Adherences and coherences of a linear point domain
Abstract Generated abstract
This note revisits Cantor’s notions of adherences and coherences for linear point domains, comparing them with W. H. Young’s treatment and with derived sets. Using the author’s notation and the concept of types of arrangement, it derives general relations between coherences of finite and transfinite order and corresponding derived sets, and shows how Young’s theorem on adherences follows directly from these relations. A sequence of constructed examples analyzes when coherences terminate, when a last coherence or nucleus exists, and how adherences may intersect higher derived sets, illustrating that arrangement types give a more systematic description of these cases.
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PROCEEDINGS
of the Tomsk Technological Institute
of Emperor Nicholas II.
Vol. 11. 1908. No. 3.
III.
V. L. Nekrasov.
ADHERENCES AND COHERENCES OF A LINEAR POINT DOMAIN
1–9.
ADHERENCES AND COHERENCES
OF A LINEAR POINT DOMAIN.
1. In the book “The Structure and Measure of a Linear Point Domain”1, in expounding G. Cantor’s doctrine of adherences and coherences, I noted that up to that time this doctrine had attracted no attention; at that time I did not have in my hands vol. XXXV of the Quarterly Journal of Mathematics, where an article by W. H. Young devoted to this question was published2; in it the author compares ad- and coherences with derived domains and gives a series of examples elucidating these rather complicated concepts.
In his exposition the author does not use Cantor’s notation, which considerably complicates the matter; as for the examples, the application of the concept of types of arrangement, as it is set forth in Chapter II of my book, will give, on the one hand, a fuller clarification of the structure of the domains cited by the author, and, on the other hand, will make it possible to give examples more convincing than those which were given by the author himself. I propose to do both in the present note, and this is of interest to me because here I have occasion to show what simplicity the use of types of arrangement brings into rather complicated questions.
2. An arbitrary, generally speaking, non-closed domain may be decomposed into two parts
\[ \mathrm{E} \equiv \mathrm{E}_u + \mathrm{E}_c, \]
of which the adherence \(\mathrm{E}_u\) consists of its isolated points, and the coherence \(\mathrm{E}_c\) contains all limiting points of \(\mathrm{E}\) that are contained in \(\mathrm{E}\); hence it follows immediately that \(\mathrm{E}_c\) is a part of \(\mathrm{E}'\)
\[ \mathrm{E}_c = \mathrm{D}(\mathrm{E}'). \tag{1} \]
Similarly, if \(\mathrm{E}_c\) is not a condensed domain,
\[ \mathrm{E}_c \equiv \mathrm{E}_{ca} + \mathrm{E}_{c^2}, \]
where \(E_{c2}\) comprises those limiting points \(E_c\) which are present in \(E_c'\); hence it is seen that \(E_{c2}=D(E_c')\); but, by virtue of (1), \(E_c'=D(E'')\); consequently
\[ E_{c2}=D(E''). \]
In exactly the same way, and in general, if
\[ \tag{2} E_{cn}=E_{cn_a}+E_{cn+1},\quad E_{cn}=D(E^{(n)}), \]
then
\[ E_{cn+1}=D(E_{cn}'),\quad E_{cn}=D(E^{n+1}), \]
whence it follows that
\[ \tag{3} E_{cn+1}=D(E^{(n+1)}). \]
On the other hand, according to the definition and \(2(2)\), one obtains
\[ \tag{4} E_{cn_a}=D(E_{cn})=D(E^{(n)}). \]
Thus one may assert that1
“every coherence of finite order of the domain \(E\) belongs to the derived set of the same order, and every adherence—to the preceding derived set.”
From (4) it follows that any adherence \(E_{cn_a}\) belongs to the corresponding derived set \(E^{(n)}\), but this does not exclude for it the possibility of having common points with \(E^{(n+1)}\), or even of being contained in it entirely, as we shall now see.
3. To explain the preceding, W. H. Young takes2 a domain of intervals of type \(\omega^3\); if we denote3
\[ Q^{(0)}=\{x_0',x_0\},\quad Q^{(1)}=\{x_i\},\quad Q^{(2)}=\{x_{ij}\},\quad Q^{(3)}=\{x_{ijk}\}, \]
then the three domains considered by the author will be
\[ \tag{5} G=Q^{(0)}+Q^{(1)}+Q^{(3)},\quad F=Q^{(0)}+Q^{(1)}+D_n(Q^{(2)})+Q^{(3)},\quad E=Q^{(0)}+Q^{(2)}+Q^{(3)}, \]
where \(D_n(Q^{(2)})\) denotes a certain part of \(Q^{(2)}\), consisting of a finite number \(n\) of points.
