different processes, establishing for each interval \(l_i\) its own particular law of the sequence of placement of intervals. Thus still broader prospects open before us with respect to the further construction of domains.

All types of such placement, in distinction from the pure types, we shall call mixed types of placement; we now pass to the investigation of such types.

124. If on two adjacent segments \(L_1\) and \(L_2\) corresponding intervals of types \(\omega_1\) and \(\omega_2\) are placed, then the placement \(\omega_0\) on the full segment \(L_1+L_2\) we shall call the sum of the types \(\omega_1\) and \(\omega_2\)

\[ \omega_0=\omega_1+\omega_2 . \tag{1} \]

A. Suppose we have two adjacent segments \(L'\) and \(L''\), on which there are placed respectively domains of intervals of types \(m\) and \(n\).

It is then obvious that, considering \(L'+L''\) as an entire segment \(L\), we would obtain the same result by taking the placement \(m+n\). The character of the placement is not changed here.

The type of such a placement of intervals, according to the accepted definition, is the sum of the types of placement on each segment.

B. Suppose further that on each of the adjacent segments there lie intervals of type \(\omega\) or \(*\omega\); such a placement we may denote by

\[ \omega+\omega,\quad *\omega+*\omega . \]

If, bearing in mind 28°, we construct a placement of type \(2\omega\) and \(2*\omega\), then we shall be convinced that

\[ \omega+\omega=2\omega,\quad *\omega+*\omega=2*\omega . \]

C. It is likewise obvious that the arrangement

\[ \tilde{\omega}+\tilde{\omega}=2\tilde{\omega} \]

and—on the basis of §88—

\[ \tilde{\omega}+\tilde{\omega}=\tilde{\omega}. \]

D. Finally—on the basis of §83—

\[ ({}^{*}\omega+\omega)+({}^{*}\omega+\omega)=2({}^{*}\omega+\omega). \]

This exhausts the sums of two identical basic types.

125. We now pass to combinations of different basic types; just as in §124, it is not hard to see that

A. First,

\[ n+\omega=\omega,\quad \omega+n\neq\omega,\quad n+{}^{*}\omega\neq{}^{*}\omega,\quad {}^{*}\omega+n={}^{*}\omega. \]

B. If we take the sum of the arrangements \({}^{*}\omega\) and \(\omega\), we obtain the type \({}^{*}\omega+\omega\), which until now we have regarded as something single and have taken as basic; it now turns out that it is nothing other than the sum of the types \({}^{*}\omega\) and \(\omega\) situated on adjacent segments; thus we may regard as basic types only \(\omega\), \({}^{*}\omega\), and \(\tilde{\omega}\).

C. Next we obtain the type \(\omega+{}^{*}\omega\). For it

\[ Q^{(0)}=\{x^0_{-1},x^0_{1}\},\quad Q^{(1)}=\{x_i^{\pm\omega}\}_{\pm1},\quad Q=Q^{(0)}+Q^{(1)},\quad E=\{x^0_0\},\quad Q'=E,\quad Q''=0. \]

Obviously,

\[ \omega+{}^{*}\omega\neq{}^{*}\omega+\omega. \]

D. Finally, it is not hard to verify that

\[ \begin{gathered} n+\tilde{\omega}\neq\tilde{\omega},\quad \tilde{\omega}+n\neq\tilde{\omega},\quad \tilde{\omega}+n\neq n+\tilde{\omega},\\ \omega+\tilde{\omega}\neq\tilde{\omega},\quad \tilde{\omega}+\omega\neq\tilde{\omega},\quad \omega+\tilde{\omega}\neq\tilde{\omega}+\omega,\\ {}^{*}\omega+\tilde{\omega}\neq\tilde{\omega},\quad \tilde{\omega}+{}^{*}\omega\neq\tilde{\omega},\quad {}^{*}\omega+\tilde{\omega}\neq\tilde{\omega}+{}^{*}\omega; \end{gathered} \]

the structure of the rows of intervals and of the point domains corresponding to these types is sufficiently clear from their notation.

126. Let us now suppose that we have a finite number \(n\) of adjacent intervals, and that on them there are placed series of intervals of identical basic types

\[ n,\ \omega,\ {}^{*}\omega,\ {}^{*}\omega+\omega,\ \tilde{\omega}, \tag{2} \]

The arrangement of intervals on the segment

\[ \mathbf{L}=\sum_{1}^{n}\mathbf{L}_{i} \]

may, in accordance with 124°, be denoted as

\[ \sum_{1}^{m}\omega_{0}, \tag{3} \]

where \(\omega_{0}\) will be one of the basic types (2).

Considering the simplest of the types (3), we can easily see that

\[ \sum_{1}^{m} n = mn,\quad \sum_{1}^{m}\omega=m\omega\ne\omega,\quad \sum_{1}^{m}{}^{*}\omega=m{}^{*}\omega\ne{}^{*}\omega, \]

\[ \sum_{1}^{m}\tilde{\omega}=m\tilde{\omega}=\tilde{\omega},\quad \sum_{1}^{m}({}^{*}\omega+\omega)=m({}^{*}\omega+\omega)\ne{}^{*}\omega+\omega. \]

Hence we see that every time we have a finite sum of identical types, we arrive either at a basic type—for \(n\) and \(\tilde{\omega}\)—or at a derived type \(m\omega\), \(m{}^{*}\omega\), \(m({}^{*}\omega+\omega)\).

Thus all mixed types (3) give nothing new and may therefore be eliminated from consideration.

127. Let us pass next to the investigation of the arrangement

\[ \sum_{1}^{\omega}\omega_{0}, \]

where \(\omega_{0}\) is one definite one of the basic types.

First of all it is necessary to note the circumstance that, since the number of intervals \(\mathbf{L}_{i}\) is made infinite, among them—on the basis of 71°—non-adjacent intervals must necessarily appear; an infinite set of adjacent intervals would be possible only in the case when the whole segment \(\mathbf{L}\), with

\[ \sum_{1}^{\omega}\mathbf{L}_{i}=\mathbf{L}, \]

extended to infinity, which we do not admit.

126. Suppose now that we have a finite series of \(n\) adjacent intervals, and that there are placed upon them series of intervals of identical basic types

\[ n,\ \omega,\ {}^{*}\omega,\ {}^{*}\omega+\omega,\ \tilde{\omega}. \tag{2} \]

The arrangement of intervals on the interval

\[ L=\sum_{1}^{n} L_i \]

we may, in agreement with 124°, denote by

\[ \sum_{1}^{m} \omega_0, \tag{3} \]

where \(\omega_0\) will be one of the basic types (2).

Considering the simplest of the types (3), we can easily convince ourselves that

\[ \sum_{1}^{m} n = mn,\quad \sum_{1}^{m} \omega = m\omega=\omega,\quad \sum_{1}^{m} {}^{*}\omega = m{}^{*}\omega = {}^{*}\omega, \]

\[ \sum_{1}^{m} \tilde{\omega}=m\tilde{\omega}=\tilde{\omega},\quad \sum_{1}^{m}({}^{*}\omega+\omega)=m({}^{*}\omega+\omega)={}^{*}\omega+\omega. \]

Hence we see that every time we have a finite sum of identical types, we arrive either at a basic type—for \(n\) and \(\tilde{\omega}\)—or at a derived type \(m\omega,\ m{}^{*}\omega,\ m({}^{*}\omega+\omega)\).

Thus all the mixed types (3) yield nothing new and may therefore be eliminated from consideration.

127. Let us next pass to the study of the arrangement

\[ \sum_{1}^{\omega} \omega_0, \]

where \(\omega_0\) is one definite one of the basic types.

First of all, it is necessary to note the circumstance that, once the number of intervals \(L_i\) becomes infinite, among them—on the basis of 71°—there must necessarily appear non-adjacent intervals; an infinite set of adjacent intervals would be possible only in the case when the whole interval \(L\), with

\[ \sum_{1}^{\omega} L_i = L, \]

extended to infinity, which we do not allow.

Since the number of intervals is infinite, and among them there occur non-adjacent intervals, the arrangement

\[ \sum_{1}^{\infty} \omega_0, \]

if no further indications are given, becomes unclear. Therefore, if we wish to deal with the arrangement

\[ \sum_{1}^{\infty} \omega_0, \]

we must first say how the intervals \(L_i\) are arranged.

Among the simplest arrangements are

\[ \tag{4} \omega,\quad {}^*\omega,\quad {}^*\omega+\omega,\quad \tilde{\omega}. \]

If the intervals are arranged in this way, then

\[ \sum_{1}^{\infty} \omega_0 \]

for the corresponding arrangements (4) will be nothing other than

\[ \omega\omega_0,\quad {}^*\omega\,\omega_0,\quad ({}^*\omega+\omega)\omega,\quad \tilde{\omega}\omega, \]

i.e. types already familiar to us.

In exactly the same way we shall obtain known types of arrangements also in the case when the arrangement of the intervals \(l_i\) belongs not to the basic types, but to derivative or even composite ones.

Thus the types

\[ \sum_{1}^{n} \omega_0,\quad \sum_{1}^{\infty} \omega_0 \]

give nothing decisively new; among the new types belonging to the category of mixed ones, we have so far obtained only

\[ \omega+n,\quad n+{}^*\omega,\quad \omega+{}^*\omega,\quad \omega_0+\tilde{\omega},\quad \tilde{\omega}+\omega_0, \]

where \(\omega_0\) is any one of (4), except \(\tilde{\omega}\); the structure of the corresponding series of intervals and point domains is extremely simple and evident, and therefore it is not worth dwelling on them further.

128. We pass then to the case where, on a finite row \(m\) of adjacent intervals, there are arranged different types

\[ \omega_1,\quad \omega_2,\quad \omega_3,\ldots,\quad \omega_m, \]

which are certain basic, derivative, or composite types of arrangement; the type of arrangement thus obtained we shall denote by

\[ \omega_0=\sum_1^m \omega_i . \]

In view of the infinite variety of the types \(\omega_i\), it is not possible to subject them to an exhaustive investigation; moreover, the study of regions of such types presents no special difficulty and arouses no special interest, since in each case the whole matter reduces to a combination of types which have already been sufficiently studied above; we can give here only a few examples.

A. If, for example, we were given

\[ \omega_1=\omega,\quad \omega_2=\omega^2,\quad \omega_3=\omega^3,\ldots,\quad \omega_m=\omega^m, \]

then we would obtain\(^1\), on the segment \(L\) with endpoints \(\{x_0^0,x_m^0\}\), a region \(P\) of type

\[ \omega+\omega^2+\omega^3+\cdots+\omega^m, \]

for which \(P^{(m)}=\{x_m^0\}\).

B. If it were

\[ \omega_1=1,\quad \omega_2=n,\quad \omega_3=\omega,\ldots,\quad \omega_m=\omega^{m-2}, \]

then the type of arrangement on \(L\) would be

\[ 1+n+\omega+\omega^2+\cdots+\omega^{m-2} =\omega+\omega^2+\cdots+\omega^{m-2}, \]

and the region \(P\) would be of the \((m-2)\)-nd kind.

129. We pass finally to the most complex type: suppose that in each of the intervals of an arbitrary arrangement \(\omega_0\) there is placed a region of intervals of a special kind.

A region of intervals \(\omega_0\) is always countable; consequently these intervals can be brought to the form \(\sum_1^\omega l_i\), where the order of the terms of the series can usually be established in a great variety of ways; there are, however, such types, namely \(-\omega\) and \({}^*\omega\), where this order is sub-

\(^1\) See A. M. 2, p. 344.

is determined by the arrangement itself; but, besides these, in all the remaining types we shall have to establish the order in which the intervals themselves follow.

Thus, suppose that this order has in one way or another been established; then each interval \(L_i\) is completely determined by its number; in this interval we again place a completely definite arrangement \(\omega_i\), which may be fundamental, derived, or composite.

After a definite type of arrangement \(\omega_i\) has been assigned to each interval \(L_i\), the general arrangement of the intervals on all the \(L_i\) becomes precisely determined; the type of arrangement thereby obtained we shall denote by

\[ \omega_1 \overset{\omega}{\underset{1}{\mathbf Z}} \omega_i = \Omega . \]

Since the number of intervals \(\omega_0\) is countable, and in each of them a countable sequence of intervals \(\omega_i\) is placed, the sequence of intervals \(\Omega\) will also be countable, as agrees with Theorem \(69^\circ\).

Concerning the notation \(\overset{\omega}{\underset{1}{\mathbf Z}}\), the following must be observed: it has been introduced in place of \(\overset{\omega}{\underset{1}{\sum}}\) under the law of arrangement indicated here, because \(\omega_0 \overset{\omega}{\underset{1}{\sum}} \omega_i\) would mean that in each of the intervals \(\omega_0\) the domain \(\overset{\omega}{\underset{1}{\sum}} \omega_i\) is placed; we had to denote in some way such a relation between \(\omega_0\) and the sequence of types \(\{\omega_i\}\) that to each free interval \(l_i\) of type \(\omega_0\) there corresponds a certain type \(\omega_i\). Thus the multiplication sign in connection with the sign \(\overset{\omega}{\underset{1}{\mathbf Z}}\) has an entirely different meaning from that of the multiplication sign followed by the summation sign.

The wealth of the types \(\Omega\) is inexhaustible; for any given \(\omega_0\) and for a sequence \(\overset{\omega}{\underset{1}{\mathbf Z}}\omega_i\) of given types \(\omega_i\), by merely changing the order of the numbering of the intervals we may obtain an uncountable set of domains. From this variety of domains it seems interesting to consider at least the simplest.

130. Let, on intervals of type \(\omega\), there be arranged the sequence of domains

\[ \omega,\ \omega^2,\ \omega^3,\ \omega^4,\ \ldots,\ \omega^n,\ \ldots, \]

where the numbering of the intervals in \(\omega\) is left natural; the type of arrangement obtained in this way will be

\[ \omega \sum_{1}^{\omega} \omega^i . \]

Let

\[ Q_i=\{x_i^0,\ x_1''\},\qquad Q_0=\sum_{1}^{\omega}\{x_i\}. \]

On each of the intervals \(l_i\) of the domain of the boundaries of intervals of type \(\omega^i\) there will be countably many, and we may renumber them in a known order. Suppose this numbering has been carried out; let us call the points of the \(n\)-th division on the interval \(l_i\) \(x_{ij}^{(n)}\), and denote

\[ \begin{gathered} \{x'_{1j}\}=Q_1^{(1)},\quad \{x'_{2j}\}=Q_2^{(1)},\quad \{x'_{3j}\}=Q_3^{(1)},\ldots,\quad \{x'_{nj}\}=Q_n^{(1)},\ldots,\\ \{x''_{2j}\}=Q_2^{(2)},\quad \{x''_{3j}\}=Q_3^{(2)},\ldots,\quad \{x''_{nj}\}=Q_n^{(2)},\ldots,\\ \{x'''_{3j}\}=Q_3^{(3)},\ldots,\quad \{x'''_{nj}\}=Q_n^{(3)},\ldots,\\ \ldots\ldots\ldots\ldots\ldots\ldots,\\ \{x^{(n)}_{nj}\}=Q_n^{(n)},\ldots, \end{gathered} \]

where here \(j\) assumes all integral positive values, and the points of each of the systems of division on \(l_i\) are assumed to have been renumbered according to some law. Let us further set

\[ Q_1=Q_1^{(1)},\quad Q_2=\sum_{1}^{2}Q_2^{(i)},\ldots,\quad Q_n=\sum_{1}^{n}Q_n^{(i)},\ldots; \]

then the domain of all boundaries and external points will be, respectively,

\[ Q=\{x_1^0\}+\sum_{0}^{\omega}Q_i,\qquad E=\{x_1^0\},\qquad P=Q+E=\sum_{1}^{\omega}Q_i . \]

It is obvious that

\[ P'=\sum_{1}^{\omega}\{x_i\}+Q_2^{(1)}+\sum_{1}^{2}Q_3^{(i)}+\ldots+\sum_{1}^{n-1}Q_n^{(i)}+\ldots+\{x_1^0\}, \]

\[ P''=\sum_{2}^{\omega}\{x_i\}+Q_3^{(1)}+\sum_{1}^{2}Q_4^{(i)}+\ldots+\sum_{1}^{n-1}Q_{n+1}^{(i)}+\ldots+\{x_1^0\}, \]

\[ \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots, \]

\[ P^{(m)}=\sum_{m}^{\omega}\{x_i\}+Q_{m+1}^{(1)}+\sum_{1}^{2}Q_{m+2}^{(i)}+\ldots+\sum_{1}^{n-1}Q_{n+m-1}^{(i)}+\ldots+\{x_1^0\}. \]

We see here that in \(P'\) the first interval \(l_1\), except for its boundary \(x_1^0\), has completely disappeared; in \(P''\) also \(l_2\) has disappeared, and so on; in \(P^{(m)}\) the interval \(l_m\) disappears.

It is obvious that for every finite \(m\) the point \(x_1^0\) belongs to the region \(P^{(m)}\); consequently this point will also enter into \(P^{(\omega)}\); it is also clear that besides it there will be no other point in \(P^{(\omega)}\). Indeed: whatever point \(\xi\), distinct from \(x_1^0\), and not belonging to the fundamental interval \(L\), may be, it will be either its left boundary or an interior point; consequently some finite distance from it to \(x_1^0\) can be indicated; it will then enter into one of the intervals \(l_i\), for example into \(l_\sigma\), or will be its boundary, and in such a case the point \(\xi\), together with all other interior points of the interval \(l_\sigma\), will no longer occur in \(P^{(\sigma)}\) or in \(P^{(\sigma+1)}\) and—therefore—cannot be a point common to all \(P^{(i)}\).

Thus in the present case

\[ P^{(\omega)}=\{x_1^0\},\qquad P^{(\omega+1)}=0; \]

we obtain in this way a region of the second kind. Since each of the \(Q_i\) is countable, \(P\) will also be countable; moreover, \(P\) is closed and, generally speaking, nowhere dense. This unique nowhere-dense region was given¹ by Cantor as an example of a region of the second kind.

131. We pass next to the investigation of the type

\[ \tag{5} \omega^2\cdot \sum_{1}^{\omega}\omega_i; \]

that is, in intervals of type \(\omega^2\) we shall place the corresponding intervals \(\omega_i\), where \(\omega_i\) are different powers \(\omega^\lambda\).

In order to carry out such an arrangement, we must establish a correspondence between the intervals \(l_i\) of type \(\omega^2\) and the powers \(\omega^\lambda\); here there is already no natural arrangement, as in \(\omega\); therefore it is necessary to indicate in advance the order in which we wish to bring the intervals of type \(\omega^2\); otherwise the type of arrangement (5) will not be completely determined.

Here, on each interval \(l_i\), a denumerable sequence of intervals \(\{l_{ij}\}\) is placed, so that the total number of intervals may be given by the double sequence \(\{l_{ij}\}\), where \(i\) and \(j\) may independently take all positive integral values, beginning with one. Since every double sequence contains a denumerable sequence of elements, we may, for example, establish the following correspondence between \(l_{ij}\) and \(\omega^\lambda\): transform

\[ \sum_{1}^{\omega} \omega_i \]

into the double sequence of powers of \(\omega\)

\[ \sum_{i=1}^{\omega}\sum_{j=1}^{\omega}\omega^{i+j}, \]

and assign to the interval \(l_{ij}\) the arrangement type \(\omega^{i+j}\). Then the arrangement type

\[ \omega^2 \cdot \sum_{i=1}^{\omega}\sum_{j=1}^{\omega}\omega^{i+j} \]

is completely determined.

Under this condition, on each of the intervals \(l_i\) we have the arrangement

\[ \omega \cdot \sum_{j=1}^{\omega}\omega^{i+j}, \tag{6} \]

since the intervals \(l_{ij}\) give on \(l_i\) an arrangement \(\omega\).

Calling the set of boundaries of the intervals of the final arrangement \(Q\), and the set of those of these boundaries which lie inside the interval \(l_i\) by \(Q_i\), we obtain the set \(P\) of non-interior points of all intervals in the form

\[ P \equiv \{x_1^0, x_1^0\} + \{x_i\}_{1}^{\omega} + \sum_{1}^{\omega} Q_i . \]

On each of the intervals \(l_i\), according to (6), we have an arrangement similar to arrangement \(130^\circ\), with the sole difference that already on the first interval \(l_{i1}\) the set \(\omega^{i+1}\) is placed instead of

\(\omega\) in 130°, in the second \(\omega^{i+2}\) instead of \(\omega^2\), etc.; in such a case, as is not hard to see, on the interval \(l_i\) the falling-away of the first interval \(l_{i_1}\) will occur not in the first derived set, but only in \(Q^{(i+1)}\); subsequently, however, the course of formation of the derived sets remains the same, so that we obtain \(Q_i^{(\omega)}=\{x_i\}\); in consequence of this

\[ P^{(\omega)}=\{x_i\}_{1}^{\omega}+\{x_1^{0}\}, \]

and—therefore—

\[ P^{(\omega+1)}=\{x_1^{0}\},\qquad P^{(\omega+2)}=0; \]

the domain of the second kind \(P\) will be countable, closed, and, generally speaking, everywhere dense.

132. In exactly the same way we can investigate the domain of non-interior points \(P\) of intervals of the type

\[ \omega^n\cdot \sum_{1}^{\omega}\omega_i, \]

with the \(\omega_i\) equal to powers of \(\omega\), if we establish a correspondence analogous to 131°, i.e. transform the simple series \(\sum_{1}^{\omega}\omega_i\) into an \(n\)-fold one; then it will turn out that

\[ P^{(\omega+n-1)}=\{x_1^{0}\},\qquad P^{(\omega+n)}=0. \]

Finally, concerning a domain of the type \(\omega\cdot\sum_{1}^{\omega}\omega_i\), we shall speak below.

133. We would obtain analogous results for the arrangements \({}^{*}\omega\); among other things, to the type

\[ {}^{*}\omega\cdot \sum_{1}^{\omega}\omega_i\cdot \sum_{1}^{\omega}\omega_{ij}\cdot \sum_{1}^{\omega}\omega_{ijk}\cdots \]

under the condition that \(\omega_1=\omega_{i1}=\omega_{ij1}=\cdots=1\), and all the remaining types equal \({}^{*}\omega\), there belongs a domain of Volterra\(^1\).

To the types

\[ {}^{*}\omega\cdot \sum_{1}^{\omega}{}^{*}\omega^i,\qquad {}^{*}\omega^{\,n-1}\cdot \sum_{1}^{\omega}{}^{*}\omega^i,\qquad \left\{{}^{*}\omega\cdot \sum_{1}^{\omega}{}^{*}\omega^i\right\}^2, \]

\(^1\) See 8°.

belong to the Mittag-Leffler domain\(^1\), for which respectively

\[ \mathrm{P}^{(\omega+1)}=0,\quad \mathrm{P}^{(*+n+1)}=0,\quad \mathrm{P}^{(2\omega+1)}=0, \]

moreover in the second type the correspondence of intervals \(\mathbf{Z}*_{\omega}^{\,i}\) is established so that \(i\) corresponds to the natural order of enumeration of the intervals of the \(n+1\)-st division, situated on the intervals of the \(n\)-th division.

134. Let us now investigate an arrangement of the type

\[ \omega,\ \overset{\omega}{\underset{1}{\mathbf{Z}}}\,(*\omega+\omega)^i, \]

i.e. on intervals of type \(\omega\) let us place the series of domains

\[ *\omega+\omega,\quad (*\omega+\omega)^2,\ \ldots,\ (*\omega+\omega)^n,\ \ldots, \]

regarding the enumeration of the intervals \(\omega\) as natural.

Let us first call

\[ \{x_1^0,\ x_1^b\}=Q_1,\quad \{x_i^0\}=Q_0, \]

then denote the points of the \(n\)-th division on the interval \(l_i\) by \(x_{ij}^{(n)}\), bringing them to the double counting series \(\{x_{ij}^{(n)}\}_{j=-\infty}^{+\infty}\); then

\[ \{x_{1j}'\}=Q_1^{(1)},\quad \{x_{2j}'\}=Q_2^{(1)},\ \ldots,\quad \{x_{nj}'\}=Q_n^{(1)},\ \ldots, \]

\[ \{x_{2j}''\}=Q_2^{(2)},\ \ldots,\quad \{x_{nj}''\}=Q_n^{(2)},\ \ldots, \]

\[ \ldots\ldots\ldots, \]

\[ \{x_{nj}^{(n)}\}=Q_n^{(n)},\ldots \]

the points

\[ Q_1^{(1)},\ Q_2^{(2)},\ Q_3^{(3)},\ \ldots,\ Q_n^{(n)},\ \ldots; \]

will enter among the boundaries of the intervals of the final arrangement.

\(^1\) See 18°.

Converting again, as in 131°, the simple sequence of powers \({}^{*}\omega+\omega\) into the double sequence \(\{({}^{*}\omega+\omega)^{i+j}\}\), let us assign \(({}^{*}\omega+\omega)^{i+j}\) to the interval \(l_{ij}\); then the type

\[ \omega^2 \cdot \sum_{i=1}^{\omega}\sum_{j=1}^{\omega}({}^{*}\omega+\omega)^{i+j} \]

is made completely definite and gives, on each of the intervals \(l_i\), the arrangement

\[ \omega \cdot \sum_{j=1}^{\infty}({}^{*}\omega+\omega)^{i+j}; \]

this arrangement on \(l_i\) will be analogous to the arrangement 134°, with the only difference that on \(l_{i1}, l_{i2}, l_{i3}, \ldots\) there will be placed intervals of the types

\[ ({}^{*}\omega+\omega)^{i+1},\quad ({}^{*}\omega+\omega)^{i+2},\quad ({}^{*}\omega+\omega)^{i+3},\ldots . \]

Let us again call \(P\) the domain of non-interior points of the intervals of the final arrangement, and \(P_i\) the domain of such points lying inside the interval \(l_i\); then

\[ P=\{x_1^0,x_1^\omega\}+\{x_i^0\}_{1}^{\omega}+\sum_{1}^{\omega}P_i; \]

it is evident that for the points on the interval \(l_i\), we shall, as in 134°, have \(P_i^{(\omega)}=\{x_i\}\), and therefore

\[ P^{(\omega)}=\{x_i\}_{1}^{\omega}+\{x_1^0\},\quad P^{(\omega+1)}=\{x_1^0\},\quad P^{(\omega+2)}=0. \]

Similarly, for every finite \(n\), for the non-interior points of the intervals of type

\[ \omega^n \cdot \sum_{1}^{\omega}\omega_i, \]

we shall have

\[ P^{(\omega+n-1)}=\{x_1^0\},\quad P^{(\omega+n)}=0, \]

provided only that we establish a correspondence between the intervals \(\omega^n\) and the powers \(({}^{*}\omega+\omega)^i\), first converting the simple sequence \(\sum_{1}^{\omega}\omega_i\) into an \(n\)-fold one.

136. Of the types of arrangement

\[ \omega\cdot \overset{\omega}{Z}_{1}\,{}^{*}\omega^{i},\quad \omega^{n}\cdot \overset{\omega}{Z}_{1}\,{}^{*}\omega^{i},\quad \omega\cdot \overset{\omega}{Z}_{1}\,\tilde{\omega}^{i},\quad \omega^{n}\cdot \overset{\omega}{Z}_{1}\,\tilde{\omega}^{i}, \]

the first present nothing special in comparison with 130°—132°; the second, on the basis of 87°, are nothing other than \(\omega\cdot\tilde{\omega}\) or \(\omega^{n}\cdot\tilde{\omega}\).

In exactly the same way we shall find nothing new in the types

\[ {}^{*}\omega\cdot \overset{\omega}{Z}_{1}\omega^{i},\quad {}^{*}\omega^{2}\,\overset{\omega}{Z}_{i=-1}\,\overset{\omega}{Z}_{j=-1}\omega^{i+j} \]

and in analogous combinations of \({}^{*}\omega\) with \({}^{*}\omega+\omega\) and \(\tilde{\omega}\).

137. Let us take also a type of arrangement

\[ \tag{7} \cdot\left({}^{*}\omega+\omega\right)\cdot \overset{\omega}{Z}_{1}\omega_{i}, \]

where \(\omega_i\) are various powers of \(\omega\); in view of the circumstance that for intervals of type \({}^{*}\omega+\omega\) there is no definite natural order of numbering, in order fully to determine (7) we must establish, first, this order and, secondly, the correspondence of the intervals and the arrangements \(\omega_i\).

Let us take the interval of greatest length in the domain \({}^{*}\omega+\omega\) as the middle interval \(l_0\), and to those following from it to the left in order assign the marks \(\bar{1}, \bar{2}, \bar{3}, \ldots\), and to the right \(1, 2, 3, \ldots\); by this the order of succession of the intervals will be established.

Assign to the interval \(l_0\) the type \(\omega^0=1\), and to the intervals \(l_{\pm i}\) the type \(\omega^i\); then

\[ \overset{\omega}{Z}_{1}\omega_i \]

will turn into the double series

\[ \overset{\pm\omega}{Z}_{0}\omega^{|i|}, \]

so that (7) will take the form

\[ \left({}^{*}\omega+\omega\right)\cdot \overset{\pm\omega}{Z}_{0}\omega^{|i|}. \]

Let us call

\[ Q^{(0)}=\{x_0',\,x_0''\},\qquad Q^{(1)}=\underset{\pm 1}{\overset{\pm\omega}{\{x_i\}}}, \]

then

\[ \begin{aligned} \{x_{1j}'\}&=Q_1^{(1)},& \{x_{2j}'\}&=Q_2^{(1)},& \{x_{3j}'\}&=Q_3^{(1)},&\ldots,& \{x_{nj}'\}&=Q_n^{(1)},&\ldots;\\ && \{x_{2j}''\}&=Q_2^{(2)},& \{x_{3j}''\}&=Q_3^{(2)},&\ldots,& \{x_{nj}''\}&=Q_n^{(2)},&\ldots;\\ \{x_{1j}^{-}\}&=Q_1^{(1)},&&& \{x_{3j}'''\}&=Q_3^{(3)},&\ldots,& \{x_{nj}'''\}&=Q_n^{(3)},&\ldots;\\ \{x_{2j}^{-}\}&=Q_2^{(1)},& \{x_{2j}^{--}\}&=Q_2^{(2)},&&&&& \ldots\ldots\ldots;&\\ \{x_{3j}^{-}\}&=Q_3^{(1)},& \{x_{3j}^{--}\}&=Q_3^{(2)},& \{x_{3j}^{---}\}&=Q_3^{(3)},&&& \{x_{nj}^{(n)}\}&=Q_n^{(n)},&\ldots,\\ \ldots\ldots\ldots\ldots\ldots\ldots\ldots&&&&&&&&&&\\ \{x_{nj}'\}&=Q_n^{(1)},& \{x_{nj}''\}&=Q_n^{(2)},& \{x_{nj}'''\}&=Q_n^{(3)},&\ldots,& \{x_{nj}^{(n)}\}&=Q_n^{(n)},&\ldots.\\ \ldots\ldots\ldots\ldots\ldots\ldots\ldots&&&&&&&&&& \end{aligned} \]

finally

\[ Q_{\pm1}^{(i)}=Q_{\pm1},\qquad \sum_{1}^{2} Q_{\pm2}^{(i)}=Q_{\pm2},\qquad \sum_{1}^{3} Q_{\pm3}^{(i)}=Q_{\pm3},\ldots,\qquad \sum_{1}^{n} Q_{\pm n}^{(i)}=Q_{\pm n},\ldots. \]

Here, for the intervals of the final placement, the only external points will be \(\{x_0',\,x_0''\}\), while all the rest will be boundaries of intervals; therefore

\[ Q=\underset{\pm1}{\overset{\pm\omega}{\{x_i\}}} +\underset{\pm1}{\overset{\pm\omega}{\sum}} Q_i, \qquad E=\{x_0',\,x_0''\}, \]

and consequently the domain of non-internal points of all the intervals will be formed thus:

\[ P=Q+E=\{x_0',\,x_0''\} +\underset{\pm1}{\overset{\pm\omega}{\{x_i\}}} +\underset{\pm1}{\overset{\pm\omega}{\sum}} Q_i. \]

Obviously, for \(n>m\) the \(m\)-th derivative \(Q_{\pm n}^{m}\) of the domain \(Q_{\pm n}\) will be as follows:

\[ Q_{\pm n}^{m}=\sum_{i=1}^{n-m} Q_{\pm n}^{(i)}+\{x_\lambda\};\quad \lambda=n+1 \text{ for } n>0,\quad \lambda=n \text{ for } n<0; \]

\[ Q_n^n=x_{n+1},\quad Q_n^{n}=x_n,\quad Q_{\pm n}^{\,n+1}=0; \]

here the signs \(m\) at the derivatives \(Q_{\pm n}\) we write without parentheses, in order to distinguish them from the domains \(Q_{\pm n}^{(m)}\); hence

\[ P^{(m)}=\{x_0',x_0^0\}+\{x_m^\omega\}+\{x_{m+1}^{\omega}\} +\sum_{n=m+1}^{\omega}\sum_{i=1}^{n-m} Q_{\pm n}^{(i)}, \]

\[ P^{(\omega)}=\{x_0',x_0^0\},\quad P^{(\omega+1)}=0; \]

the domain \(P\) will be of the second kind, countable, closed, and, generally speaking, sparsely distributed.

138. Some peculiarity is further presented by an arrangement of the type

\[ (*\omega+\omega)^2 \sum_{1}^{\omega} \omega_i; \]

again in view of the circumstance that here it is necessary to establish a correspondence between the intervals \((*\omega+\omega)^2\) and the types \(\omega_i\) answering to them.

Let us choose on \(*\omega+\omega\), as also in 137, the middle—the greatest—interval \(l_0\); we shall do the same also for the intervals situated on each of the \(l_i\), so that \(l_{i0}\) will be the greatest of all the intervals on \(l_i\). Then the whole aggregate of intervals can be represented in the form \(\{l_{ij}\}_{i=0,j=0}^{\pm\omega}\).

Having represented the intervals \((*\omega+\omega)^2\) in the form of such a double series, we shall assign to each \(l_{ij}\) the degree \(\omega^{|i|+|j|}\). We shall then obtain quite a definite type

\[ (*\omega+\omega)^2 \cdot \sum_{i=0}^{\pm\omega}\sum_{j=0}^{\pm\omega} \omega^{|i|+|j|} \]

and, reasoning analogously to the preceding case, we arrive at the conclusion that for it

\[ P^{(\omega+1)}=\{x_0',x_0^0\},\quad P^{(\omega+2)}=0. \]

In exactly the same way one could obtain similar results by subjecting to investigation

\[ ({}^*\omega+\omega)^n\cdot \sum_{1}^{\omega}\omega_i \]

with the former value of \(\omega_i\), and then

\[ ({}^*\omega+\omega)\cdot \sum_{1}^{\omega}\omega_i \quad\text{and}\quad ({}^*\omega+\omega)^n\sum_{1}^{\omega}\omega_i, \]

where \(\omega_i\) already represent powers of \({}^*\omega\) or of \({}^*\omega+\omega\).

139. Passing next to the type

\[ \tilde{\omega}\cdot \sum_{1}^{\omega}\omega^i, \tag{8} \]

we must again first of all establish a law of correspondence between the intervals of \(\tilde{\omega}\) and the types \(\omega^i\).

We shall establish this correspondence as follows: let \(l_1\) be the largest of all free intervals of \(\tilde{\omega}\); \(l_2\) and \(l_3\) the largest of the intervals, respectively, on the portions \((x_0^0,x_{-1})\) and \((x_1,x_0^0)\); \(l_4,l_5,l_6,l_7\) the largest of the intervals on the portions \((x_0^0,x_{-2})\), \((x_2,x_1)\), \((x_1,x_3)\), \((x_3,x_0^0)\); and so on; then the intervals will be arranged in groups consisting successively of \(2^0, 2, 2^2, 2^3,\ldots,2^n,\ldots\) intervals, so that the numbers of the intervals in the \((n+1)\)-st group will be

\[ \sum_{0}^{n-1}2^i+1,\quad \sum_{0}^{n-1}2^i+2,\quad \ldots,\quad \sum_{0}^{n-1}2^i+2^n=\sum_{0}^{n}2^i, \]

and these numbers are assigned to the intervals in order, beginning from the left. Having chosen in this way a numbering of the intervals, we assign to each \(l_i\) the type of distribution \(\omega^i\); then (8) becomes completely determined.

Let us call \(\{x_0^0,x_0^0\}=Q^{(0)}\), \(\{x_i^{\pm\omega}\}=Q^{(1)}\); further let us call the domain of exterior points \(\tilde{\omega}\),

which lie inside \(L\), through \(\overline{E}^{(1)}\), and through \(E^{(1)}\) the domain of all external points; then

\[ E^{(1)}=\overline{E}^{(1)}+Q^{(0)},\qquad P^{(1)}=Q^{(1)}+E^{(1)}, \]

where \(P^{(1)}\) is a perfect domain \(\mathfrak{C}\). Denote, as before,

\[ \tag{9} \begin{aligned} \{x'_{1j}\}&=Q^{(1)}_1,\quad \{x'_{2j}\}=Q^{(1)}_2,\quad \{x'_{3j}\}=Q^{(1)}_3,\ldots,\quad \{x'_{nj}\}=Q^{(1)}_n,\ldots,\\ \{x''_{2j}\}&=Q^{(2)}_2,\quad \{x''_{3j}\}=Q^{(2)}_3,\ldots,\quad \{x''_{nj}\}=Q^{(2)}_n,\ldots,\\ &\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots,\\ \{x^{(n)}_{nj}\}&=Q^{(n)}_n,\ldots; \end{aligned} \]

let us further denote

\[ Q^{(1)}_1=Q_1,\quad \sum_1^2 Q^{(i)}_2=Q_2,\ldots,\quad \sum_1^n Q^{(i)}_n=Q_n,\ldots; \]

in this case \(Q\), the domain of the boundaries of the intervals of the final arrangement, is thus composed:

\[ Q=\sum_1^\omega \{x_i\}+\sum_1^\omega Q_n, \]

and the domain of external points will be

\[ E=E^{(1)}+\sum_1^\omega \{x_i\}. \]

Thus the domain \(P\) of non-internal points of the intervals of the final arrangement will prove to be

\[ P=Q+E=\{x_i\}_{\pm1}^{\pm\omega}+E^{(1)}+\sum_1^\omega Q_n = P^{(1)}+\sum_1^\omega Q_n, \]

where \(P^{(1)}\) is perfect, and \(\displaystyle \sum_{n=1}^{\omega} Q_n\) is a countable reducible domain of the Cantor–Bendixson theorem.

On each of the intervals \(l_i\), we have an arrangement \(\omega^i\), whose properties were investigated by us above; for them

\[ Q_m^m=\{x_m\},\qquad Q_m^{m+1}=0, \]

where the derivatives of \(Q_m\) are again denoted by \(m\) without parentheses. Hence

\[ P^m = P^{(1)} + \sum_{n=m+1}^{\infty}\bigl[Q_n^m - \{x_n\}\bigr],\quad P^\omega = P^{(1)}, \]

since, in this case,

\[ Q_n^m = \{x_n\} + \sum_{1}^{n-m} Q_n^{(i)}, \]

therefore

\[ P^m = P^{(1)} + \sum_{n=m+1}^{\infty}\sum_{i=1}^{n-m} Q_n^{(i)}, \]

and after each passage from \(P^m\) to \(P^{m+1}\) the points of the domains situated along the diagonals in table (9) fall away: the points of the domains \(Q_1^{(1)}, Q_2^{(2)}, Q_3^{(3)}, \ldots, Q_n^{(n)}, \ldots\) will not enter into \(P'\); into \(P''\) the next diagonal row, and so on.

The domain \(P\), obtained by the method indicated above, will be uncountable, since its composition includes \(P^{(1)}\), closed and nowhere dense.

140. Domains of type

\[ \delta^n \cdot \sum_{1}^{\infty}\omega_i \tag{10} \]

are not subject to investigation, since, by virtue of 87°, \(\delta^n=\delta\), and consequently type (10) reduces to the one examined above.

Of the domains of type

\[ \delta \cdot \sum_{1}^{\infty} *\omega^i,\quad \delta^n \cdot \sum_{1}^{\infty} *\omega_i, \]

the first presents no peculiarities in comparison with the domain of 139°, and the second reduces to the first.

141. We proceed further to domains determined by the type

\[ \delta \cdot \sum_{1}^{\infty}(*\omega+\omega)^i, \]

where we shall leave the law of correspondence between the intervals \(\delta\) and the powers of \(*\omega+\omega\) the same as in 139°.