It is obvious that all the derived sets for \(G,F\), and \(E\) will be one and the same:
\[ G'\equiv F'\equiv E'\equiv \{x_{0i}\}+Q^{(1)}+Q^{(2)},\quad G''\equiv F''\equiv E''\equiv \{x_0\}+Q^{(1)}, \]
\[ G'''\equiv F'''\equiv E'''\equiv \{x_0\}. \]
A. For the first of the domains (5) we have
\[ G_a=\{x_0\}+Q^{(3)},\quad G_c=\{x_0\}+Q^{(1)},\quad G_{ca}=Q^{(1)},\quad G_{c^2}=\{x_0\}, \]
whereby from the comparison of \(G_{ca}\) and \(G''\) it follows that
\[ G_{ca}=D(G''). \]
B. For the domain \(F\)
\[ F_a=\{x_0\}+Q^{(3)},\quad F_c=\{x_0\}+Q^{(1)}+D_n(Q^{(2)}); \]
\[ F_{ca}=Q^{(1)}+D_n(Q^{(2)}),\quad F_{c^2}=\{x_0\}, \]
whence it is clear that
\[ D(F_{ca})=D(F''). \]
C. Finally, for \(E\)
\[ E_a=\{x_0\}+Q^{(3)},\quad E_c=\{x_0\}+Q^{(2)},\quad E_{ca}=Q^{(2)},\quad E_{c^2}=\{x_0\}, \]
and consequently1
\[ D\{E_{ca},E''\}=0. \]
D. The domain \(T_n\) mentioned by W. H. Young is a domain of type \(\omega^n\), and the composition of the adhesions and coherences obtained from it may be analyzed in an analogous manner.
Taking \(n=2m\) and denoting again by \(Q^{(0)}\) and \(Q^{(i)}\), for \(i=1,2,3,\ldots,2m\), respectively the boundaries of the fundamental interval and the boundaries of its \(i\)-th division, let us take, for example, a domain \(E\) of the following composition
\[ E=Q^{(0)}+Q^{(2)}+Q^{(4)}+\cdots+Q^{(2m)}, \]
i.e. we exclude the boundaries of all odd divisions; then, on the one hand,
\[ E'=\{x_1\}+\sum_{1}^{2m-1} Q^{(i)},\quad E''=\{x_0\}+\sum_{2}^{2m-2} Q^{(i)},\ldots, \]
\[ E^{(m)}=\{x_0\}+\sum_{1}^{m}Q^{(i)},\quad E^{(m+1)}=\{x_0\}+\sum_{1}^{m-1}Q^{(i)},\ldots,\quad E^{(2m)}=\{x_0\}, \]
and with the other
\[ \mathrm E_a=\{x_0\}+\mathrm Q^{(2m)},\quad \mathrm E_{ca}=\mathrm Q^{(2m-2)},\ldots,\quad \mathrm E_{c^{m-1}a}=\mathrm Q^{(2)},\quad \mathrm E_{c^m a}=\{x_0\}, \]
\[ \mathrm E_c=\{x_0\}+\sum_{1}^{m-1}\mathrm Q^{(2k)},\quad \mathrm E_{c^2}=\{x_0\}+\sum_{1}^{m-2}\mathrm Q^{(2k)},\ldots,\quad \mathrm E_{c^m}=\{x_0\},\quad \mathrm E_{c^{m+1}}=0; \]
whence it is plainly seen how the points of the domain \(\mathrm E\) are distributed among the adherences and coherences and their derivatives, and what, in the present case, is the interdependence between the former and the latter.
4. Let, for \(\mathrm E\), there exist coherences \(\mathrm E_{c^n}\) for every finite \(n\); then one of the following two alternatives is possible:
\[ \underset{1}{\overset{\omega}{\mathrm D}}\{\mathrm E_{c^n}\}=0,\qquad \underset{1}{\overset{\omega}{\mathrm D}}\{\mathrm E_{c^n}\}=\mathrm E_{c^\omega}; \]
in the first case all the \(\mathrm E_{c^n}\) have no common points, and this is conceivable, since the \(\mathrm E_{c^n}\) are in general not closed\({}^{1}\); in the second—the common points of all the coherences form a coherence of order \(\omega\).