Keeping the previous notation

\[ \{x_i^{(0)}, x_i^{\prime\prime(0)}\}=Q^{(0)},\quad \{x_i\}_{-1}^{+\omega}=Q^{(1)},\quad E^{(1)},\quad P^{(1)}=Q^{(1)}+E^{(1)}, \]

we shall call, as \(j\) varies from \(\pm 1\) to \(\pm \omega\),

\[ \{x_j'\}=Q_1^{(1)},\quad \{x_{2j}'\}=Q_2^{(1)},\ldots,\quad \{x_{nj}'\}=Q_n^{(1)},\ldots, \]

\[ \{x_{2j}^{\prime\prime}\}=Q_2^{(2)},\ldots,\quad \{x_{nj}^{\prime\prime}\}=Q_n^{(2)},\ldots, \]

\[ \ldots\ldots\ldots\ldots\ldots, \]

\[ \{x_{nj}^{(n)}\}=Q_n^{(n)},\ldots, \]

\[ \ldots, \]

and, moreover,

\[ Q_1^{(1)}=Q_1,\quad \sum_{1}^{2} Q_2^{(i)}=Q_2,\ldots,\quad \sum_{1}^{n} Q_n^{(i)}=Q_n,\ldots \]

In the present case \(Q\), the domain of the boundaries of the intervals of the final arrangement, will be

\[ Q=\sum_{1}^{\omega} Q_i^{(i)}, \]

whereas the domain of the external points will have the following composition,

\[ E=E^{(1)}+Q^{(1)}+\sum_{n=1}^{\omega}\left[Q_n-Q_n^{(n)}\right] =P^{(1)}+\sum_{n=1}^{\omega}\sum_{1}^{n-1}Q_n^{(i)}. \]

The domain \(P\) of non-internal points of the intervals of the final arrangement is obtained, in this case, in the form

\[ \tag{11} P=P^{(1)}+\sum_{n=1}^{\omega}\sum_{i=1}^{n}Q_n^{(i)} =P^{(1)}+\sum_{1}^{\omega}Q_n. \]

The arrangement \((*_{\omega}+\omega)^i\) situated on each of the intervals \(l_i\) gives

\[ Q_n^m=\sum_{1}^{n-m}Q_n^{(i)}+\{x_{-n},x_n\},\quad Q_n^n=\{x_{-n},x_n\},\quad Q_n^{n+1}=0; \]

as a consequence of this,

\[ \mathrm{P}^{m}=\mathrm{P}^{(1)}+\sum_{n=m+1}^{\omega}\{Q_{n}^{m}-Q_{n}^{n}\} \]

or

\[ \mathrm{P}^{m}=\mathrm{P}^{(1)}+\sum_{n=m+1}^{\omega}\sum_{i=1}^{n-m}Q_{n}^{(i)},\qquad \mathrm{P}^{(\omega)}=\mathrm{P}^{(1)}. \]

The set \(\mathrm{P}\) will here be uncountable, closed, and sparsely scattered; in expression (11)

\[ \sum_{1}^{\omega} Q_{n}=R \]

is a reducible Cantor–Bendixson set, for which \(D(R,R^{(\omega)})=0\).

142. Proceeding further to the investigation of the types

\[ \omega^{\omega}\cdot \sum_{1}^{\omega}\omega_i;\qquad ({}^{*}\omega+\omega)^{\omega}\sum_{1}^{\omega}\omega_i;\qquad \tilde{\omega}^{\omega}\cdot \sum_{1}^{\omega}\omega_i, \]

where the \(\omega_i\) are various powers of \(\omega\), \({}^{*}\omega\), or \({}^{*}\omega+\omega\), we must first of all suppose that the set of intervals

\[ \omega^{\omega},\quad ({}^{*}\omega+\omega)^{\omega},\quad \tilde{\omega}^{\omega} \tag{12} \]

will be sparsely scattered; otherwise, when intervals free of points of the set are absent, there can be no question of placing in them various sets of the sum \(\sum_{1}^{\omega}\omega_i\).

Then the arrangements (12) have much in common with one another and differ from one another only in secondary features; therefore it is possible to investigate directly the type

\[ \Omega\cdot \sum_{1}^{\omega}\omega_i. \tag{13} \]

where \(\Omega\) is any of the composite types (12).

The free intervals of the type \(\Omega\) are, by virtue of 112°, the free intervals of a certain perfect set, while in the remaining part of the segment the non-inner points of the free intervals are often scattered over this perfect set; therefore the placement of the intervals \(\omega_i\) in

the free intervals \(\Omega\) reduce essentially to a distribution of the type
\(\widetilde{\omega}\cdot \sum_{1}^{\omega}\omega_i\), if only we have in mind exclusively the intervals themselves; here the establishment of the correspondence between the intervals \(l_i\) of the domain \(\widetilde{\omega}\) and the types \(\omega_i\) proceeds in the same way as in 139°. If, however, we take into account the \(Q\)-domain of the boundaries of all systems of division, then an essential difference appears between \(\widetilde{\omega}\sum \omega_i\) and \(\Omega\sum \omega_i\).

The domain \(Q\) for (13) will consist: 1) of the boundaries \(Q_i\) of the various divisions \(\omega_i\), lying inside \(l_i\); 2) of some boundaries of the free intervals of the distribution \(\Omega\), and finally 3) of a series of external points, often scattered over the perfect domain corresponding to the distribution \(\Omega\); the points of the last two categories form the domain \(Q_\omega\); for the domain

\[ Q \equiv \sum_{1}^{\omega} Q_i + Q_\omega = Q_0 + Q_\omega \]

we have

\[ Q' \equiv M'(Q_0', Q_\omega') \equiv S + R \]

where \(S \equiv Q_\omega'\) is perfect and \(R\) is a reducible domain consisting of the limiting points of \(Q_0\) lying inside the intervals of the perfect domain.

143. We could further continue the combination of domains given above, taking as elements domains of transfinite and mixed type, whereby ever more and more complicated types of distribution would be obtained.

But we do not think that such an investigation is worth making; the preceding is quite sufficient to see how the combination of only three basic types \(\omega\), \({}^{*}\omega\), and \(\widetilde{\omega}\) can yield a distribution of arbitrary complexity.

144. We shall add here a few more words concerning the classification of the points of a domain, which we already touched upon in 22°. In constructing, by means of various combinations of the basic types, point domains that differ greatly from one another in character, we are dealing with a great variety of kinds of limiting points, appearing according to the frequency of the process of the basic distributions; therefore it is desirable to classify the points of a domain in connection with its type.

Cantor’s classification in §22 raises no objections so long as the matter concerns closed domains; but as soon as we pass to non-closed domains, as we shall now see, the situation changes, and it becomes necessary to introduce a small modification into Cantor’s classification.

For a closed domain \(P\), expression (1) of §22 takes the form

\[ P=\sum [P^{(\alpha)}-P^{(\alpha+1)}]+P^{(\Omega)}, \]

and in it there is already contained the distribution of the points into various classes. The simplest of them are the isolated points forming the domain \(P-P'\); these are points or limit points of zero order. Further, the isolated points of the first derivative, \(P'-P''\), will be points of first order; the points of the \(n\)-th, \(\omega\)-th, \(\alpha\)-th derivative—points of the \(n\)-th, \(\omega\)-th, \(\alpha\)-th order. Finally, the points of the domain \(P^{(\Omega)}\) we shall call points of condensation; in an arbitrary neighborhood of such points the points of the domain will be uncountable.

As the various derivatives are found, the points of order \(0\), \(1\), …, \(n\), and so on, successively drop out, until we reach the isolated points of the last derivative \(P^{(\alpha)}\), after which there follows an already closed and dense-in-itself, i.e. perfect, derivative \(P^{(\alpha+1)}=P^{(\Omega)}\).

Depending on the composition of the domain \(P\), these or other points appear in it. Let \(P\) be countable; then for it \(P^{(\Omega)}\) must reduce to \(0\), and the points of highest order will be the points \(P^{(\alpha)}\); in the particular case when \(\alpha=\omega\) or \(\alpha=n\), we shall have, as the last in complexity among the limit points, points of \(\omega\)-th or \(n\)-th order. If \(P\) is uncountable, such points will be points of condensation. It is obvious that the higher the order of a limit point, the more densely the points of \(P\) crowd in its neighborhood, although the number of points of \(P\) in this neighborhood is always countable.

It is clear that the absence of points of some order entails the absence of points of all higher orders, while points of condensation may nevertheless be present. But, on the other hand, it is possible for points of all orders to be absent; this will occur for a perfect domain, where all points of the domain are points of condensation.

The preceding classification of points contains nothing arbitrary and nothing accidental; it is determined by the very nature of things, by the structure of the domain itself.

If, instead of a closed domain, we are dealing with a non-closed domain \(P\), we complete it to \(M(P, P')\) and assign to the points \(P'\) the orders which they have in \(M(P, P')\): this alone constitutes the change that we introduce into Cantor’s classification.

145. In order to see by an example what difference this change makes, let us consider the domain of intervals \(( *\omega + \omega)^3\), denote by \(Q^{(0)}, Q^{(1)}, Q^{(2)}, Q^{(3)}\) the boundaries of the corresponding divisions, and take the domain \(R \equiv Q^{(0)} + Q^{(1)} + Q^{(3)}\); for it the corresponding closed domain will be

\[ P \equiv \sum_{0}^{3} Q^{(i)} \]

and—according to the definition—

\[ P - P' \equiv Q^{(3)}, \quad P' - P'' \equiv Q^{(2)}, \quad P'' - P''' \equiv Q^{(1)}, \quad P''' \equiv Q^{(0)}, \quad P'''' = 0, \]

where the points of the third division will be points of order \(0\), those of the second division—of order \(1\), those of the first division—of order \(2\), and the boundaries of the segment—of order \(3\).

For the domain \(R\), from which the points \(Q^{(2)}\) have been thrown out, the entire internal structure remains the same as for \(P\); for it—according to Cantor—the points \(R_a \equiv Q^{(3)}\) are likewise points of genus \(0\), while as for \(P_{ca}\), these will be the isolated points of the domain \(R - R_a \equiv Q^{(0)} + Q^{(1)}\), i.e. the points \(Q^{(1)}\); thus

\[ P_{ca} \equiv Q^{(1)}, \quad P_{ca}^{2} \equiv Q^{(0)}, \]

where the points of the first division here become points of the first genus, and the boundaries of the interval—points of the second genus.

If we make a very slight change in the composition of the domain, namely, instead of \(R\), take the domain

\[ E = Q^{(0)} + Q^{(1)} + \{x''\} + Q^{(3)}, \]

where \(x''\) is one of the points of the second division, then this will already be sufficient to change the genus of the limiting points.

Indeed: in the present case

\[ P_a = Q^{(3)}, \quad P_{ca} = \{x''\}, \quad P_{ca}^{2} = Q^{(1)}, \quad P_{ca}^{3} = Q^{(0)}, \]

i.e. the points of the first division will here already be points of the second genus, and the boundaries of the interval—of the third genus.

Thus, while preserving the known structure of a domain, the absence of only a single point in its composition is enough to violate the classification of points established by Cantor; therefore this classification cannot be recognized as natural and must be altered in a corresponding way; the modification indicated above does not suffer from this defect and may therefore be recognized as a natural classification, arising from the essence of the matter.

6. The structure of an arbitrary closed domain.

146. We shall now pass to the inverse problem: to ascertain to what extent, for an arbitrary closed domain, a definite type of distribution can be indicated.

In order to answer this question, we shall have to consider separately countable and uncountable domains; as for perfect domains, there is nothing to say about them, since their type has been determined; we shall begin the investigation with a sparsely distributed closed countable domain, for which, according to D 66°, \(P = P_i + P'\).

Let \(L\) be the fundamental interval of the domain \(P\); since \(P\) is closed, its boundaries must be points of \(P\). Since \(P'\) is countable, then—according to 13°—there exists a number \(\lambda\) of the first or second class and of the first kind, for which the corresponding derivative \(P^{(\lambda)}\) is equal to zero; since \(\lambda\) is a number of the first kind, there exists for it the immediately preceding number \(\lambda - 1\), which we shall call \(\alpha\); then \(\lambda = \alpha + 1\), and \(P^{(\alpha+1)} = 0\), whereas \(P^{(\alpha)}\) consists of a finite number of points.

We shall conduct the further investigation assuming successively \(\alpha = n, \omega, \omega + n, \omega \cdot 2, \ldots\), and moreover we shall first take the case when the highest derivative distinct from zero contains only one or two points, namely—either both boundaries of \(L\), or only one of them.

147. Thus, let us consider the simplest case, when \(\alpha = n\) is finite; since—by assumption—\(P^{(n)}\) contains either both points \(Q^{(0)}\) or only one of them, then \(P^{(n)} = Q^{(0)}\) or \(P^{(n)} = D(Q^{(0)})\); in the second case, if \(\xi\)—one of the points of \(Q^{(0)}\)—is not a point of \(P^{(n)}\), it must be an isolated point either of \(P\), or of one of the derivatives

\(P^{(n-\nu)}\); this circumstance must necessarily occur, since the domain \(P\) is closed, and—consequently—the endpoints of the fundamental interval must belong to it; otherwise the interval could be diminished; in the general case let \(\xi\) be an isolated point of \(P^{(n-\nu)}\), where \(\nu\) may be equal to \(1, 2, 3, \ldots, n\), if \(P^{(0)}\) is understood as \(P\).

The points \(P^{(n)}\) serve as limit points for the isolated points of the preceding derivative \(P^{(n-1)}\); let us call the domain of such points \(Q^{(1)}\); these points serve as endpoints of the intervals \(\{l'_i\}\) situated on \(L\); the type of this arrangement will be one of the types \({}^{*}\omega+\omega\), \(\omega\), \({}^{*}\omega\), according as both endpoints of \(L\) are points of \(P^{(n)}\), or only the right one, or only the left one; let us call this type of arrangement \(\omega_1\).

In the composition of the derivative \(P^{(n-1)}\), besides its isolated points \(Q^{(1)}\), there will also enter the points \(P^{(n)}\), so that

\[ P^{(n-1)} = Q^{(1)} + P^{(n)}. \]

Each of the boundaries \(Q_1 = Q^{(0)} + Q^{(1)}\), except for the single point \(\xi\), will, as a point of \(P^{(n-1)}\), be a limit point for the isolated points of \(P^{(n-2)}\); let us call the domain of these latter points \(Q^{(2)}\); the points \(Q^{(2)}\) will be situated on each of the intervals \(l'_i\) of the first division and will serve as endpoints of the intervals \(\{l'_{ij}\} = \{l''_k\}\). The arrangement of the \(l'_{ij}\) on \(l'_i\) may be one of \(\omega\), \({}^{*}\omega\), \({}^{*}\omega+\omega\), or a finite type \(m\), but on different \(l'_i\) the arrangements may be different; therefore the type of arrangement of all the intervals \(\{l'_{ij}\}\) must be denoted as \(\omega_1 \cdot \prod_{1}^{\omega} \omega_2^{(i)} = \omega_2\), where each \(\omega_2^{(i)}\) is one of the types \(\omega\), \({}^{*}\omega\), \({}^{*}\omega+\omega\), \(m\). The composition of the derivative \(P^{(n-2)}\) will now be determined as

\[ P^{(n-2)} = Q^{(2)} + P^{(n-1)} = \sum_{1}^{2} Q^{(i)} + P^{(n)}. \]

The endpoints of the intervals after the second arrangement may be, besides \(\xi\) and the points \(Q^{(2)}\), also some points \(Q_1\), namely those which in this arrangement will be one-sided limit points; let us call \(Q_2\) the domain of such endpoints. Analogously to the preceding, we may reasonitsiaq

and further: the points \(Q_2\), except for one \(\xi\), being points of \(\mathrm{P}^{(n-2)}\), turn out to be limit points for the isolated points of \(\mathrm{P}^{(n-3)}\), which make up the region \(Q^{(3)}\); since in each of the intervals \(l_i''\) the arrangement of the intervals \(l_{kl}''\) may be not the same, the type of arrangement of the intervals \(l_{kl}'' \ldots \{l_m'''\}\) we must denote by \(\omega_2 \cdot \sum_1^\omega \omega_3^{(k)}=\omega_3\); in this case

\[ \mathrm{P}^{(n-3)}=\mathrm{P}^{(n-2)}+Q^{(3)}=\sum_1^3 Q^{(i)}+\mathrm{P}^{(n)}. \]

The points \(Q^{(3)}\), together with some points \(Q_2\), will be the boundaries of the intervals of the third division \(\{l_m'''\}\), in each of which a special series of intervals can be placed, so that for \(\mathrm{P}^{(n-4)}\) we obtain the type \(\omega_3 \cdot \sum_1^\omega \omega_4^{(m)}=\omega_4\); and so on.

Since, by hypothesis, \(\xi\) is an isolated point of the derived set \(\mathrm{P}^{(n-\nu)}\), all isolated points of the latter will be expressed through \(Q^{(\nu)}+\{\xi\}\), and \(\mathrm{P}^{(n-\nu)}\) will then take the form

\[ \mathrm{P}^{(n-\nu)}=\mathrm{P}^{(n-\nu+1)}+Q^{(\nu)}+\{\xi\} =\sum_1^\nu Q^{(i)}+\mathrm{P}^{(n)}+\{\xi\} =\sum_0^\nu Q^{(i)}, \]

and the type of the region \(\mathrm{P}^{(n-\nu)}\) will now be \(\omega_{\nu-1}\cdot \sum_1^\omega \omega_{\nu}^{(i)}=\omega_\nu\).

After \(\mathrm{P}^{(n-\nu)}\), all preceding derived sets \(\mathrm{P}^{(n-\nu)}\) will coincide with

\[ \sum_0^\nu Q^i, \]

so that it will turn out that

\[ \mathrm{P}'=Q^{(n-1)}+\mathrm{P}''=\sum_0^{n-1}Q^i,\qquad \mathrm{P}=\mathrm{P}'+Q^{(n)}=\sum_0^n Q^{(i)}, \]

whereby the final type of arrangement of the intervals of the region \(\mathrm{P}\) will be \(\omega_{n-1}\cdot \sum_1^\omega \omega_n^{(i)}=\omega_n\); thus, for the given region \(\mathrm{P}\), the type of arrangement \(\omega_n\) can be indicated quite definitely.

148. We proved the preceding theorem under the assumption that \(P^{(n)}\) consists only of one or two endpoints of the fundamental interval \(L\); let, in the general case,

\[ P^{(n)}=\{x_1^0,x_2^0,x_3^0,\ldots,x_m^0\}, \]

where \(m\) must be finite, since otherwise there would exist a subsequent derivative \(P^{(n+1)}\) different from zero.

By the points \(P^{(n)}\) divide the interval \(L\) into a finite number of parts \(L_i\), not exceeding \(m+1\); we shall reduce the problem to the preceding one: on each of the intervals \(L_i\) satisfying the conditions of 147, we construct a domain of intervals of type \(\omega_{n_i}^{(i)}\), and the type of distribution for the whole \(L\) will be given by the expression \(\sum \omega_{n_i}^{(i)}\). In this way the problem will be solved also in the general case; thus:

“For every countable closed domain \(P\) of the first kind there can be indicated a definite type of distribution.”

149. Let now \(\alpha=\omega\), i.e. for the domain \(P\) its derivative \(P^{(\omega)}\) consists of a finite number of points.

A. Suppose first that these points are one or two, namely the endpoints of \(L\); since \(P\) is closed, both endpoints of the fundamental interval \(L\) belong to \(P\).

Choose from \(P\) a sequence of points \(\{x_i\}\) for which the points \(P^{(n)}\) will be limiting points; these points determine on \(L\) intervals \(\{l_i'\}\) of type \(\omega_0\), belonging to one of the distributions \(\omega\), \({}^{*}\omega\), \({}^{*}\omega+\omega\). The points of \(P\) lying inside each of the intervals \(l_i'\) form a domain \(P_i\) of the first kind, since the only points of \(P^{(\omega)}\) are the endpoints of the fundamental interval; therefore, for each of the intervals \(l_i'\), there can—according to 147 and 148—be indicated a definite type of distribution \(\omega_{n_i}^{(i)}\), so that the total distribution will have the type

\[ \omega_0\cdot\sum \omega_{n_i}^{(i)}=\omega', \]

which is nothing other than one of the types

\[ ({}^{*}\omega+\omega)\sum_{0}^{\pm\omega}\omega_{n_i}^{(i)},\qquad \omega\sum_{1}^{\omega}\omega_{n_i}^{(i)},\qquad {}^{*}\omega\sum_{1}^{\omega}\omega_{n_i}^{(i)}. \]

It is obvious that from some definite value of \(n\), for example from \(n_\nu\), the index \(n_i\), i.e. the kind of the domain \(P_{n_i}\), must increase without bound; if this were not so, the domain \(P\) would be of the first kind, and \(P^{(\omega)}\) would have to be absent.

B. If \(P^{(\omega)}\) consisted not of one or two, but of \(m\) points, then these points would divide the interval \(L\) into at most \(m+1\) parts \(L_i\), whose boundaries would be the points \(P^{(\omega)}\). The part \(P_i\) of the domain \(P\) situated inside \(L\) would satisfy the conditions of A 149°, and to it there would correspond the type of arrangement \(\omega_i'\); in that case the entire arrangement on \(L\) would have the type \(\sum \omega_i' \equiv \omega''\).

Thus, for \(\alpha=\omega\), for the given domain \(P\) a definite type of arrangement can also be indicated.

150. Let, further, \(\alpha=\omega+n\), and let the domain \(P^{(\omega+n)}\) consist, indifferently, of one or several points. Since

\[ P^{(\omega+n)}=\{P^{(\omega)}\}^{(n)}=R^{(n)}, \]

where by \(R\) is denoted the closed domain \(P^{(\omega)}\), then—according to 147°–148°—we can construct a domain of intervals \(\{l_i\}\) of type \(\omega_n\), which will give, as non-interior points, the points of the domain \(R\).

Each of these points is a point of \(P^{(\omega)}\); consequently, in each free interval \(l_i\) of the domain \(R\) there is placed a domain \(P_i^{(\omega)}\) of type A 149°; moreover, on the interval \(l_i\) either both boundaries, or only one of them, may be limiting points of \(P^{(\omega)}\). In the latter case the corresponding point will be a limiting point of the domain \(P^{(\omega)}\) on the adjacent interval.

Since on different intervals \(l_i\) the domains \(P_i^{(\omega)}\) may be identical or different, the final type of arrangement in the present case will be \(\omega_n\cdot\omega'\) or \(\omega_n\sum_{1}^{\omega}\omega_i'\).

151. In exactly the same way we can determine the type of arrangement and construct a domain of intervals in the case when \(\alpha=\omega\cdot2\). Considering again

\[ P^{(2n)}=\{P^{(\omega)}\}^{(\omega)}=R^{(\omega)}, \]

we can—bearing in mind B 149°—construct a domain of intervals \(\{l_i\}\) of type \(\omega''\), whose non-interior points will be the points \(R\).

Each of the points \(R\) is a point of the domain \(P^{(\omega)}\); consequently, in each of the free intervals \(\{l_i\}\) there is again placed a domain

of type \(\omega'\), analyzed in 149°; therefore the final arrangement will have the type

\[ \omega'' \mathop{Z}\limits_{1}^{\omega} \omega_i'=\omega'''. \]

It is clear that in this way we can go further as well, constructing and determining the type of domains if \(\alpha=\omega\cdot 2+n,\ \omega\cdot 3,\ldots\), whatever the number \(\alpha\) may be. We now arrive at the following conclusion:

“Every closed countable domain can be constructed as a domain of non-interior points of intervals of, generally speaking, a mixed type.”

152. Passing to uncountable domains \(P\), we shall have to rely on the Cantor–Bendixson theorem, which expresses \(P\) in the form \(P=R+S\), where \(R\) is a countable domain, while \(S\) is perfect, and \(D(R^{(\alpha)},R)=0\).

On the basis of this theorem we first construct the perfect domain \(S\), whose free intervals \(\{l_i\}\) will have the arrangement type \(\tilde{\omega}\); the points of \(R\) can be placed only in these intervals, and \(R\) decomposes into a countable series of parts \(R_i\); each of them will be a domain of the first kind, and—consequently—its arrangement type is determined on the basis of 147°–151°; if we call this type \(\Omega_i\), then the arrangement type of the free intervals of the domain \(P\) must be denoted as

\[ \tilde{\omega}\mathop{Z}\limits_{1}^{\omega}\Omega_i. \]

Comparing this result with 152°, we now arrive at the conclusion that

“Every closed domain can be constructed as a domain of non-interior points of a series of intervals of some definite type.”

153. Thus, since closed domains are under discussion, the theory of arrangement types makes it possible, on the one hand, to construct—according to a given type—a domain of intervals possessing known properties, and, conversely, for a given domain to determine the arrangement types, i.e. to characterize by a known symbol any given closed domain. Such a characterization is also possible for many non-closed domains, namely for those which are the result of complex arrangement types.

The basic types \(\omega,\ *\omega,\ \tilde{\omega}\) serve here as the instrument by means of which we can attain the entire result indicated above.

As for the application of the preceding theory to non-closed domains, we should not expect in advance to obtain results as simple as those expressed in Theorem 152; we are unable to state, for a non-closed domain, a similar general theorem, although in particular cases this problem may perhaps be solved. “The theoretically most important point domains are closed and perfect; moreover, they occur most often in analysis and geometry”¹, and therefore the definite answer that we were able to give above concerning their structure acquires special significance.

CHAPTER III.

Recent Works.

154.¹ For an arbitrary, possibly countable, sequence of intervals \(\{l\}\), Young defines² the domain of interior points \(G\); since its complementary domain is closed, \(G\) may be given as the domain of interior points of a sequence of intervals not overlapping one another.

If we have an infinite system of such sequences of intervals \(\{l^{(n)}\}\), then the points lying inside at least one interval of each sequence form the interior limiting domain³; this domain may be given so that the intervals \(\{l^{(n+1)}\}\) will be situated on the intervals \(\{l^n\}\); then one obtains normal sequences of intervals.

Condition A. “Let there exist a sequence of increasing numbers

\[ \nu_1,\ \nu_2,\ \nu_3,\ \ldots\ \nu_n,\ \ldots \]

such that one can indicate an interval of the \(\nu_1\)-th normal sequence to which correspond two intervals of the \(\nu_2\)-th sequence; on each of these there are two intervals of the \(\nu_3\)-th sequence, and so on.”

If this is so, then between the limiting points of the boundaries of the intervals and the points of the interval \((0,1)\), expressed in the binary system, one can establish a one-to-one correspondence, except for a countable sequence of points; if we exclude those limiting points which serve as boundaries of intervals \(l^{(n)}\) superposed upon one another, and whose number is countable (they correspond to terminating binary fractions), then the remaining points will be interior limiting points, and their domain is uncountable.

Thus, if condition A is satisfied, the inner limiting domain \(P\) is uncountable, and—therefore¹—it contains in its composition a condensed part.

Further—if the inner limiting domain \(P\) contains a dense-in-itself part, then the corresponding normal intervals must also be dense-in-themselves; and in that case condition A is satisfied, and the domain \(P\) is uncountable. From this we draw the conclusion that

B. “If the inner limiting domain \(P\) contains dense-in-themselves parts, then it is uncountable”; and further

C. “A countable inner limiting domain must be scattered.”

Let, then, \(P = R + E\), where \(R\) is the separable part, and \(E\) the condensed part; let us assign the perfect domain \(E'\) to the interval \((0,1)\), in such a way that to each complementary interval of \(E'\) there corresponds one point of the interval \((0,1)\); to the domain \(E\), dense-in-itself with respect to \(E'\), there corresponds \(T\), dense-in-itself with respect to \((0,1)\); to a point \(x\) from the composition of \(E\), lying inside the normal interval \(l_x^{(n)}\), there corresponds a point \(\xi\), which will likewise lie inside the corresponding interval \(\lambda_\xi^{(n)}\), or else will be its boundary, if \(x\) was the boundary of a complementary interval of \(E'\). To non-overlapping intervals \(l^{(n)}\) there correspond all the more intervals \(\lambda^{(n)}\) of the same kind, and to overlapping \(l^{(n)}\)—analogous intervals \(\lambda^{(n)}\); therefore to the normal intervals \(\{l^n\}\) there will correspond normal intervals \(\{\lambda^n\}\), for which \(\xi\) may be either an inner limiting point or a boundary. Since \(T\) is dense-in-itself with respect to \((0,1)\), then, as we saw above, the whole domain \(\{\xi\}\), and the part of it consisting of inner points, will be uncountable; consequently the domains \(E \equiv \{x\}\) and \(P\) will also be uncountable; thus we have:

D. “If the inner limiting domain is not separable, then it is uncountable”²;

from D it follows immediately:

E. “If the inner limiting domain is countable, then it is separable.”

We now arrive at two conclusions:

F. “Every domain containing condensed or dense-in-themselves parts can be an inner limiting domain only in the case when it is uncountable.”

G. “A countable set can be an inner limiting set only on the condition that it is rarefied and separated.”

From this it follows, among other things, that

H. “The set of both rational and algebraic irrational numbers cannot be an inner limiting set.”

155. Given any set \(P\), we may include its points \(\{x\}\) in a normal sequence of intervals \(\{l_x^{(m)}\}\); the inner limiting sequence \(P^{(0)}\), obtained by an infinite diminution of all the intervals \(l^{(m)}\), will not necessarily be identical with \(P\); it may contain points absent from \(P\).

Having theorems F and G, we are in a position in many cases to predict in advance what the difference between \(P^{(0)}\) and \(P\) will be—a difference upon which, of course, the law of construction of the intervals may have an influence; thus a separated set \(P\) with an uncountable derivative \(P'\) may be an inner limiting set, since it is situated in the free intervals of some perfect set, where intervals containing \(P\) may also be placed. As regards the size of the sets \(P\) and \(P^{(0)}\), then, not counting the preceding case, \(P^{(0)}\) and \(M(P,P')\) are equipotent.

Generally speaking, the set \(P^{(0)}\) is not closed; it will probably not include 1) the limiting points of the infinite sequence of intervals and 2) the boundaries, superposed one upon another, of the normal sequence of intervals, if they exist.

Thus, inner limiting sets can be only the following sets:

a) an everywhere rarefied homogeneous¹ uncountable set,

b) any separated set,

c) a rarely rarefied dense homogeneous uncountable set,

d) a set consisting of a homogeneous uncountable part \(U\) and a separated part, situated in the free intervals of the perfect set \(U'\); this will be the most general case of an inner limiting set.

Hobson² arrives at the same result, though by a very complicated path; as a conclusion to the preceding, we shall quote his words³:

“Determination of the necessary and sufficient conditions that any given set may be an inner limiting set,

is thus reduced to the determination of a criterion only for a homogeneous uncountable domain; moreover, the case of a sparsely scattered domain can be reduced, by the method of correspondence, to that of a densely scattered domain. In connection with 162°, this reduces to the determination of the conditions under which a densely scattered homogeneous uncountable domain may be a domain of the second category; but Hobson did not succeed in making this determination¹.

156. Veblen’s article of 1904 is devoted² to clarifying the connection between Borel’s theorem³, taken without restriction as to the countability of the sequence of intervals, and Dedekind’s definition of number:

“If all the numbers of the interval \(AB\) split into two domains \(\{X\}\) and \(\{x\}\), and every number \(X\) is greater than every \(x\), then there exists some number \(\xi\) satisfying the condition \(X > \xi > x\).”

Veblen asserts that Borel’s theorem is a consequence of Dedekind’s definition, and conversely; the first part of this assertion follows from the fact that Borel’s first proof⁴ rests on the existence of the number \(B_{\omega}\), which is ensured by Dedekind’s definition. As for the second part, Veblen proves it as follows:

Let Borel’s theorem be true, and suppose that the point \(\xi\) is absent; then the points \(X\) have no lower bound, and the points \(x\) no upper bound, owing to which about each \(X\) or \(x\) one can construct an interval whose boundary points and interior points are only points \(X\) or \(x\), respectively; since on \(AB\) there are only points \(X\) and \(x\), the intervals so constructed satisfy the conditions of Borel’s theorem, and from them \(n\) intervals \(\{l_i\}\) may be chosen, covering all the points of \(AB\). Taking the largest interval that covers the point \(A\) and belongs to the domain \(\{x\}\), we obtain, as its right boundary, the point \(x_1\) of the same domain; the largest interval among the \(\{l_i\}\) covering \(x_1\) will determine, as its right boundary, the point \(x_2\); and so on. Since the number of intervals is finite, and there is not a single point left uncovered by them, we shall arrive at the point \(B\), which must also prove to be a point of the domain \(\{x\}\), contrary to the supposition; thus the denial of the point \(\xi\) is inadmissible.

Borel’s theorem, as we saw above¹, can be proved in a more general form for a closed domain; Veblen, giving a new proof of this theorem, observes that it is valid only for a closed domain. For a non-closed domain, the absence of even a single limiting point already entails the impossibility of including the points of the domain in a finite number of intervals; Veblen proves this as follows: suppose that for a domain \(P\), situated on the segment \((x_0, X_0)\), there is lacking one limiting point \(\xi\); let \(X\) be all the points of the interval \((x_0, X_0)\) lying to the right of \(\xi\), and \(x\) the points to the left of \(\xi\); then by all possible intervals \((x_0, x)\), \((X, X_0)\) all the points of the domain \(P\) are covered, since for every point \(p\) of it to the left of \(\xi\) there is a point \(x\) lying between \(p\) and \(\xi\), and for a point \(p\) to the right of \(\xi\) there exist points \(x\) between \(\xi\) and \(p\); but among the intervals \((x_0, x)\) and \((X, X_0)\) there is no finite number of them that would satisfy the Borel–Young theorem.

157. In 1904 Young gives² still another generalization of Borel’s theorem. A small segment (region)³ \(l\), associated with some definite point \(P\) situated inside it, is called an element (tile) \(l_p\), and the point \(P\) is the point of attachment; its position, the form, and the magnitude of the segment characterize the element; two elements, coinciding in form and magnitude, are different if their points of attachment are different; ordinarily an element may be arbitrarily diminished at its edges without depriving it of its properties. An element covers all points lying inside it, but the points of its boundary are not considered covered.

Theorem A. “If a series of elements is given which can be diminished, and whose points of attachment fill a closed segment \(S\), then one can determine a finite number of elements possessing the following properties⁴:

a) the length of each element is less than \(\varepsilon\);

b) each point of \(S\) will be covered by one or more elements;

c) each point of attachment \(P_i\) is covered by the unique element \(l_{p_i}\);

d) the sum of the intervals covered by more than one element is less than \(\varepsilon'\),

where \(\varepsilon\) and \(\varepsilon'\) are arbitrarily small positive numbers.”

Since the elements can be diminished, we shall take them already smaller than \(\varepsilon\); by Borel’s theorem one can indicate a finite number \(n\) of elements covering \(S\), and all the remaining elements we shall discard.

Let \(A, B\) be the endpoints of the principal interval \(S\); from among the elements covering \(A\), choose that one whose associated point lies to the left of all the others; if there are several of them, take \(l_{p_1}\), the one which extends farther to the right than the others; we diminish, if necessary, the remaining elements so that they do not reach \(P_1\), while at the same time leaving no point of \(S\) uncovered.

Among the remaining \(n-1\) elements there will be one which covers the points of the element \(l_{p_1}\); let \(l_{p_2}\) be that one of them whose associated point lies as far to the left as possible, and, if there are several such, the one which extends as far as possible to the right; we diminish the remaining \(n-2\) elements so that they do not reach the point \(P_2\); moreover, if one of the elements turns out to be covered by another, we discard the former completely. If \(P_2\) is covered by \(l_{p_1}\), we diminish \(l_{p_1}\) so much that the point \(P_2\) lies outside \(l_{p_1}\). If the common part of \(l_{p_1}\) and \(l_{p_2}\) is not less than \(\frac{\varepsilon}{n}\), we diminish \(l_{p_2}\) so as to make this part less than \(\frac{\varepsilon}{n}\), and so on; since we are dealing with only \(n\) elements, we see that all the conditions of the theorem will be fulfilled.

In the same year Young gives\(^1\) a more general form of Theorem A, proving it, moreover, independently of Borel’s theorem.

Theorem B. “If a sequence of elements is given which may be diminished arbitrarily and whose associated points fill a measurable\(^2\) domain \(S\), then it is possible to determine a finite or countable sequence of elements possessing the following properties:

a) the length of each element \(l_i\) is less than \(\varepsilon\),

b) every point of \(S\) is covered by one or more elements,

c) every associated point \(P_i\) is covered by the single element \(l_i\),

d) the sum of the elements differs from \(M(S)=s\) by less than \(\varepsilon'\).”

Let us enclose\(^3\) \(S\) in a sequence of intervals \(\lambda_i\) such that

\[ s < \sum \lambda_i < s + \frac{\varepsilon'}{2}, \]

where all points of \(S\) will be interior to \(\{\lambda_i\}\); then diminish the elements so that they are less than \(\varepsilon\), and so that they do not

\(^1\) Proceedings of Lond. Mathem. Soc., (2) 2, p. 67.
\(^2\) See below, 171°.
\(^3\) See below, 172°.

covered the exterior points for \(\{\lambda_i\}\); then the sum of the intervals covered by elements will lie between \(s\) and \(s+\dfrac{\varepsilon'}{2}\). Next choose from among the elements a countable sequence

\[ l',\ l'',\ l''',\ldots, \tag{1} \]

covering\(^1\) the domain \(S\).

Taking \(l'\) from the sequence (1), denote it by \(l_{p_1}\), and construct to the right and to the left of it, following the methods of Theorem A, the intervals \(l_{p_2}, l_{p_3}, \ldots, l_{p_{-2}}, l_{p_{-3}}, \ldots\) with only such a difference that the common parts of the elements \(l_{p_1}\) and \(l_{p_2}\), \(l_{p_{-1}}\) and \(l_{p_{-2}}\) are less than \(\dfrac{\varepsilon'}{2^4}\), the common parts of \(l_{p_2}\) and \(l_{p_3}\), \(l_{p_{-2}}\) and \(l_{p_{-3}}\) are less than \(\dfrac{\varepsilon'}{2^5}\), and so on; then we shall obtain a finite or countable sequence of elements, while the sum of the twice-covered parts of the fundamental interval will be less than \(\dfrac{\varepsilon'}{2^2}\). If not all points of \(S\) turn out to be covered, we take in the sequence (1) the first element not yet used and covering points still uncovered up to now, \(l^{(\mu)}\), to the right and to the left of which a new finite or countable sequence of elements will arise, only now instead of \(\dfrac{\varepsilon'}{2^4}=\dfrac{\varepsilon'}{2^{1+3}}\) we take already \(\dfrac{\varepsilon'}{2^{\mu+3}}\); the twice-covered parts here will be less than \(\dfrac{\varepsilon'}{2^{\mu+1}}\). Similarly a third sequence of elements may be obtained, beginning from \(l^{(\nu)}\), for \(\nu>\mu\), with \(\dfrac{\varepsilon'}{2^{\nu+3}}\) instead of \(\dfrac{\varepsilon'}{2^{1+3}}\) and with twice-covered parts less than \(\dfrac{\varepsilon'}{2^{\nu+1}}\); and so on.

Thus, using the sequence (1), we shall obtain a sequence of elements satisfying the conditions of the theorem, while the sum of the twice-covered intervals will be less than

\[ \dfrac{\varepsilon'}{2^2}+\dfrac{\varepsilon'}{2^{\mu+1}}+\dfrac{\varepsilon'}{2^{\nu+1}}+\cdots<\dfrac{\varepsilon'}{2}; \]

and since, moreover, the sum of the intervals covered by elements lay between \(s\) and \(s+\dfrac{\varepsilon'}{2}\), the sum of the elements themselves will be contained between \(s\) and \(s+\varepsilon'\).

158. A further development of Young’s idea was obtained in 1905 in a fine paper devoted to limiting domains and the theory of measure.\(^2\)

\(^1\) See above, Theorem B, p. 69.
\(^2\) Proc. (2) 2, p. 16.

For a given sequence of intervals we usually contrasted the interior points of these intervals with their boundaries and exterior points; thus, at the very outset we are dealing, on the one hand, with a countable sequence of open intervals and, on the other, with their non-interior points, forming a closed domain.

If the given intervals \(\{\lambda_i\}\) are diminished, each interval being also allowed to split into a finite or countable number of new intervals, the closed domain \(P_1\) determined by the intervals \(\lambda_i\) is likewise transformed into a closed domain \(P_2\), containing the former domain \(P_1\); if \(G_1\) and \(G_2\) are the interior points of the intervals \(\{\lambda_i\}\) and \(\{\lambda_i'\}\), then

\[ G_2 = D(G_1), \quad P_2 = M(P_1). \]

Let us imagine that the process of diminishing the intervals \(\lambda_i\) and of the growth of \(P_i\) continues without end; we shall then obtain domains

\[ G_1, G_2, G_3, \ldots, G_n, \ldots; \quad P_1, P_2, P_3, \ldots, P_n, \ldots, \tag{1} \]

for which always

\[ G_{n+1} = D(G_n), \quad P_{n+1} = M(P_n). \tag{2} \]

The points which will be common to all the domains \(\{G_i\}\) constitute the ordinary inner limiting domain \(G_0\); another domain \(P\) we obtain by taking all the points of all the closed domains \(P_n\); this will be the ordinary outer limiting domain; thus

\[ G_0 = \overset{\infty}{\underset{1}{D}}(G_n), \quad P = \overset{\infty}{\underset{1}{M}}(P_n). \tag{3} \]

From this definition it follows immediately that

A. “The exterior points of a countable sequence of separate intervals form an ordinary inner limiting domain”; and further

B. “A countable sequence of continuous closed intervals is an ordinary outer limiting domain.”

If there are domains (1) under condition (2), but the \(G_n\) are not the interior points of continuous intervals, and the \(P_n\) are not closed, the domains (3) are called generalized inner and outer limiting domains; it is self-evident that in particular cases we need not consider simultaneously and jointly the domains \(\{G_i\}\) and \(\{P_i\}\); we may be interested either only in \(\{G_n\}\) and \(G_0\), or only in \(\{P_n\}\) and \(P\).