Since, for each \(n\), 2 (2) holds, we have
\[ \mathrm E_{c^\omega}=\underset{1}{\overset{\omega}{\mathrm D}}\{\mathrm E_{c^n}\} =\underset{1}{\overset{\omega}{\mathrm D}}\{\mathrm E^{(n)}\} =\mathrm D\{\mathrm E^{(\omega)}\}, \]
i.e. “the coherence of order \(\omega\) is part of the derivative of order \(\omega\).”
Having the possibility of passing from \(n\) to \(n+1\) and from a sequence of numbers to their limit, we are entitled, for any \(\alpha\), where \(\alpha\) is a number of the first or second class, either admitting an immediately greater number \(\alpha+1\) or not, to assert that
\[ \tag{6} \mathrm E_{c^\alpha}=\mathrm D\{\mathrm E^{(\alpha)}\}, \]
and finally, that
\[ \mathrm E_{c^\Omega}=\mathrm D\{\mathrm E^\Omega\}, \]
where \(\Omega\) is the first number of the third class.
It follows from this that Theorem 2 has a general meaning.
From (4) and (6) there follows immediately Theorem 29 of W. H. Young\({}^{2}\):
“the points of each adherence are limiting for all preceding adherences,”
so that the author’s proof becomes superfluous.
\({}^{1}\) ib., p. 58–59.
\({}^{2}\) ib., p. 62.
5. The process of finding adherences and coherences breaks off at the moment when one of the coherences \(E_{c^\alpha}\) proves to be condensed or to consist of isolated points; in the first case we have
\[ E_{c^\alpha u}=0,\quad E_{c^\alpha+1}=E_{c^\alpha}=E_{c\Omega}, \tag{7} \]
and in the second
\[ E_{c^\alpha u}=E_{c^\alpha},\quad E_{c^\alpha+1}=0. \tag{8} \]
Since every coherence forms part of the corresponding derivative, it is clear that, if the sequence of derivatives breaks off, then with it the further formation of coherences will also cease, but not conversely: derivatives may exist while there are no coherences, as we saw in D 3° and shall see again below; moreover, it is possible that the domain and its adherences will include points belonging to further derivatives and even to \(E^{(2)}\).
In case (8) the domain \(E\) decomposes into a denumerable sequence of adherences and must therefore itself be denumerable; in case (7) there is further joined to this sequence the condensed domain \(E_{c^\alpha}\), which is the last coherence1 (ultimate coherenz).
All condensed parts of the domain are contained in the last coherence; if it is non-denumerable, then in \(E_{c^\alpha}\) one can distinguish a denumerable part \(U\) and a non-denumerable part \(V\); this non-denumerable condensed part of the domain W. H. Young calls2 its base or nucleus (nucleus).
6. To explain the preceding, W. H. Young takes3 a Cantor domain of type
\[ \omega \sum_{1}^{\omega}\omega^{i}; \]
A. not including in it the right endpoint \(x_0\) of the basic interval.
Using my notation and, for uniformity of form, not including in the domain also the left endpoints of the intervals \(x_0\) and the points of the first division \(\{x_i\}\), we obtain an unclosed domain of the form
\[ E=\sum_{1}^{\omega}Q_i=\sum_{i=1}^{\omega}\sum_{j=1}^{j=i}Q_i^{(j)}; \]
for it
\[ E_u=\sum_{1}^{\omega}Q_i^{(i)},\quad E_{cu}=\sum_{1}^{\omega}Q_{i+1}^{(i)},\ldots,\quad E_{c^n u}=\sum_{1}^{\omega}Q_{i+n}^{(i)},\ldots; \]
\[ E_c \equiv \sum_{i=2}^{\omega}\sum_{j=1}^{i-1} Q_i^{(j)},\quad E_{c^2} \equiv \sum_{i=3}^{\omega}\sum_{j=1}^{i-2} Q_i^{(j)},\ldots,\quad E_{c^{\,n+1}} \equiv \sum_{i=n+2}^{\omega}\sum_{j=1}^{i-n-1} Q_i^{(j)},\ldots; \]
\[ \tag{9} E' \equiv \{x_0\}+\sum_{1}^{\omega}\{x_i\}+E_c,\quad E'' \equiv \{x_0\}+\sum_{2}^{\omega}\{x_i\}+E_{c^2},\ldots, \]