It is also clear that, when the ordinary limit domains \(\{G_i\}\) and \(\{P_i\}\) are considered jointly, we have

\[ G_n + P_n = L,\quad G_0 + P = L, \tag{4} \]

i.e., the corresponding points of the two domains give all points of the fundamental interval.

159. We assumed in § 158 that the intervals \(\lambda_i^{(n)}\) were open; let us now take a series of corresponding closed intervals \(\{\overline{\lambda}_i^{(n)}\}\) and the domains \(E_n\) of exterior points of these intervals determined by them; denote further

\[ G_n + Q_n = \overline{G}_n. \]

Here, as before,

\[ G_{n+1} = D(G_n),\quad E_{n+1} = M(E_n); \]

let us determine in what relation to \(G_0\) and \(P\) stand the domains

\[ \overline{G}_0 = \underset{1}{\overset{\infty}{D}}(\overline{G}_n),\quad E = \underset{1}{\overset{\infty}{M}}(E_n). \]

The domain \(\overline{G}_n\) differs from \(G_n\) by the presence of the endpoints of the intervals \(\lambda_i^{(n)}\); in passing from \(\{\lambda^{(n)}\}\) to \(\{\lambda^{(n+1)}\}\), the latter intervals may either lie entirely within the former, or have one common endpoint with them; if the second case is repeated in the series \(\{\lambda^{(n)}\}\) for every \(n\), the corresponding endpoint will enter into the composition of the limiting domain \(\overline{G}_0\), but will not enter into \(G_0\); if such an accident is excluded, no endpoint will enter into \(\overline{G}_0\) unless it also entered into \(G_0\). Thus the inner limiting domain \(G_0\) and the limiting domain \(\overline{G}_0\) can differ only by some series of endpoints of intervals, so that the domain \(\overline{G}_0 - G_0\) is always finite or countable; whenever the case indicated above is excluded by the construction of the domain, \(\overline{G}_0 = G_0\). Hence it follows that, exactly likewise, the domain \(P - E = Q\) is also finite or countable, and that \(P = E\) if the second case is impossible.

We retain the name ordinary inner limiting domain only for the domain \(G_0\), contrary to Young¹.

160. Before proceeding further in the exposition of the properties of limiting domains, we shall give below several necessary theorems.

A. “If, on the fundamental interval of a closed domain \(G\), an arbitrary closed segment is taken, then the part of \(G\) lying on it will be closed.”

Let \(L\) be the principal interval, and \(\Lambda\) its segment, the endpoints of which may turn out to be interior points of the free intervals \(\{l_i\}\) of the closed domain \(G\); if this happens, we shall diminish \(\Lambda\) to \(\Lambda_0\) so that the endpoints of \(\Lambda_0\) are points \((x_0,\xi_0)\) of the domain \(G\); on \(\Lambda_0\) there will lie a certain series of free intervals \(\{l_{m_j}\}\), and their non-interior points, being points of \(G\), form a closed domain.

B. “If \(G_1\) and \(G_2\) are closed domains without common points, then each of them is situated inside a finite number of free intervals of the other”¹.

Let \(\{l_i'\}\) be the free intervals of the domain \(G_1\); since \(D(G_1,G_2)=0\), the points of \(G_2\) can only be interior points of \(l_i'\); if they were situated in an infinite series of intervals

\[ l_{m_1}',\ l_{m_2}',\ \ldots,\ l_{m_n}',\ \ldots, \tag{1} \]

then for their endpoints and interior points there would exist some limit point \(\xi\), which would also be a limit point for the points of \(G_2\) situated on \(\{l_{m_i}\}\); therefore, since \(G_2\) is closed, \(\xi\) would be a point of \(G_2\) and, consequently, would be a common point of \(G_1\) and \(G_2\), which is excluded by assumption; thus the existence of the infinite series (1) is impossible. The same would also hold for the intervals \(\{l_i''\}\).

C. “If \(G_1, G_2\) are closed domains without common points, then \(G_1+G_2\) is also closed,” since every limit point of \(G_1+G_2\), being a limit point for an infinite series of points of at least one of \(G_1, G_2\), will belong to that domain, and hence also to \(G_1+G_2\).

D. “If \(G_1\) and \(G_2\) are closed, then \(D(G_1,G_2)\) and \(M(G_1,G_2)\) are also closed”².

E. “If \(G_{n+1}=D(G_n)\), and all \(G_n\) are closed, then the interior limiting domain

\[ G_0=\overset{\infty}{D}(G_n) \]

is also closed”³.

161. For limiting domains we have the following theorems:

A. “The sum of a finite number of ordinary interior or exterior limiting domains is a domain of the same kind.”

The first follows from the definition, and the second from Theorem D.

B. “The inner limiting domain for a sequence of ordinary inner limiting domains is an ordinary inner limiting domain”¹.

Let

\[ G_n=\underset{i=1}{\overset{\infty}{\mathrm D}}(G_{ni}), \]

where the \(G_{ni}\) are defined by sequences of continuous open intervals, and

\[ G=\underset{1}{\overset{\infty}{\mathrm D}}(G_n); \]

for brevity Young writes \(K>L\) if \(K=\mathrm M(L)\), and \(K<L\) if \(K=\mathrm D(L)\); thus, if \(G\) is an inner limiting sequence, we must have \(G_{n+1}<G_n\). In connection with the definition of \(G_r\) and \(G_{r+1}\),

\[ G_{rs}>G_{r,s+1},\qquad G_{r+1,s}>G_{r+1,s+1}; \]

moreover, \(G_{r,s+1}\) and \(G_{r+1,s+1}\) consist of continuous intervals lying respectively on the intervals \(G_{rs}\) and \(G_{r+1,s}\); if we take \(\mathrm D(G_{rs},G_{r+1,s})\), then these will be their common intervals²; the points of those parts of the intervals \(G_{r+1,s}\) which do not enter here do not enter \(G_{rs}\) or \(G_r\), and consequently cannot enter \(G_{r+1}\) either, since \(G_r>G_{r+1}\); therefore, without changing the essence of the matter, we may always discard in \(G_{r+1,s}\), and together with it in all \(G_{r+1,s+m}\), those intervals or those parts of intervals which are not in \(G_{rs}\); it will then turn out that the domain \(G_{r+1,s}\), modified in this way, will lie on the intervals \(G_{rs}\); and since moreover the domain \(G_{r+1,s+1}\) lies on the intervals \(G_{r+1,s}\), it follows that \(G_{r+1,s+1}\) will also lie on the intervals \(G_{rs}\); thus, for domains consisting of continuous intervals, we obtain exactly the same table as in Young³.

Taking in it the sequence of diagonal domains

\[ G_{11},\ G_{12},\ \ldots,\ G_{mm},\ \ldots, \tag{2} \]

we see, in view of the preceding, that the intervals \(G_{r+1,r+1}\) always lie on the intervals \(G_{rr}\), and in this case the sequence (2) defines an ordinary inner limiting domain

\[ G_0=\underset{1}{\overset{\infty}{\mathrm D}}(G_{nn}), \]

which is identical with \(G\).

C. “The outer limiting domain for a series of ordinary outer limiting domains is an ordinary outer limiting domain”¹.

D. “The difference of two closed domains may be represented either as an ordinary outer, or as an ordinary inner limiting domain.”

The second part of Young’s theorem is proved² sufficiently fully; as for the first, for it a proof is desirable that is not the one given by the author, which, owing to its excessive brevity, is not immediately convincing.

Let us include the points of the smaller domain \(H\) inside intervals smaller than \(c\); then—according to the Borel–Young theorem³—a finite number of such intervals \(\{\lambda_j^{(c)}\}\) with the same property can be given; after their removal from \(L\) there will remain likewise a finite number \(m\) in construction of non-adjacent intervals \(\{l_i^{(c)}\}\), whose boundary points do not include the points \(H\), and on which will lie the part \(G^{(c)}\) of the larger domain \(G\); on each of the \(l_i^{(c)}\) there is situated a certain part \(G_i^{(c)}\) of the domain \(G^{(c)}\), which, as a part of \(G\), according to A 160°, must be closed; closed, then, will also be

\[ \sum G_i^{(c)} = G^{(c)}. \]

For \(c' < c\) the domain \(G^{(c')}\) is also closed, and \(G^{(c')} > G^{(c)}\); if \(c\) decreases without bound, we obtain the ordinary outer limiting domain

\[ G \equiv M\bigl(G^{(c)}\bigr). \]

On the other hand, if \(K^{(c)}\) is the domain of inner points of the intervals \(\{\lambda_j^{(c)}\}\), \(K^{(c')} < K^{(c)}\), and the intervals \(K^{(c')}\) cannot go outside the intervals \(K^{(c)}\), since \(c' < c\) (the points of the domain \(H\) may be taken as midpoints of the corresponding intervals), and in such a case the domain

\[ D\{K^{(c)}\} \equiv K^{(0)} \]

is an ordinary inner limiting domain, which—by virtue of R 51°—is nothing other than \(H\).

Only the points \(G\) will enter into the composition of \(\bar G\), and in addition—each of them, except for the points \(K^{(0)} \equiv H\); indeed: each of the points \(x\)

the domain \(G-H\) will not enter into \(K^{(0)}\) and, consequently, it must be a non-interior point of some series of intervals \(\{ \lambda^{(e_x)}_j\}\), and then of all subsequent series for \(e<e_x\); in that case it will enter into \(G^{(e_x)}\) and into \(\overline G\). Hence \(\overline G = G-H\), and the theorem is proved.

E. “If from an ordinary exterior or interior limiting domain one subtracts a closed domain, then the difference will be, respectively, an ordinary exterior or interior limiting domain”¹.

F. “The difference of ordinary interior (exterior) and exterior (interior) limiting domains is an ordinary interior (exterior) limiting domain”².

Such are the extremely important theorems concerning limiting domains with which Young completes their theory; as for the questions relating to the measure of a domain and solved by Young in the same article, they will be discussed in Chapter IV.

162. We have already spoken, in §155, of the article by Hobson³, written by him simultaneously with Young’s work mentioned above in §§158–161; what is also of interest in it is the comparison of limiting domains with domains of the first and second category.

If \(P_n\) are nowhere dense domains, then \(P=\mathrm{M}_{1}^{\infty}(P_n)\), as we saw in §37⁴, is called a domain of the first category; if we take a sequence of domains

\[ P_1=Q_1,\quad \mathrm{M}_{1}^{2}(P_i)=Q_2,\ldots,\quad \mathrm{M}_{1}^{n}(P_i)=Q_n,\ldots, \tag{1} \]

then, obviously, \(Q_{n+1}=\mathrm{M}(Q_n)\), and \(\mathrm{M}_{1}^{\infty}(Q_n)=P\); thus a domain of the first category \(P\) is nothing other than a special case of a generalized exterior limiting domain, a special case because the domains \(Q_n\) are subject to the restriction that, like the \(P_n\), they are nowhere dense.

If one imposes on the domains \(P_n\) the further new condition that they are all closed, then, by virtue of D §160, all the \(Q_n\) will also be closed; \(P\) will then be an ordinary exterior limiting domain.

Hobson⁵ at once introduces this latter restriction, absent in Baire⁶ and unnecessary in connection with the applications of the theory of do-

to the theory of functions. In order to distinguish the general case of domains of the first category from Hobson’s domains, the latter may be called ordinary domains of the first category.

A domain complementary to a domain of the first category \(P\) is a domain of the second category, and, by virtue of 158°, it is an inner limiting domain; thus every domain of the second category belongs to the number of inner limiting domains, but it is only a special case of them: a domain of the second category is always uncountable, whereas inner limiting domains, as we have seen above¹, may also be countable and even finite.

If upon \(P_n\), and consequently upon \(Q_n\), the condition is imposed that it be closed, they will be defined by the non-inner points of a series of intervals, whereas the domain \(K_n\) complementary to \(Q_n\) will consist of the inner points of these intervals; in such a case

\[ Q = \prod_{1}^{\infty} (Q_n) \quad \text{and} \quad K = \sum_{1}^{\infty} (K_n) \]

turn out to be ordinary outer and inner limiting domains; since \(K\) is the domain complementary to \(Q\), \(K\) is a domain of the second category. To distinguish it from the general case, we shall call such domains ordinary domains of the second category.

Thus every domain of the second category belongs to the number of inner limiting domains, and every ordinary domain of the second category is included among ordinary inner limiting domains; therefore² “the investigation of the nature and properties of inner limiting domains will shed light on the nature of domains of the second category and will consequently have an influence on the theory of pointwise-discontinuous functions.”

A countable domain is an ordinary domain of the first category; its complement is an ordinary domain of the second category; therefore the domain of both nonmeasurable and transcendental numbers is an ordinary domain of the second category and, consequently, an ordinary inner limiting domain.

Hobson says³ that “it would be desirable to find necessary and sufficient conditions under which an arbitrarily given everywhere dense and homogeneous uncountable domain would be an inner limiting domain”⁴; this is equivalent to determining the necessary and suffic—

there are no conditions for an arbitrary domain to be a domain of the second category*.

163. For an ordinary inner limiting domain Hobson gives also¹) two theorems, which can also be proved in a more general form:

A. “If \(P\) and \(Q\) are generalized or ordinary inner limiting domains, then \(D(P,Q)\) and \(M(P,Q)\) will also be domains of the corresponding kind.”

Let the domains composing \(P\) and \(Q\) be \(P_n\) and \(Q_n\),

\[ P=\overset{\infty}{D}_{1}(P_n),\quad Q=\overset{\infty}{D}_{1}(Q_n); \]

let us further denote

\[ D(P_n,Q_n)=R_n,\quad M(P_n,Q_n)=S_n; \]

by hypothesis

\[ P_{n+1}=D(P_n),\quad Q_{n+1}=D(Q_n). \]

a. The domain \(R_{n+1}\) consists of the common points of the domains \(P_{n+1}\) and \(Q_{n+1}\), which are contained entirely, respectively, in \(P_n\) and \(Q_n\); consequently the points of \(R_{n+1}\) will be among the common points also for the domains \(P_n\) and \(Q_n\), i.e. \(R_{n+1}=D(R_n)\); whence it follows that

\[ R=\overset{\infty}{D}(R_n) \]

will be an inner limiting domain; every point common to \(P\) and \(Q\) will enter into it, since this point enters into each of \(P_n\) and \(Q_n\) and, consequently, into \(R_n\); on the other hand, no point \(\xi\) not belonging to \(D(P,Q)\) can be found in \(R\); indeed, it will not enter into \(P_n\) or into \(Q_n\), and—consequently—into \(P_n,Q_n\) for \(n\) equal to the larger of the numbers \(n_1\) and \(n_2\); therefore it will not be in \(R_n\) either; this point thus cannot be a point of \(R\); whence it follows that \(R=D(P,Q)\), and for a generalized inner limiting domain the theorem is proved; as for the ordinary domain, for it \(P_n\) and \(Q_n\) are given by inner points of continuous intervals; therefore, as is not difficult to see²), \(R_n=D(P_n,Q_n)\) will be given by intervals of the same kind: first of all, in \(R_n\) there cannot figure isolated points, which may be common boundaries of adjacent intervals of one \(P_n\) and another \(Q_n\), since these intervals are open; \(R_n\) can consist only of continuous intervals which belong to one of the domains \(P_n\) and \(Q_n\), if they are situated entirely on intervals of the other domain, and of intervals which will be common parts of intervals of \(P_n\) and \(Q_n\).

¹) ib. p. 326, p. 319.
²) Cf. the proof of theorem B 161°.

Thus the domain \(R \equiv D(P,Q)\), defined by a sequence of domains \(R\) of solid intervals, will turn out to be an ordinary inner limiting domain.

b. The domain \(S_{n+1}\) is composed of all distinct points \(P_{n+1}\) and \(Q_{n+1}\), which enter respectively into \(P_n\) and \(Q_n\), and consequently also into \(M(P_n,Q_n) \equiv S_n\); thus it turns out that

\[ S_{n+1} \equiv D(S_n), \quad \text{and} \quad S \equiv \overset{\infty}{\underset{1}{D}}(S_n) \]

will be an inner limiting domain.

Every point \(M(P,Q)\) belongs to all the domains of at least one of the sequences \(\{P_n\}\) or \(\{Q_n\}\); therefore it necessarily enters into all \(S_n\), and consequently also into \(S\). On the other hand, every point of \(S\) occurs in all \(S_n\), and consequently it must be a point of every domain of at least one of the sequences \(\{P_n\}\) and \(\{Q_n\}\); it must therefore enter at least one of the domains \(P\) or \(Q\), and—consequently—it will be a point of the domain \(M(P,Q)\). Thus \(S \equiv M(P,Q)\), which proves the theorem for generalized domains.

For ordinary domains, when \(P_n\) and \(Q_n\) are defined by solid intervals, the domain \(M(P_n,Q_n) \equiv S_n\) will possess the same property; indeed, it is composed, first, of the intervals of \(P_n\) and \(Q_n\) which have no common points; second, of the intervals of one domain continued by intervals of the other, if such intervals partially overlap one another; and finally, third, of the intervals of one of \(P_n\) or \(Q_n\) upon which intervals of the other domain equal to them or smaller are wholly superposed. Adjacent intervals will in this case be in positions independent of one another, since their common boundary does not enter into either of the domains \(P_n\) and \(Q_n\); and if \(S_n\) is defined by solid intervals, then \(S\) will be an ordinary inner limiting domain.

B. “If \(P\) and \(Q\) are generalized or ordinary outer limiting domains, then \(D(P,Q)\) and \(M(P,Q)\) will also be domains of the corresponding kind.”

By hypothesis, for \(P \equiv \overset{\infty}{\underset{1}{M}}(P_n)\) and \(Q \equiv \overset{\infty}{\underset{1}{M}}(Q_n)\) we have

\[ P_{n+1}=M(P_n), \quad Q_{n+1}=M(Q_n), \tag{3} \]

where for the domains \(R_n, S_n\) let the former meaning be retained.

a. Every point \(R_n\) enters simultaneously into \(P_n\) and \(Q_n\); it will enter, by virtue of (3), into \(P_{n+1}\) and \(Q_{n+1}\), and—consequently—into \(R_{n+1}\);

Thus \(R_{n+1}=M(R_n)\), and \(R=M_1^\infty(R_n)\) is an outer limiting domain. Each point \(D(P,Q)\) belongs to all domains \(P_n\) for \(n>n_1\) and to all \(Q_n\) for \(n>n_2\); for \(n\) greater than or equal to the greater of the numbers \(n_1\) and \(n_2\), this point will then enter simultaneously into \(P_n\) and \(Q_n\), and consequently into \(R_n\), and in that case it will also appear in \(R\); conversely: each point \(R\) appears in all \(R_n\) for \(n>n_0\); consequently, under the same condition it will enter into \(P_n\) and \(Q_n\) simultaneously, and in that case it will be found in \(P\) and \(Q\), and in \(D(P,Q)\); thus \(R=D(P,Q)\); by this the theorem is proved for a generalized outer limiting domain; as for the ordinary domain, for it \(P_n\) and \(Q_n\), and consequently—by virtue of D 160°—also \(R_n\), will be closed, whence \(R\) will be an ordinary outer limiting series.

b. Each point \(S_n\) belongs to at least one of the domains \(P_n\) and \(Q_n\); therefore it enters into at least one of \(P_{n+1}\) or \(Q_{n+1}\), and consequently into \(S_{n+1}\); thus

\[ S_{n+1}=M(S_n),\quad \text{and}\quad S=M_1^\infty(S_n) \]

is an outer limiting domain.

Each point \(M(P,Q)\) belongs to at least one of the domains \(P,Q\) and, consequently, to the domains \(P_n\) or \(Q_n\) of at least one of the series \(\{P_n\}\) or \(\{Q_n\}\) for \(n>n_0\); for such values of \(n\) it enters into \(S_n\) and consequently into \(S\); on the other hand, each point \(S\) appears in all \(S_n\) for \(n>n_0\); for these same values of \(n\) it will then appear at least in one of the domains \(P_n\) and \(Q_n\), and by virtue of this at least in one of the domains \(P,Q\), and consequently in \(M(P,Q)\). Thus we find that \(S=M(P,Q)\), and the theorem for generalized domains \(P\) and \(Q\) is proved. In the case of ordinary domains, \(P_n\) and \(Q_n\), and consequently \(S_n\), are closed, and in that case the outer limiting domain \(S\) will be ordinary.