\[ E^{(n+1)} \equiv \{x_0\}+\sum_{n+1}^{\omega}\{x_i\}+E_{c^{\,n+1}},\ldots,\quad E^{(\omega)} \equiv \{x_0\}. \]
We see from this that, in forming adherences and coherences, the rows along the diagonal in the first table successively fall away,¹) that there exist adherences and coherences of any finite order, but that no coherence of order \(\omega\) exists, since it could only be the point \(\{x_0\}\), which is excluded by the supposition. The series (9) shows the relations between the coherences and derivatives of one and the same order. Thus \(E\) can here be represented in the form of the sum of its adherences
\[ E \equiv \sum_{1}^{\omega} E_{c^i a}. \]
Further, W. H. Young takes²) the domain
\[ F \equiv \{x_0\}+\sum_{1}^{\omega} Q_i^{(i)}, \]
for which
\[ F_a \equiv \sum Q_i^{(i)},\quad F_c \equiv \{x_0\};\quad F_{ca} \equiv \{x_0\},\quad F_{c^2}=0, \]
whereas the derivatives of \(F\) coincide with the corresponding derivatives (9), and in particular \(F^{(\omega)} \equiv \{x_0\}\); thus here we have
\[ F_{c\omega}=0,\quad F_{ca}=F_c=F^{(\omega)}. \]
7. As an example of a domain for which there exists \(Y_c^{2}\), which W. H. Young denotes by \(Y_{c*}\), he, taking a series of intervals of type \(\tilde{\omega}\), includes³) in the composition of the domain their exterior points \(E\) and a finite number \(n\) of boundaries \(D_n(Q)\)
\[ \tag{10} Y \equiv E+D_n(Q); \]
¹) See p. 183, 130°.
²) p. 59, ex. 4.
³) ib., p. 60; cf. pp. 132–134, 78°.
then
\[ \mathrm{Y}_a=0,\quad \mathrm{Y}_c=\mathrm{Y}=\mathrm{Y}_c^\circ,\quad \mathrm{Y}'=\mathrm{E}+\mathrm{Q}=\mathrm{Y}^{(\Omega)}, \]
so that
\[ \mathrm{Y}=\mathrm{Y}_c^\circ=\mathrm{D}\bigl(\mathrm{Y}^{(\Omega)}\bigr). \]
As for the domains \(\mathrm{Y}_2,\mathrm{Y}_3,\ldots,\mathrm{Y}_n\) of example 6 and the domain \(\mathrm{Y}\) of example 7, all these domains, contrary to the opinion of \(W. H. Young\)\(^1\), are of one and the same type and give nothing new in comparison with the domain (10). Domains of new types are obtained only in the case when in the intervals \(\tilde{\omega}\) one places “\(\mathrm{T}_1,\mathrm{T}_2,\ldots,\mathrm{T}_n\) or still more complicated domains”\(^2\). I shall now give examples of such domains below.
8. In order to show that \(\mathrm{E}_c^\circ\) may fail to exist, and hence all the more \(\mathrm{E}\) may have common points with \(\mathrm{E}^{(\Omega)}\), \(W. H. Young\) gives an indication\(^3\) which can be realized,
A. by using, for instance\(^4\), a domain of intervals of the type
\[ \tilde{\omega}\sum_{1}^{\omega}\omega^i \]
and taking
\[ \mathrm{Y}=\mathrm{D}_n(\mathrm{E})+\sum_{1}^{\omega}\mathrm{Q}_i^{(i)}, \]
where \(\mathrm{D}_n(\mathrm{E})\) is a finite number of exterior points of the domain \(\tilde{\omega}\). For the domain \(\mathrm{Y}\)
\[ \mathrm{Y}_a=\sum_{1}^{\omega}\mathrm{Q}_i^{(i)},\quad \mathrm{Y}_c=\mathrm{D}_n(\mathrm{E})=\mathrm{Y}_{ca},\quad \mathrm{Y}_{c^2}=0, \]
whereas
\[ \mathrm{Y}^{(\omega)}=\mathrm{E}+\mathrm{Q}=\mathrm{Y}^{(\Omega)}, \]
where \(\mathrm{Q}\) is the domain of the boundaries of the intervals \(\tilde{\omega}\), and consequently \(\mathrm{E}+\mathrm{Q}\) is perfect. We see from this that
\[ \mathrm{Y}_{ca}=\mathrm{D}\{\mathrm{Y},\mathrm{Y}^{(\Omega)}\}. \]
B. The domain
\[ \tilde{\mathrm{Q}}=\sum_{1}^{\omega}\{\mathrm{Q}_n-\mathrm{Q}_n^{(n)}\} \]
\(^1\) ib. p. 60; cf. pp. 132–134, 87°.