From theorems A and B it follows immediately:

C. “If \(P\) and \(Q\) are generalized or ordinary inner or outer limiting domains without common points, then \(P+Q\) is a domain of the corresponding kind,”

since in this case \(P+Q=M(P,Q)\); and further

D. “If \(P,Q,R,\ldots,U\) is a finite series of generalized or ordinary inner or outer limiting domains without common points, then \(P+Q+R+\ldots+U\) is a domain of the corresponding kind.”

Let us note further, incidentally, that to the question of Schoenflies’\(^{1}\) Hobson \(^{2}\) replies in the negative, giving as an example a domain \(P\) consisting of the sum of a finite number of parts \(\sum P_i\), situated on adjacent intervals \(\{\lambda_i\}\), with some of the intervals \(P_i\) being of the first category and others of the second.

164. In § 1 we said\(^{3}\) that Bolzano must be acknowledged as the progenitor of the modern theory of domains; but Kasner in 1905 noted\(^{4}\) the fact, of extraordinary importance, that this honor belongs to Galileo Galilei: namely—in the “Discorsi e dimostrazioni matematiche,” published in 1638\(^{5}\), there is a conversation among three interlocutors concerning continuity and divisibility; in it one of the interlocutors, Simplicio, says that, admitting infinite divisibility, we must admit that one infinity can be greater than another, since a long segment contains more points than a short one; in the opinion of Salviati (the author), the difficulty arises here because we, investigating infinity with our finite mind, ascribe to it properties derived by us from the finite and the bounded; but this is not just, for the properties “greater,” “less,” “equal” are not applicable to infinity, just as one cannot speak of greater, lesser, and equal infinities. Salviati then gives the example of the numbers of the natural series and their squares, which in the series of numbers occur more and more rarely as they recede from the beginning, but which can be put into mutual one-to-one correspondence with the numbers of the natural series.

Thus Galileo Galilei, arriving here at the fundamental ideas of the modern theory of domains, still does not venture to operate with infinities and even rejects the possibility of such operations; this is hardly surprising if one recalls that even Cantor did not at once decide to publish the results of his investigations\(^{6}\).

165. In 1905, Borel’s theorem attracted the attention of Riesz, who generalized\(^{7}\) its proof, in Borel’s words\(^{8}\), almost

\(^{1}\) See above, p. 61.
\(^{2}\) Proceed. (2) 2, p. 326.
\(^{3}\) P. 2.
\(^{4}\) Bull. of the Am. Math. Soc., XI, p. 499.
\(^{5}\) See also Ostwald Klassiker, No. 11, p. 29.
\(^{6}\) M. A., 17, p. 358.
\(^{7}\) C. R., CXL, p. 224.
\(^{8}\) ib. p. 298.

identical with the proof of Lebesgue’s¹; on the other hand, it is simpler in comparison with the latter, but in a certain respect it is interesting to report² Borel’s letter to Baire:

Every point \(x\) of the fundamental interval \((0,1)\) lies inside a whole series \(\{(\alpha_x,\beta_x)\}\) of Borel intervals; if \(h_x\) is the smaller of the numbers \(x-\alpha_x\), \(\beta_x-x\), and if we define \(\varepsilon_x=\operatorname{oGr} h_x\), then for the point \(x\) there will exist either at least one interval whose nearest boundary is removed from \(x\) by a distance equal to \(\varepsilon_x\), or else a series of intervals with the nearest boundary at a distance exceeding \(\varepsilon_x-k\), for arbitrary small \(k\); \(\varepsilon_x\) is thus a function of \(x\) and, evidently, will moreover be a discontinuous function.

If for the interval \((0,1)\) we find \(\eta=\operatorname{uGr}\varepsilon_x\), then \(\eta=\varepsilon_{x_0}\) for some definite point \(x_0\), and consequently \(\eta\) is different from 0. For \(0<\eta'<\eta\), for each point \(x\) there can be specified an interval \(l_x\) with the nearest boundary removed from \(x\) by more than \(\eta'\); and from the series \(\{l_x\}\) one can select a finite number of intervals \(l_x\) satisfying Borel’s theorem.

166. In Lennes’s article³, devoted to the “Heine–Borel” theorem for three-dimensional space, the author gives the “continuity theorem”⁴, which for linear domains may be expressed as follows:

“If every point \(x\) of a closed bounded domain \(P\) lies inside at least one interval of the domain of intervals \(\{l\}\), then there exists such a positive number \(\delta\) that every interval
\[ > \left(x-\frac{\delta}{2},\,x+\frac{\delta}{2}\right) > \]
will lie wholly inside at least one of the intervals \(\{l\}\).”

The author says that in the last year many persons have come upon this theorem, and that the theorem cited by Lennes may have been partly inspired by Bolza; for a linear domain Wederburn used it, but it has not yet appeared in print.

CHAPTER IV.

The Theory of the Measure of a Set.

167. Let the point set \(\Gamma\) consist of all points of a sequence of closed, continuous, and mutually non-overlapping intervals\(^1\) \(\{\lambda_i\}\), whose number may be finite or countable; suppose also that the length of the fundamental interval \(L\), on which the intervals \(\lambda_i\) are situated, is finite and equal to \(l\); then

\[ \sum \lambda_i=\lambda_0 \]

will be called the measure of the set \(\Gamma\).

Let now all or several of the intervals \(\{\lambda_i\}\) be open; let us denote by \(\{\lambda'_i\}\) such a sequence of intervals, in contrast to the closed intervals \(\{\lambda_i\}\), and call the set of the corresponding points \(\Gamma^{(1)}\); let \(\Gamma\) be the set obtained by adjoining to \(\Gamma^{(1)}\) all endpoints of intervals not belonging to \(\{\lambda'_i\}\); then

\[ \Gamma = M(\Gamma^{(1)}). \]

On the other hand, within each interval \(\lambda'_i\) construct an interval \(\lambda''_i\) in such a way that \(\lambda_i-\lambda''_i=\varepsilon \lambda_i\), where \(\varepsilon\) is some small positive number; then one obtains a sequence of closed continuous intervals \(\{\lambda''_i\}\), for which

\[ \sum \lambda''_i=(1-\varepsilon)\lambda_0, \]

and all of whose points lie inside the corresponding intervals \(\lambda_i\) and \(\lambda'_i\), so that, if \(\overline{\Gamma}\) denotes the set of these points, then

\[ \overline{\Gamma}=D(\Gamma^{(1)}). \]

In this way we construct for \(\Gamma^{(1)}\) two sets \(\Gamma\) and \(\overline{\Gamma}\), an outer and an inner one, of which one contains \(\Gamma^{(1)}\), while the other is a part of it, and, moreover, such that their measures will be respectively \(\lambda_0\) and \((1-\varepsilon)\lambda_0\); in view of the circumstance that \(\varepsilon\) may be taken arbitrarily small, and that consequently the measure

\(^1\) Such intervals—obviously—will not be adjacent.

differs arbitrarily little from the measure of the inner domain, we define for the domain \(\Gamma^{(1)}\) the measure \(l_0^{(1)}\) as

\[ \operatorname{uGr}\lambda_0 = l_0^{(1)} - \operatorname{oGr}(1-\varepsilon)\lambda_0, \]

i.e., since \(\lambda_0\) is constant here,

\[ l_0^{(1)}=\operatorname{oGr}(1-\varepsilon)\lambda_0=\lambda_0. \]

We could have arrived at the result just derived by another route, arguing as follows: the length of the interval \(\lambda_i\), whether its endpoints are included among the points of the domain or not, will be one and the same; consequently the value of \(\sum \lambda_i\) will also be one and the same under both assumptions. But—in our view—such reasoning would not be entirely rigorous: at the very outset it would make the assumption that it is possible to sacrifice a countable set of points without changing the value of the measure of a point domain, an assumption which can by no means be regarded as beyond dispute; therefore we have avoided this not entirely rigorous method of definition.

From the preceding it follows:

A. “The measure of a domain of points defined by a sequence of solid intervals remains one and the same, whether the individual intervals, or even all the intervals, are closed or not.”

168. Having thus established the measure \(M(\Gamma)\) of a domain \(\Gamma\) given by a sequence of solid intervals, indifferently—closed or open—we now pass to the complementary domain \(G = L-\Gamma\), which, generally speaking, does not consist of solid intervals; for it we define the measure \(l_0\) by putting \(l_0 = l-\lambda_0\); if \(G\), like \(\Gamma\), consisted of points of solid intervals, then this last definition would coincide for it with the definition of 167°.

If all intervals \(\{\lambda_i\}\) of the domain \(\Gamma\) are open, \(G\) is a closed domain; if all \(\lambda_i\) are closed, \(G\) is the domain of the outer points of the corresponding closed domain; in the general case \(G\), including all outer points, will also contain \(D(Q)\), where \(Q\) is the domain of the boundaries of the intervals \(\lambda_i\).

From this definition of the measure of a point domain, as a consequence of theorem A, we first of all have:

B. “The measure of a closed domain is equal to the measure both of the domain of outer points of the sequence of its free intervals and of the intermediate domains differing from the latter by a finite or countable number of boundaries.”

Such are the fundamental propositions which must be laid at the foundation of the whole doctrine of measure; from such a definition of measure, which must be acknowledged as the simplest and most natural, there develops the entire further theory of the measurement of sets.

169. For the measure \(l_0\) of a closed set \(P\) we have a series of theorems:

C. “The measure of a closed set cannot exceed the length of the fundamental interval.”

If the closed set \(P\) is dense, then \(P=L\), and \(M(P)=l\); if, however, it is nowhere dense, then for it the sum of the free intervals \(\sum \lambda_i>0\), and consequently \(M(P)=l-\sum \lambda_i<l\).

D. “All points of a closed set can be included in a finite series of intervals with measure \(l_0+\varepsilon\), arbitrarily close to \(l_0\)”\(^{1}\).

E. “If the fundamental interval \(l\) is divided into a finite number \(m\) of parts \(\{\lambda_i^{(m)}\}\), and the sum \(\sum \bar{\lambda}_i^{(m)}=\lambda_m\) is taken of those intervals whose boundary and interior points are points of the set \(P\), then, as \(m\) increases without bound, \(\lim \lambda_m=l_0\)”\(^{2}\).

F. “If about each point \(x\) of the set \(P\) one constructs a finite interval of one and the same length \(\varepsilon\), then the sum of the parts of \(L\) covered by these intervals tends, as \(\varepsilon\) decreases, to the limit \(l_0\)”\(^{3}\).

G. “The measure \(l_0\) serves as such a limit also in the case when these intervals \((x-\varepsilon_x,\ x+\varepsilon_x)\) are different for different points of the set”\(^{4}\).

In theorems F and G we regard—obviously—the given closed set \(P\) as an ordinary inner limiting set.

From the preceding theorems it follows that, when a closed set is in question, all the definitions of the measure of a set known up to now give one and the same result; the difference between them begins only from the moment when we pass to non-closed sets.

H. “If \(G_1, G_2\) are closed sets without common points, then

\[ M(G_1+G_2)=M(G_1)+M(G_2). \]

This follows from theorem B, 160°; hence it follows directly further:

\(^{1}\) See above—Harnack, 26°, p. 27; Osgood, 34°, p. 52; Young, 50°, p. 90.
\(^{2}\) See above Stolz, 19°, p. 26.
\(^{3}\) See above—theorem Q of Young, 50°, p. 91; Cantor, 17°, p. 23.
\(^{4}\) See—theorem E of Lindelöf, 42°, p. 70.

I. “If \(\{G_i\}\) is a finite number \(m\) of closed regions without common points, then

\[ \mathbf{M}\left(\sum_{1}^{m} G_i\right)=\sum_{1}^{m}\mathbf{M}(G_i). \]

If the number of component regions is infinite, the theorem, without new definitions, loses its force, since then the sum-region ceases to be closed.

J. “The measure of one of two closed regions \(G_1\) and \(G_2\) without common points cannot exceed the sum of the free intervals of the other region.”

As we have seen\(^1\), each of \(G_1, G_2\) is disposed on a finite number of free intervals of the other region; thus the region \(G_2\) lies on the intervals \(\lambda'_{m_1}, \lambda'_{m_2}, \ldots, \lambda'_{m_k}\) of the series of free intervals \(\{\lambda'_i\}\) of the region \(G_1\) and is therefore divided into \(k\) closed regions \(G_{2j}\); according to theorem C, \(\mathbf{M}(G_{2j}) \leq \lambda'_{m_j}\), and therefore—in connection with theorem I—

\[ \mathbf{M}(G_2)=\sum_{j=1}^{k}\mathbf{M}(G_{2j})\leq \sum_{j=1}^{k}\lambda'_{m_j}<\sum_{1}^{\infty}\lambda'_i; \]

the same also applies to the region \(G_1\).

If \(\Gamma_1\) is the region complementary to \(G_1\), then it consists of the interior points of the series of solid intervals \(\{\lambda'_i\}\); bearing this in mind, theorem J may be paraphrased as follows:

K. “If the closed region \(G_2\) consists of the interior points of a series of solid intervals \(\{\lambda'_i\}\), then \(\mathbf{M}(G_2)\) cannot exceed

\[ \sum \lambda'_i. \]

170. Bearing in mind theorems A and B, both the regions given by solid closed or open intervals and their complementary regions we shall call primary; their measure is determined either as the sum of the solid intervals, or as the complement of this sum to the length of the fundamental interval; the measure of a region not belonging to the primary ones will in its definition rest upon the measure of primary regions, as we shall see below; for the present we shall state the theorem necessary for the subsequent exposition:

L. “If \(G\) and \(H\) are two primary regions, of which \(G \equiv D(H)\), then \(\mathbf{M}(G)\leq \mathbf{M}(H)\).”

\(^1\) See B 160°.

Let us examine separately each of the possible cases:

a. If \(G\) and \(H\) are given by rows of solid intervals \(\{\lambda_i\}\) and \(\{l_i\}\), and moreover, by assumption, the intervals \(\lambda_i\) must be situated on the intervals \(l_i\), it is clear that \(\sum \lambda_i < \sum l_i\), and the theorem is proved.

b. If neither \(G\) nor \(H\) is given by solid intervals, and if \(\overline{G}\), \(\overline{H}\) are the corresponding closed sets, then, according to theorem B 168°,
\(M(G)=M(\overline{G})\), \(M(H)=M(\overline{H})\). Since \(G=D(H)\), it must be that \(\overline{G}=D(\overline{H})\); indeed: every point of \(G\) is at the same time a point of \(H\); consequently, every limiting point of \(G\) will enter into the composition of \(\overline{H}\), so that \(\overline{G}\) proves to be \(D(\overline{H})\). If \(\{\lambda_i\}\) and \(\{l_i\}\) are the free intervals of \(\overline{G}\) and \(\overline{H}\), respectively, then—obviously—all the \(l_i\) will at the same time be free from the points of \(\overline{G}\), so that
\[ \sum \lambda_i > \sum l_i, \]
and consequently \(M(\overline{G})<M(\overline{H})\), and at the same time
\[ M(G)<M(H). \]

c. Finally, let \(H\) consist of solid intervals \(\{l_i\}\), while \(G\) contains no such intervals; let first \(G\) be a closed set, and let \(\{\lambda_i\}\) be its free intervals; we must prove the inequality \(M(G)<M(H)\), i.e.
\[ l-\sum \lambda_i < \sum l_i, \]
or, finally,
\[ \sum l_i+\sum \lambda_i > l; \]
suppose that this is not so, but, on the contrary, that
\[ \sum l_i+\sum \lambda_i < l; \]
then, as is known¹, outside the intervals \(\{\lambda_i,l_i\}\) there is an uncountable set of points \(\{\xi\}\). If these points lie outside the intervals \(\{\lambda_i\}\), then they must enter into the closed set
\(L-\{\lambda_i\}=G\), and consequently into \(H\), since \(G=D(H)\); if the points \(\{\xi\}\) were only boundaries of the intervals \(l_i\), then they would be countable; and since the set \(\{\xi\}\) is uncountable, an uncountable number of points \(\xi\) must lie inside the intervals \(\{l_i\}\), which contradicts the definition of the set \(\{\xi\}\).

We shall obtain the analogous result if \(G\) is the set of exterior points of a closed set \(\overline{G}\) with free intervals \(\{\lambda_i\}\); here, by the definition of \(\overline{G}\), the set \(\overline{G}-G\) consists of the boundaries of the intervals \(\{\lambda_i\}\) and will always be countable. As above, we must prove the inequality

\[ \mathbf{M}(G)=\mathbf{M}(\bar G)=l-\sum \lambda_i<\sum l_i; \]

assuming again

\[ \sum \lambda_i+\sum l_i<l, \]

we shall have an uncountable domain \(\{\xi\}\), lying outside the intervals \(\{\lambda_i,l_i\}\); since the points \(\{\xi\}\) lie outside \(\{\lambda_i\}\), they must enter into \(G\), and consequently into \(G \equiv D(H)\), since the domain \(\bar G-G\) is countable; the uncountable part of the domain \(\{\xi\}\) entering into \(H\) must, therefore, consist entirely of interior points of the intervals \(\{l_i\}\), which is impossible.

Since the supposition that \(G\) consists of continuous intervals, while \(H\) contains no such intervals in its composition, is inadmissible in substance, the three cases examined exhaust all possibilities, and the theorem is established.

171. Primary domains, apart from the closed one and a number of a finite number of closed intervals, belong to open domains; in this way the measure is defined at first for certain special cases of domains of this kind. If from these special cases we wish to pass to the general one, we shall have to introduce new definitions, which are a development of those ideas that we already used in §167.

If an open domain \(Q\) does not belong to the number of primary ones, we can construct primary domains—on the one hand \(P\equiv D(Q)\), which are divisors of \(Q\), and on the other hand \(R\equiv M(Q)\), which contain \(Q\); the domain \(P\) we shall call the inner, and \(R\) the outer primary domain; since by the condition it turns out that \(P\equiv D(R)\), then—by virtue of theorem L—

\[ \mathbf{M}(P)<\mathbf{M}(R). \tag{1} \]

Constructing, in all possible ways, the inner and outer domains \(P\) and \(R\), we shall obtain various values of \(\mathbf{M}(P)\) and \(\mathbf{M}(R)\); we call respectively the inner and outer measure of the domain \(P\)

\[ \mathrm{oGr}\,\mathbf{M}(P)=\mathbf{M}_i(Q),\qquad \mathrm{uGr}\,\mathbf{M}(R)=\mathbf{M}_e(Q). \]

From (1) it follows that \(\mathrm{oGr}\,\mathbf{M}(P)<\mathrm{uGr}\,\mathbf{M}(R)\), i.e. for every domain

\[ \mathbf{M}_i(Q)<\mathbf{M}_e(Q). \tag{2} \]

In the case when both these limits coincide, their common value is defined as the measure of the open domain

\[ \operatorname{uGr} M(P)=M(Q)=\operatorname{oGr} M(R), \]

and domains possessing this property are called measurable.

There now arises the question: by what considerations should one be guided in constructing the domains \(P\) and \(R\)? Since each of them is, first of all, a primary domain, it is necessary to investigate precisely which domains of this kind must be used for constructing the inner domain \(P\), and which for the domain \(R\). Let us begin with the domain \(P\), which, being primary, must enter into the composition of \(Q\).

The domain \(Q\), generally speaking, does not contain solid intervals; consequently, for \(P\) one may take only either closed domains that are parts of \(Q\), or the exterior points of such domains; the measure of both the one and the other—according to theorem B—is the same, but closed domains, as richer in points, differ less from the domain \(Q\); therefore it is natural to take, as the domains \(P\) contained in \(Q\), closed domains. This decision does not impair generality also in the case when the domain \(Q\) is mixed, i.e. consists, first, of a series of solid intervals and, secondly, of a point part situated on another series of intervals; in this case closed domains are the most convenient and, in their composition, the closest to the domain \(Q\); these closed domains will here consist, on the one hand, of a finite series of closed intervals and, on the other, of the sum of a finite number of closed domains, each situated on one of the intervals complementary with respect to the first intervals.

For the construction of the exterior domain \(R\) we have two possibilities: we may take the domain \(\overline{Q}\) obtained from the closure of \(Q\), or the exterior points of such a domain, if \(Q\) contains none of the boundaries of the free intervals, whereby the measures of both such domains are equal to one another; or we may construct around the points of \(Q\) a series of intervals \(\{\lambda_i\}\). The measure of the domain \(Q\) is not subject to change, and consequently its lower limit is \(M(Q)\) itself; as for the measure of the series of intervals, by choosing the latter in one way or another we may considerably vary its value. Therefore it is necessary for us to compare \(M(\overline{Q})\) and the lower limit \(\sum \lambda_i\).