\(^2\) ib. p. 60, ex. 7.
\(^3\) ib. p. 60, ex. 8.
\(^4\) see p. 193, 139°.
is composed of the limiting points of the domain
\[ \tilde{\omega}\sum_{1}^{\infty}\omega^{i}, \]
lying inside the intervals \(\tilde{\omega}\); if we take
\[ X = D_m(E)+D_n(Q)+\sum_{1}^{\infty}Q_i^{(i)}, \]
where \(D_m\) and \(D_n\) are finite parts of \(E\) and \(Q\), then
\[ X_a=\sum_{1}^{\infty}Q_i^{(i)},\quad X_c=D_m(E)+D_n(Q)=X_{ca},\quad X_{c^2}=0, \]
whereas
\[ X^{(\omega)}=E+Q=X^{(2)}, \]
and consequently
\[ D\{X_{ca}\}=D\{X^{(2)}\}, \]
i.e.,1 “some, but not all, points of the last adherence enter into \(X^{(2)}\), whereas \(X_{c^2}\) does not exist.”
9. Comparing the last coherence and the remainder \(V\) of the domain,2 W. H. Young did not give explanatory examples; to fill this gap,
A. Let us call \(Q\) the domain of the boundaries of intervals3 of type \((*\omega+\omega)^\omega\); for it
\[ Q_a=0,\quad Q_c=Q=Q_{c^\omega},\quad V=0, \]
i.e. the last coherence is identical with the domain itself, and the remainder is absent.
B. Taking further a domain of type
\[ \tag{11} \tilde{\omega}\{(*\omega+\omega)^\omega+\sum_{1}^{\infty}(*\omega+\omega)^i\}, \]
we renumber the intervals \(\tilde{\omega}\) in the order of their magnitude \(\{l_i\}\), and assign the domain \((*\omega+\omega)^\omega\) to the largest interval \(l_1\), while the domains \((*\omega+\omega)^i\) we place in the intervals \(l_{i-1}\); defining the type (11) in this way, we include in the composition of the domain \(G\) the exterior points \(E\) of the domain \(\tilde{\omega}\), all the boundaries \(Q\) of the intervals \((*\omega+\omega)^i\), and the boundaries \(R\) of all divisions of the domain \((*\omega+\omega)^\omega\); then
\[ \mathrm{G}=\mathrm{E}+\mathrm{Q}+\mathrm{R},\quad \mathrm{G}^{(\omega)}=\mathrm{E}' + \mathrm{R}'-\{x_{-1},x_1\}=\mathrm{G}^{(\Omega)}, \]
\[ \mathrm{G}_{c\omega}=\mathrm{E}+\mathrm{R}=\mathrm{G}_{c\Omega}, \]
where \(x_{-1}, x_1\) are the boundaries of the interval \(l_1\), included both in \(\mathrm{E}'\) and in \(\mathrm{R}'\); and \(\mathrm{G}^{(\omega)}\) is a perfect1 domain.
Here \(\mathrm{E}+\mathrm{R}\) is the last coherence, while its uncountable part \(\mathrm{E}\) is the skeleton of the domain \(\mathrm{G}\), whereas \(\mathrm{R}\) is a countable domain.
C. Finally, if we take a closed domain \(\mathrm{H}\) of boundaries and exterior points of intervals of type
\[ \tilde{\omega}\,\underset{1}{\overset{\omega}{\mathrm{Z}}}(*_\omega+\omega)^i, \]
then for it
\[ \mathrm{H}_{c\omega}=\mathrm{H}_{c\Omega}=\mathrm{P}=\mathrm{H}^{(\omega)}=\mathrm{H}^{(\Omega)}, \]
where \(\mathrm{P}\) is the perfect domain of boundaries and exterior points \(\tilde{\omega}\); in this example the last coherence and the skeleton of the domain turn out to be identical.
The foregoing, it seems to me, gives a sufficiently convincing example of how one may use the concept of types of arrangement, and how simply and quickly it leads to the goal in clarifying even rather complicated definitions.