As we saw in 51°, the intervals \(\lambda_i\) can, in certain cases, be chosen in such a way that from them there will successively be sifted out not only isolated points \(\lambda_i\) not belonging to \(Q\), but even limit points of \(Q\) not contained in \(Q\); in some cases the latter points cannot be sifted out, but the former can be excluded in any case. Thus, \(\operatorname{uGr}\sum \lambda_i\), while containing no points foreign to \(\overline Q\), may—in certain cases—not contain even the limit points missing from \(Q\); thus it is clear that \(\operatorname{uGr}\sum \lambda_i\), as a measure, in comparison with \(Q\), differing in its composition from \(Q\), must be taken as the outer measure of the domain. That such a decision has serious grounds will be shown by even the example of a countable everywhere dense domain; as is known, for it the intervals \(\{\lambda_i\}\) can be constructed so that \(\sum \lambda_i\) is arbitrarily small, and consequently \(\operatorname{uGr}\sum \lambda_i=0\), whereas \(\overline Q\) here coincides with the fundamental interval, and \(M(\overline Q)=l\). Thus:

The inner measure of an open domain \(Q\) is defined as the upper bound of the measure of closed domains contained in \(Q\), and the outer measure—as the lower bound of the measure of a sequence of intervals that contain the points of \(Q\).

172. From this definition there immediately follows the following consequence:

“If \(M_i(P)=a,\ M_e(P)=b\), then the domain \(P\) contains among its components closed parts with measure greater than \(a-\varepsilon\), and can be included in a sequence of intervals with measure less than \(b+\varepsilon'\), where \(\varepsilon,\varepsilon'\) are arbitrarily small positive numbers.”

From the same definition it follows further that for a closed domain \(S\) one must have \(M_i(S)=M(S)\), and—on the basis of theorems F and G 169°—also \(M_e(S)=M(S)\).

In exactly the same way, for a domain \(\Gamma\) consisting of the points of a sequence of closed intervals \(\{\lambda_i\}\), \(M_e(\Gamma)=M(\Gamma)\); on the other hand, the points of the sequence of intervals \(\sum_1^n \lambda_i\) give a closed domain, and \(\operatorname{oGr}\sum_1^n \lambda_i=M(\Gamma)\), so that \(M_i(\Gamma)=M(\Gamma)\).

Taking into account consideration 167°, we then find that also for the domain \(\Gamma\) determined by an arbitrary series of intervals,

\[ \mathbf{M}_{i}(\Gamma)=\mathbf{M}(\Gamma); \]

it is likewise obvious that here

\[ \mathbf{M}_{e}(\Gamma)=\mathbf{M}(\Gamma). \]

From the definition of the outer measure it follows further that, for an ordinary inner limiting domain \(G_{0}\), given by domains \(G_{n}\), we have

\[ \mathbf{M}_{e}(G_{0})=\underset{n\ldots\infty}{\mathrm{uGr}}\sum \lambda_{i}^{(n)} =\lim \mathbf{M}(G_{n}), \tag{3} \]

where the intervals \(\lambda_{i}^{(n)}\) must be those to which the domain \(G_{0}\) owes its origin, since otherwise a domain richer in points may appear which, by virtue of theorem L, may have a greater lower bound.

From the definition of the inner measure it follows that, for an ordinary outer limiting domain \(G\), given by closed domains \(G_{n}\),

\[ \mathbf{M}_{i}(G)=\underset{n\ldots\infty}{\mathrm{oGr}}\mathbf{M}(G_{n}) =\lim \mathbf{M}(G_{n}). \tag{4} \]

Next we need a lemma\(^1\) and Young’s theorem\(^2\):

Lemma. “If \(\Lambda^{(n)}=\{\,l_{i}^{(n)}\,\}_{i=1}^{i=m_{n}}\), for arbitrarily increasing \(n\), is a sequence of systems of intervals with finite \(m_{n}\), and

\[ \Lambda^{(n+1)}=\mathrm{D}\bigl(\Lambda^{(n)}\bigr), \]

and

\[ \underset{n\ldots\infty}{\mathrm{uGr}}\sum_{i=1}^{m_{n}} l_{i}^{(n)}=g>0, \]

then there exists a closed domain

\[ \Lambda_{0}=\prod_{1}^{\infty}\mathrm{D}\bigl(\Lambda^{(n)}\bigr), \]

for which

\[ \mathbf{M}(\Lambda_{0})=g.” \]

M. “If \(\Lambda^{(n)}=\{\,l_{i}^{(n)}\,\}\) is a sequence of systems of intervals, with

\[ \Lambda^{(n+1)}=\mathrm{D}\bigl(\Lambda^{(n)}\bigr),\qquad \underset{n\ldots\infty}{\mathrm{uGr}}\sum_{i=1}^{\infty} l_{i}^{(n)}=g>0, \]

then there exists a domain

\[ G_{0}=\prod_{1}^{\infty}\mathrm{D}\bigl(\Lambda^{(n)}\bigr), \]

for which

\[ \mathbf{M}_{i}(G_{0})=g.” \]

From theorem M in connection with (3) it follows that, for an ordinary inner limiting domain,

\[ \mathbf{M}_{i}(G_{0})=\mathbf{M}_{e}(G), \]

i.e.

N. “An ordinary inner limiting domain is measurable.”

\(^1\) Proc. (2) 2, p. 18, theorems 1 and 2.
\(^2\) ib. p. 19, theorem 3.

Then Young proves¹ that for an ordinary outer limiting set \(M_e(G)=\lim M(G_n)\); and this, in connection with (4), gives the conclusion that also

O. “An ordinary outer limiting set is measurable.”

Since the set of outer points \(P\) of a sequence of closed intervals is² a special case of an ordinary inner limiting set, for it—by virtue of theorem O—we shall have \(M_e(P)=M_i(P)\); since, as we likewise had in 172° for closed sets and for sets of continuous intervals, we may assert that

P. “For primary sets the inner and outer measures coincide and are equal to the measure of the set.”

It is not hard to see that, in addition to theorem L, the theorem may be given

Q. “If \(G\) and \(H\) are arbitrary sets, and \(G \subseteq D(H)\), then

\[ M_i(G)\leq M_i(H),\quad M_e(G)\leq M_e(H). \]

This follows from the fact that for \(G\) and \(H\) the corresponding primary sets will constitute, the former—a part of the latter; and from this it follows immediately that

R. “If \(G\) and \(H\) are measurable sets, and \(G \subseteq D(H)\), then

\[ M(G)\leq M(H). \]

In particular, we may further assert that

“Every countable set \(G\) is measurable, and its measure is equal to \(0\),”

since for it \(M_e(G)=0\); finally,

“The set of irrational fractions is measurable, and its measure is equal to \(1\),”

which follows from the fact that these fractions form an ordinary inner limiting set for a sequence of intervals whose sum is equal to 1.

173. The definition of \(M_i(P)\) and \(M_e(P)\), the inner and outer measure, given above, was given independently of each other by Lebesgue and Young³; that both definitions are identical for outer measure is immediately clear; as for \(M_i(P)\),⁴ the inclusion of an additional set \(\Pi\) in the sequence of intervals \(\{\lambda_i\}\) determines, on the basic

interval \(L\) a closed domain \(E\), consisting exclusively of points \(P\); with all possible choices of \(\{\lambda_i\}\), made with the aim of finding \(\operatorname{uGr}\sum \lambda_i\), we shall obtain new closed parts of \(P\), more and more rich in points and therefore—in virtue of theorem L—with nondecreasing measure; since \(M(E)=l-\sum \lambda_i\), then

\[ l-\operatorname{uGr}\sum \lambda_i=\operatorname{oGr} M(E), \]

and this is precisely \(M_i(P)\).

174. Using the notion of inner measure, and having in view D and E \(160^\circ\), we can state here¹ two theorems concerning closed domains:

S. “If \(G_1\) and \(G_2\) are closed, then

\[ M(G_1)+M(G_2)=M\bigl[D(G_1,G_2)\bigr]+M\bigl[M(G_1,G_2)\bigr]”. \]

T. “If \(G_{n+1}=D(G_n)\), and all \(G_n\) are closed, then

\[ M\left[D\left(\begin{array}{c} \infty\\ 1 \end{array}G_n\right)\right]=\lim M(G_n)”. \]

Dealing with arbitrary domains, we may assert that

U. “If \(G=G_1+G_2\), then

\[ M_i(G_1+G_2)\geq M_i(G_1)+M_i(G_2), \]

\[ M_e(G_1+G_2)\leq M_e(G_1)+M_e(G_2)”. \]

Let

\[ M_i(G_1)=a_1,\quad M_i(G_2)=a_2,\quad M_e(G_1)=b_1,\quad M_e(G_2)=b_2. \]

a. Suppose

\[ M_i(G)=a<a_1+a_2 \tag{1} \]

and call \(a_1+a_2-a=\alpha\); assume first that \(G\) is closed; having in view corollary \(172^\circ\), we can choose such closed divisors \(\bar G_1\) and \(\bar G_2\) of the domains \(G_1\) and \(G_2\), that

\[ M(\bar G_1)=a_1-\varepsilon_1,\quad M(\bar G_2)=a_2-\varepsilon_2, \tag{2} \]

where each of the numbers \(\varepsilon_1\) and \(\varepsilon_2\) is less than \(\dfrac{\alpha}{2}\); in such a case, for the closed domain \(\bar G_1+\bar G_2\), which is a divisor of \(G_1+G_2=G\), we—in virtue of theorem L—have

\[ \mathbf{M}(G_1)+\mathbf{M}(G_2)=\mathbf{M}(G_1+G_2)<\mathbf{M}(G) \]

or

\[ a_1+a_2-(\varepsilon_1+\varepsilon_2)<a=a_1+a_2-\alpha, \]

whence it follows that \(\varepsilon_1+\varepsilon_2>\alpha\), contrary to the assumption.

Let now \(G\) be not closed; take \(\overline{G}\)—a closed extender of \(G\) with measure \(a-\varepsilon\), and now take each of the numbers \(\varepsilon,\varepsilon_1,\varepsilon_2\) to be less than \(\dfrac{\alpha}{4}\).

According to Theorem D 160°, \(D(\overline{G},\overline{G}_1)=\overline{G}_1\), \(D(\overline{G},\overline{G}_2)=\overline{G}_2\), and also \(\overline{G}_1+\overline{G}_2\) will be closed; on the basis of Theorem S—

\[ \mathbf{M}\bigl[D(\overline{G},\overline{G}_1)\bigr] +\mathbf{M}\bigl[M(\overline{G},\overline{G}_1)\bigr] = \mathbf{M}(\overline{G})+\mathbf{M}(\overline{G}_1); \tag{3} \]

but, obviously, \(M(\overline{G},\overline{G}_1)\), being a closed extender of \(G\), has measure not exceeding \(a\); therefore (3) gives us

\[ \mathbf{M}(\overline{G}_1)+a>a-\varepsilon+a_1-\varepsilon_1,\quad \mathbf{M}(\overline{G}_1)>a_1-(\varepsilon+\varepsilon_1); \]

in exactly the same way

\[ \mathbf{M}(\overline{G}_2)>a_2-(\varepsilon+\varepsilon_2), \]

whence

\[ \mathbf{M}(\overline{G}_1+\overline{G}_2) = \mathbf{M}(\overline{G}_1)+\mathbf{M}(\overline{G}_2) > a_1+a_2-(2\varepsilon+\varepsilon_1+\varepsilon_2); \tag{4} \]

on the other hand, \(\overline{G}_1+\overline{G}_2\) constitutes a closed extender of \(G\); therefore

\[ \mathbf{M}(\overline{G}_1+\overline{G}_2)\leq a; \tag{5} \]

from (4) and (5) we have

\[ a_1+a_2-(2\varepsilon+\varepsilon_1+\varepsilon_2)<a, \]

whence \(2\varepsilon+\varepsilon_1+\varepsilon_2>\alpha\), contrary to the assumption.

Thus, in both cases, the impossibility of inequality (1) has been proved.

b. Suppose that

\[ b=\mathbf{M}_e(G)>\mathbf{M}_e(G_1)+\mathbf{M}_e(G_2)=b_1+b_2, \]

and call \(b-(b_1+b_2)=\beta\); include the points of each of \(G_1\) and \(G_2\) in an infinite sequence of intervals with measures respectively equal to \(b_1+\eta_1,\ b_2+\eta_2\), where each of the numbers \(\eta_1\) and \(\eta_2\) is taken

less than \(\dfrac{\varepsilon}{4}\); then the domain \(G\) will be contained in a series of intervals whose sum will not exceed \(b_1+b_2+\eta_1+\eta_2\); thus it follows that

\[ b=\mathbf{M}_e(G)\leq b_1+b_2+\eta_1+\eta_2<b_1+b_2+\frac{\varepsilon}{2}, \]

which is contrary to the assumption.

175. Passing further to the very important question concerning the addition of measures, we shall cite here a series of Young’s theorems:

V. “If \(G \equiv G_1+G_2\), and \(G_1\) is closed, then

\[ \mathbf{M}_i(G)=\mathbf{M}(G_1)+\mathbf{M}_i(G_2),\quad \mathbf{M}_e(G)=\mathbf{M}(G_1)+\mathbf{M}_e(G_2) \]”\(^{1}\).

W. “If \(G_{n+1}\equiv \mathbf{D}(G_n)\), then
\[ \mathbf{M}_i\left[\prod_1^\infty \mathbf{D}(G_n)\right]=\lim \mathbf{M}_i(G_n) \]”\(^{2}\).

X. “If \(G_{n+1}\equiv \mathbf{M}(G_n)\), then
\[ \mathbf{M}_e\left[\sum_1^\infty \mathbf{M}(G_n)\right]=\lim \mathbf{M}_e(G_n) \]”\(^{3}\).

From the last two theorems it follows\(^{4}\)

Y. “The inner and outer limiting domain for a series of measurable domains \(\{G_n\}\) is measurable and has measure \(\lim \mathbf{M}(G_n)\).”

Since the domains \(G_n\) are measurable, let us put

\[ \mathbf{M}_i(G_n)=\mathbf{M}_e(G_n)=\mathbf{M}(G_n)=a_n. \tag{1} \]

a. For the inner limiting domain
\[ G_0\equiv \prod_1^\infty \mathbf{D}(G_n) \]
—according to theorem W—

\[ a=\mathbf{M}_i(G_0)=\lim \mathbf{M}_i(G_n)=\lim a_n; \]

the numbers \(a_n\), by theorem R, do not increase, and consequently \(a\leq a_n\); in this case, for arbitrarily small \(\varepsilon\), there can be determined a number \(\mu\) such that

\[ a_n<a+\frac{\varepsilon}{2},\qquad n>\mu; \tag{2} \]

the domain \(G_n\), being measurable, can be included—by definition—in a series of intervals with measure \(a_n+\dfrac{\varepsilon}{2}<a+\varepsilon\); since

\(^{1}\) Proc. (2) 2, p. 24, lemma 2; pp. 28–29, 9°–10°; p. 44, 18°; p. 46, 19°
\(^{2}\) ib. pp. 25–26, theorem 3′; p. 28, theorem 6.
\(^{3}\) ib. p. 44, theorem 6′.
\(^{4}\) ib. p. 45, 18°, corollary.

\(G\) is contained in \(G_n\), then the points of \(G\) are also included in a series of intervals with measure less than \(a+\varepsilon\), with arbitrarily small \(\varepsilon\), so that
\[ \mathrm{M}_e(G_0)=a=\mathrm{M}_i(G_0). \]

b. For the outer limiting domain
\[ G=\underset{1}{\overset{\infty}{M}}(G_n) \]
— according to Theorem X —
\[ b=\mathrm{M}_e(G)=\lim \mathrm{M}_e(G)=\lim b_n, \]
where the numbers \(b_n\) do not decrease; in this case \(b \geq b_n\), and consequently, for sufficiently large \(n\), \(b-b_n<\dfrac{\varepsilon}{2}\), or
\[ b_n>b-\frac{\varepsilon}{2}, \qquad n>\nu. \tag{3} \]

Since the domains \(G_n\) are measurable, closed domains of measure greater than \(b_n-\dfrac{\varepsilon}{2}\) are possible within \(G_n\); and in this case, in view of (3), closed domains of measure greater than \(b-\varepsilon\) are possible within \(G\); but \(\varepsilon\) is arbitrarily small, consequently
\[ \mathrm{M}_i(G)=b=\mathrm{M}_e(G). \]

176. From Theorem V we see that a closed domain, when added to an arbitrary domain, gives a domain whose inner and outer measures are equal to the sums of the corresponding measures; on the other hand, from Theorem U it follows that the sum of two arbitrary domains does not have this property, or—at least—we do not yet have data to assert this; thus, for a closed domain the characteristic circumstance is that, in combination with an arbitrary domain, it “possesses the theorem of inner and outer addition.”

Hence the question arises whether only a closed domain has this property, or whether, on the contrary, it belongs, if not to all, then at least to some open domains; Young proves that it is precisely the second supposition that holds.

The classes of domains possessing the theorem of inner and outer addition Young calls, respectively, the inner and outer additive classes; to this class, as we have just seen, belong closed domains; we are now in a position to extend these classes, bearing in mind the following theorem of Young’s¹:

I. “To the inner and outer additive classes belong both the inner and the outer limiting regions of regions of the class.”

From this theorem it follows,¹ further:

II. “To both additive classes belong the ordinary outer and inner limiting regions.”

As a special case of an ordinary outer limiting region, to the additive classes belongs a sequence of closed continuous intervals; on this basis the second part of Theorem II follows from its first part.

III. “For a generalized outer limiting region \(G\), if the regions \(G_n\) determining it belong to the inner additive class,

\[ \mathbf{M}_i(G)=\lim \mathbf{M}_i(G_n)^2.” \]

From Theorem I, analogously to II, the following theorem may be derived:

IV. “For a generalized inner limiting region \(G\), if the regions \(G_n\) determining it belong to the outer additive class,

\[ \mathbf{M}_e(G)=\lim \mathbf{M}_e(G_n).” \]

By virtue of the theorems D, E, F 160², to both additive classes belong, in addition to the inner and outer limiting regions of regions of the class, also the difference of closed regions, the difference of ordinary inner and outer limiting regions and closed regions, the difference of ordinary inner and outer limiting regions, or conversely; moreover³

V. “To both classes belong

\[ G_1+G_2,\quad G_1-G_2,\quad D(G_1,G_2),\quad M(G_1,G_2), \tag{1} \]

if \(G_1, G_2\) belong to the corresponding classes.”

It is clear that in the first two cases of (1) it is assumed, respectively, that

\[ D(G_1,G_2)=0,\quad G_2-D(G_1). \]

Thus it turns out that all regions obtained by the ordinary processes belong simultaneously both to the inner and to the outer additive classes, so that these regions form one single additive class, characterized by the fact

¹ ib. p. 32 and p. 47, corollary 1 to Theorems 8 and 8′.
² ib. p. 32, corollary 2.
³ ib. pp. 33–34, p. 47, Theorems 10, 10′, 11, 11′, 13, 13′.

under the condition that all the domains included in it possess the theorems of both internal and external composition. Young proves two further theorems.

VI. “If a domain \(G\) of an internally or externally additive class is divided into two domains \(G_1\) and \(G_2\) under the condition

\[ \mathbf{M}_i(G)=\mathbf{M}_i(G_1)+\mathbf{M}_i(G_2),\quad \mathbf{M}_e(G)=\mathbf{M}_e(G_1)+\mathbf{M}_e(G_2) \]

respectively, then \(G_1\) and \(G_2\) belong to the corresponding class”\(^{1}\).

VII. “For domains of internally and externally additive classes, respectively,

\[ \mathbf{M}_i(G_1)+\mathbf{M}_i(G_2) = \mathbf{M}_i\bigl[D(G_1,G_2)\bigr] + \mathbf{M}_i\bigl[M(G_1,G_2)\bigr], \]

\[ \mathbf{M}_e(G_1)+\mathbf{M}_e(G_2) = \mathbf{M}_e\bigl[D(G_1,G_2)\bigr] + \mathbf{M}_e\bigl[M(G_1,G_2)\bigr].”^{2} \]

177. In the case where \(\mathbf{M}_i(P)=\mathbf{M}_e(P)\), the domain is called\(^{3}\) measurable. If \(\mathbf{M}_i(P)=a\), then, in order that \(P\) be measurable, \(\mathbf{M}_e(P)\) must also be equal to \(a\), and consequently for \(P\) it must be possible to include it in a series of intervals whose sum is less than \(a+\varepsilon\) for arbitrarily small \(\varepsilon\). If \(\mathbf{M}_e(P)=b\), then, in order that \(P\) be measurable, \(\mathbf{M}_i(P)\) must also be equal to \(b\), and consequently \(P\) must contain closed terms with measure greater than \(b-\varepsilon\)\(^{4}\). Bearing this in mind, one can establish\(^{5}\) that

VIII. “Every domain of an internally or externally additive class is measurable,”

whence—by virtue of 176°—there follows

Fundamental theorem. “Domains of an additive class are measurable.”

This theorem of Young’s is of exceptional importance; it shows the full value of studying domains from the point of view of their possession of the theorem of internal and external composition; it follows from it that among measurable domains there belong\(^{6}\):

1) the domain of external and non-internal points of a series of intervals;

2) the domain of points of a series of intervals, closed or open;

3) generalized internal and external limiting domains;

4) internal and external limiting domains of a series of additive domains;

\(^{1}\) ib., p. 33, theorem 12; p. 47, theorem 12′.
\(^{2}\) ib., p. 34, theorem 13; pp. 47–48, theorem 13′.
\(^{3}\) See (“T.”), 233, 171°.
\(^{4}\) See 172°.
\(^{5}\) Proc. (2) 2, p. 42, theorem 20; pp. 43–44, theorem 21.
\(^{6}\) ib., pp. 48–49.

5) sums and differences of additive domains;

6) common divisors and least multiples of additive domains;

thus, starting from domains of a class and performing on them all the usual operations, we again obtain domains of the class, so that these domains form a corps¹.

If we could assert that there are no open domains which could not be obtained by the processes indicated above, the extremely difficult question of the measurement of domains and of their classification would be settled once and for all; in order to meet it, Young gives the theorem²:

IX. “Every domain with inner measure \(a\) is either a closed domain, or an ordinary exterior limiting domain, or the sum of one or another domain and an open domain with inner measure equal to zero.”

It follows from this that, if we could prove that these last open domains belong to the inner additive class, then the problem of measurement would be exhausted, since then—in virtue of Theorems IX and V—every open domain would possess an inner decomposition theorem, and—in virtue of Theorem VIII—would be measurable.

Since every countable domain may be regarded as an ordinary exterior limiting domain, the preceding problem can concern only an open inner-null³ uncountable domain; further—every uncountable domain \(P\) may be represented in the form \(R + D(S)\), where \(S\) is perfect and \(R\) is a countable domain lying in the free intervals of \(S\); therefore the question will stand only for \(D(S)\), i.e. for the uncountable open inner-null complement of the perfect domain \(S\). Then the domain \(D(S)\) may be decomposed into two parts \(D(E) + D(Q)\), where \(E\) is the domain of the exterior points of a set of disjoint intervals, and \(Q\) is the domain of their endpoints; since \(Q\) and consequently \(D(Q)\) are countable, \(D(E) + D(Q)\) possesses an inner decomposition theorem if \(D(E)\) possesses it, while, as we saw in A 158°, \(E\) is an ordinary inner limiting domain; thus, study requires \(D(E) \ne E\), i.e. an uncountable inner-null complement of the domain of the exterior points of a set of disjoint intervals.

If \(\Delta(E)\) is a supplementary domain for \(D(E)\), then \(D(E)\cdot E = \Delta(E)\); since \(E\) belongs to the inner additive class, then—in virtue

of Theorem V—then \(D(E)\) will also enter there, if \(\Delta(E)\) belongs to this class; thus the investigation of \(D(E)\) can be reduced to the study of \(\Delta(E)\); in particular, if \(\Delta(E)\) is countable, then it, and consequently also \(D(E)\), will belong to the inner additive class; therefore we may suppose \(\Delta(E)\) to be uncountable.

Thus, for the derivative \(D(E)\) of the preceding case, for a domain of points of highest order of a sequence of isolated intervals, there is still the condition that its complementary domain \(\Delta(E)\) must be uncountable; this derivative \(D(E)\) we shall call an elementary domain.

If such an elementary domain can always be obtained by one of the processes indicated above, then the question of the measure of a domain is finally solved: all domains will turn out to be measurable; therefore what now stands next in order is only the investigation of whether the elementary domain indicated above belongs to the inner additive class or not.

LITERATURE.

Abbreviations: A. d. L.—Atti della R. Accademia dei Lincei; A. d. M.—Annali di Matematica; A. Éc. N.—Annales de l’École Normale; A. M.—Acta Mathematica; Am. J.—American Journal of Mathematics; B. A. M. S.—Bulletin of the American Mathematical Society; B. M.—Bibliotheca Mathematica; C. R.—Comptes Rendus de l’Académie des Sciences de Paris; Cr. J.—Journal für reine und angewandte Mathematik; G. d. M.—Giornale di Matematiche; G. N.—Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen; J. d. M.—Journal de Mathématiques; J. d. M. V.—Jahresbericht der Deutschen Mathematiker-Vereinigung; L. B.—Berichte über die Verhandlungen der K. S. Gesellschaft der Wissenschaften zu Leipzig; M. A.—Mathematische Annalen; M. B.—Sitzungsberichte der mathematisch-physikalischen Classe der K. B. Akademie der Wissenschaften zu München; M. of M.—The Messenger of Mathematics; P. L. M. S.—Proceedings of the London Mathematical Society; Ph. M.—Philosophical Magazine; Quat. J.—Quaterly Journal of Mathematics; R. Ist. L.—Rendiconti del Istituto Lombardo; R. P.—Rendiconti del Circolo Matematico di Palermo; W. B.—Sitzungsberichte der K. Akademie der Wissenschaften in Wien; Z. f. M.—Zeitschrift für Mathematik.

1. G. Galilei. Discorsi e dimostrazioni matematiche.
     [Unterredungen und mathematische Demonstrationen über zwei
     neue Wissenszweige, die Mechanik und Fallgesetze betreffend.
     Ostwald’s Klassiker, No. 11, 24, 25. 1890–1891].

2. Bolzano. Rein analytischer Beweis des Lehrsatzes, dass zwischen
     je zwei Werthen, die ein entgegengesetztes Resultat gewähren,
     wenigstens eine reelle Wurzel der Gleichung liege.

3. Bolzano. Paradoxien des Unendlichen.

4. B. Riemann. Ueber die Darstellbarkeit einer Function durch eine
     trigonometrische Reihe.

5. G. Cantor. Ueber die einfachen Zahlensysteme.
     Z. f. M. 14, 121.　　　　　　　　　　　　　　　　　　　　　　　　　*

6. G. Cantor. Ueber trigonometrische Reihen.
     M. A. 4, 139; A. M. 2, 329.　　　　　　　　　　　　　　　　　　　*

Hankel. Untersuchungen über die unendlich oft oscillirenden und
  unstetigen Functionen.
  M. A. 20, 61.

1. Dedekind. Continuity and irrational numbers.

G. Cantor. On the extension of a theorem from the theory of trigonometric series.
M. A. 5, 123; A. M. 2, 395.

Heine. The elements of function theory.
Cr. J. 74, 172.

1. G. Cantor. On a property of the totality of all real algebraic numbers.
   Cr. J. 77, 258; A. M. 2, 305.

2. Ascoli. On the concept of the definite integral.
   A. d. L. (2) 2, 862

P. du Bois-Reymond. Attempt at a classification of arbitrary functions of real arguments according to their changes in the smallest intervals.
Cr. J. 79, 21. *

Darboux. Memoir on discontinuous functions.
A. Éc. N. (2) 4, 57. *

Smith. On the integration of discontinuous functions.
P. L. M. S. 6, 140.

1. G. Cantor. A contribution to the theory of manifolds.
   Cr. J. 84, 242; A. M. 2, 311.

2. Ascoli. New researches on Fourier series.
   A. d. L. (3) 2, 384.

Dini. Foundations for the theory of functions of real variables.
[Foundations for a theory of functions of a variable real quantity. 1892].

1879–1882. G. Cantor. On infinite linear point-manifolds.
M. A. 15, 1; 17, 355; 20, 113.
A. M. 2, 349, 357, 361.

1. P. du Bois-Reymond. The proof of the fundamental theorem of integral calculus.
   M. A. 16, 115.

G. Cantor. Remark on trigonometric series.
M. A. 16, 113. *

Pincherle. Essay on an introduction to the theory of analytic functions according to the principles of Prof. C. Weierstrass.
G. d. M. 18, 178, 317.

1881 Volterra. Some observations on discontinuous dotted functions.
G. d. M. 19, 75.

On the principles of integral calculus.
G. d. M. 19, 337. *

1. P. du Bois-Reymond. The general theory of functions.

G. Cantor. On a new and general condensation principle for the singularities of functions.
M. A. 19, 538. *

Harnack. Simplification of the proofs in the theory of Fourier series.
M. A. 19, 235.

Veltmann. The Fourier series.
Z. f. M. 27, 193.

On the arrangement of infinitely many singularities of a function.
Z. f. M. 27, 176.

On the theory of point sets.
Z. f. M. 27, 313.

1. Ascoli. The limiting curves of a given family of curves.
   A. d. L. (3) 18, 521.

Bendixson. Some theorems of the theory of sets of points.
A. M. 2, 415.

G. Cantor. Foundation of a general theory of sets.
A. M. 2, 381; M. A. 21, 545.

On various theorems of the theory of sets of points situated in a continuous space of \(N\) dimensions.
A. M. 2, 409.

1883–1884. G. Cantor. On infinite linear point manifolds.
M. A. 21, 51, 545; 23, 548.
A. M. 2, 372, 381.

1. G. Cantor. On the power of perfect sets of points.
   A. M. 4, 381.

Harnack. The general theorems on the relation between functions of a real variable and their derivatives. Part 2.
M. A. 24, 217.

Note on the mapping of a continuous linear manifold onto a discontinuous one.
M. A. 23, 235.

Mittag-Leffler. On the analytic representation of monogenic uniform functions of one independent variable.
A. M. 4, 1.

Phragmén. Proof of a theorem from the theory of manifolds.
A. M. 5, 17.

1. Scheeffer. Zur Theorie der stetigen Funktionen einer reellen Veränderlichen.
   A. M. 5, 183, 279.

Stolz. Ueber einen zu einer unendlichen Punktmenge gehörigen Grenzwerth.
M. A. 23, 157.

1. G. Cantor. Ueber verschiedene Theoreme aus der Theorie der Punktmengen in einem \(n\)-fach ausgedehnten stetigen Raume.
   A. M. 7, 105.

Harnack. Ueber den Inhalt der Punktmengen.
M. A. 25, 241.

Phragmen. Ueber die Begrenzungen von Continua.
A. M. 7, 43. *

1. Dedekind. Was sind und was sollen die Zahlen?

Pasch. Ueber einige Punkte der Functionentheorie.
M. A. 30, 132.

Peano. Applicazioni geometriche del calcolo infinitesimale.

1887–1888. G. Cantor. Mittheilungen zur Lehre vom Transfiniten.
Zeitschrift für Philosophie, 91, 92.

1. Ascoli. Riassunto della memoria „Le curve limite di una varietà data di curve“ ed osservazioni critiche alla medesima.
   R. Ist. L. (2) 21, passim.

H. C. Schwarz. Ein Beitrag zur Theorie der Ordnungstypen. *

Stolz. Ueber zwei Arten von unendlich kleinen und von unendlich grossen Grössen.
M. A. 31, 601.

1. Arzelà. Funzioni di linee.
   A. d. L. (4) 5, 312.

W. F. Meyer. Zur Lehre vom Unendlichen. *

1. Veronese. Fondamenti di geometria a più dimensioni e a più specie di unità rettilinee.

1891–1892. Hurwitz. Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche.
M. A. 39, 279.

1. G. Cantor. Ueber eine elementare Frage der Mannigfaltigkeitslehre.
   J. d. M. V. 1, 75.

C. Jordan. Cours d’analyse.
Deux éd., t. I.
Remarques sur les intégrales définies.
J. d. M. (4) 8, 69.

1. Vivanti. Historical note on the theory of sets.
     B. M. (2) 6, 9. *

2. Borel. On some points in the theory of functions.
     A. Éc. N. (3) 12, 9.

Levi-Civita. On infinities and actual infinitesimals as analytic elements.
  Atti d. R. Istituto Veneto. *

1895–1897. G. Cantor. Contributions to the foundation of transfinite set theory.
  M. A. 46, 481; 49, 207.

1. Couturat. On mathematical infinity.

Osgood. On non-uniform convergence and termwise integration of series.
  G. N., 333.

Schoenflies. On the mapping of cubes of different dimensions onto one another.
  G. N., 255. *

Veronese. On some observations concerning infinite segments and actual infinitesimals.
  M. A. 47, 421.

1. H. Burkhardt. Lectures on the theory of functions.

Killing. On transfinite numbers.
  M. A. 48, 425. *

Osgood. Non-uniform convergence and the integration of series term by term.
  Am. J. 19, 155.

Stolz. Two limiting values, of which the upper integral is a special case.
  W. B. 106, 2ᵃ, 153.

1. Borel. Lectures on the theory of functions.

V. L. Nekrasov. Toward the theory of functions of a real variable.
  Proceedings of the X Congress of Naturalists and Physicians, 427.

1. Baire. On the theory of sets.
     C. R. 129, 946. *

On functions of real variables.
  A. d. M. (3) 3, 1.

Borel. Concerning the new infinity.
  Revue Philosophique, 382.

Pringsheim. On the theory of double integrals and Green’s and Cauchy’s integral theorems.
  M. B. 29, 39.

1. Schoenflies. Mengenlehre.

Encyklopädie d. Math. W., 1, 184.

On the distribution of points of continuity and discontinuity for pointwise discontinuous functions of a real variable.

G. N., 187. *

On a theorem of analysis of S. Sittus.

G. N., 232. *

1. Baire. New proof of a theorem on discontinuous functions.

Bulletin de la Soc. Math. de France, 28, 173. *

Osgood. Second note on analytic functions of several variables.

M. A. 53, 461. *

Schoenflies. The development of the theory of point manifolds.

J. d. M. V. 8.

de Stefano. On some points of the theory of functions of real variables.

G. d. M. 38, 178.

1. Bernstein. Investigations in set theory.

[M. A. 61, 117. 1905.]

Hausdorff. On a certain kind of ordered sets.

L. B. 53, 460. *

Schoenflies. Transfinite numbers, the axiom of Archimedes, and projective geometry.

J. d. M. V. 5, 75. *

Schröder. On Cantorian theorems.

J. d. M. V. 5, 81. *

1902 Hardy. A theorem concerning the infinite cardinal numbers,

Quat. J. 35, 87. °

The cardinal number of a closed set of points.

M. of M. 33, 67. *

Jourdain. A general theorem on the transfinite cardinal numbers of aggregates of functions.

Ph. M. (6) 6, 323. *

The cardinal number of the aggregate of integrable functions.

M. of M. 33, 78. *

Lebesgue. Integral, length, area.

A. d. M. (3) 7, 231.

1. Levi. On the theory of aggregates.
     R. Ist. L. (2) 35, 86.

Whitehead. On cardinal numbers.
  Am. J. 24, 367. *

W. H. Young. On the analysis of linear sets of points.
  Quat. J. 35, 102.

On the density of linear sets of points.
  P. L. M. S. 34, 285.

On the division of discontinuous functions and the distribution of their points of continuity.
  W. B. 112, 1507.

Zermelo. On the addition of transfinite cardinal numbers.
  G. N. 1901, 34.

1. Borel. Contribution to the arithmetical analysis of the continuum.
     J. d. M. (5) 9, 329.

On the approximation of numbers by rational numbers.
  C. R. 136, 1654.

A theorem on measurable sets.
  C. R. 137, 966.

E. Lindelöf. On some points of set theory.
  C. R. 137, 697.

W. H. Young. A note on the condition of integrability of a function of a real variable.
  Quat. J. 35, 189.

A note on unclosed sets of points defined as the limit of a sequence of closed sets of points.
  P. L. M. S. 35, 283.

On closed sets of points defined as the limit of a sequence of closed sets of points.
  ib., 35, 269.

Overlapping intervals.
  ib., 35, 384.

Sets of intervals on the straight line.
  ib., 35, 243.

On the theory of non-closed point sets.
  L. B. 287.

1903–1906. Schoenflies. Contributions to the theory of point sets.
  M. A. 58, 195; 59, 129; 62, 286. *

1. Bernstein. Remark on set theory.
     G. N., 557. *

1. Hausdorff. Der Potenzbegriff in der Mengenlehre.
   J. d. M. V. 13, 569. *

Hobson. On inner limiting sets of points in a linear interval.
P. L. M. S. (2) 2, 316.

Jourdain. On the cardinal numbers of number classes in general.
Ph. M. (6) 7, 294. *

On the transfinite cardinal numbers of well-ordered aggregates.
ib. (6) 7, 61. *

On transfinite cardinal numbers of the exponential form.
ib. (6) 9, 42. *

Lebesgue. Leçons sur l’intégration et la recherche des fonctions primitives.

Schoenflies. Ueber den Beweis eines Haupttheorems aus der Theorie der Punktmengen.
G. N. 1903, 21.

Veblen. The Heine-Borel theorem.
B. A. M. S. (2) 10, 436.

W. H. Young. On an extension of the Heine Borel theorem.
M. of M. 33, 129.

On closed sets of points and Cantor’s numbers.
P. L. M. S. (2) 1, 230.

On sequences of sets of intervals containing a given set of points.
ib. (2) 1, 262.

Open sets and the theory of content.
ib. (2) 2, 16.

The tile theorem.
ib. (2) 2, 67.

The upper and lower integration.
ib. (2) 2, 52.

Zermelo. Beweis, dass jede Menge wohlgeordnet werden kann.
M. A. 59, 514. *

Zoretti. Sur les ensembles parfaits et les fonctions uniformes.
C. R. 138, 674. *

1. Baire. Leçons sur les fonctions discontinues.

Bernstein. Die Theorie der reellen Zahlen.
J. d. M. V. 14, 447.

Über die Reihe der transfiniten Ordnungszahlen.
M. A. 60, 187. *

1. Bernstein. On the continuum problem.
     ibid. 60, 463.
     On set theory.
     J. d. M. V. 14, 193.

Borel. Lessons on functions of real variables and expansions in series of polynomials.
  Some remarks on the principles of set theory.
  M. A. 60, 194.
  On a property of closed sets.
  C. R. 140, 298.

Faber. On the denumerability of the rational numbers.
  M. A. 60, 196.

Hobson. On the failure of convergence of Fourier’s series.
  P. L. M. S. (2) 3, 48.
  On the general theory of transfinite numbers and order types.
  ibid. (2) 3, 170.

Jourdain. On a proof that every aggregate can be well-ordered.
  M. A. 60, 465.
  On the general theory of functions.
  Cr. J. 128, 169.
  The definition of a series similarly ordered to the series of all ordinal numbers.
  M. of M. 35, 56.

Kasner. Galileo and the modern concept of infinity.
  B. A. M. S. (2) 11, 499.

König. On the foundations of set theory and the continuum problem.
  M. A. 61, 156.
  On the continuum problem.
  ibid. 60, 177.

Riesz. On discontinuous sets.
  C. R. 141, 650.
  On a theorem of Borel.
  ibid. 140, 224.
  On multiple order types.
  M. A. 61, 406.

Schoenflies. On well-ordered sets.
  ibid. 60, 181.

Vitali. On functions with zero integral.
  R. P. 20, 136.

1. W. H. Young. Linear content of a plane set of points.
     P. L. M. S (2) 3, 461.  °
     On a perfect plane set.
     M. of M. 34, 160.  
     Ordinary inner limiting set in the plan or higher space.
     P. L. M. S. (2) 3, 371.  *°

2. Dixon. On “well-ordered” aggregates.
     ib. (2) 4, 18.  *°

Hardy. The continuum and the second number class.
  ib. (2) 4, 10.  °

Hobson. On the arithmetic continuum.
  ib. (2) 4, 21.  °

Jourdain. The development of the theory of transfinite numbers.
  Archiv der Mathematik, (3) 10.  *°

Korselt. Paradoxes of set theory.
  J. d. M. V. 15, 215.  *
  On logic and set theory.
  ib. 15, 266.  *

Lennes. Note on the Heine-Borel theorem.
  B. A. M. S. 12, 395.

Russel. On some difficulties in the theory of transfinite numbers and order types.
  P. L. M. S. (2) 4, 29.  °

Schoenflies. The relations of set theory to geometry and function theory.
  J. d. M. V. 15, 557.  *
  On the logical paradoxes of set theory.
  ib. 15, 19.  *

Weber. Elementary set theory.
  ib. 15, 173.

1. Hessenberg. Powers of transfinite ordinal numbers.
     ib. 16, 130.  ;

1. ib. p. 55. ↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩↩

2. ib. p. 58; theorem 19. ↩↩↩↩↩↩↩↩↩↩↩↩↩↩

3. i.e. with inner measure equal to zero. ↩↩↩↩↩↩↩↩↩↩↩↩

4. See issue (6), p. 63. ↩↩↩↩↩↩

5. Pr. (2) 2. p. 318. ↩

6. Annali di Mat. (3)3, p. 66, and not 67, as Hobson indicates. ↩