UDC 519.52

MATHEMATICS

Ya. L. KREININ

ON \(C\Phi\)-KERNELS OF SETS EFFECTIVELY DIFFERENT FROM \(\Phi\)-SETS

(Presented by Academician P. S. Novikov on 13 IV 1965)

As in \((^{3-5})\), here we study certain questions of descriptive set theory by means of P. S. Novikov’s idea of effective difference \((^{1-3})\). In \((^{4,5})\) it was proved that if a set \(T\) of a space \(E\) is effectively different from all \(\Phi\)-sets of this space, then both \(T\) and its complement \(E - T = CT\) contain discontinua, and also contain absolute \(C_\sigma\)-sets not separable, respectively, from \(E - T\) and \(T\) by any absolute \(F_\sigma\)-sets. Here it will be proved that \(T\) contains kernels of considerably more complicated structure—the so-called \(C\Phi\)-kernels, i.e., absolute \(C\Phi\)-sets not separable from \(E - T\) by any absolute \(\Phi\)-sets.

\(1^\circ\). To the symbols \(\Pi(E)\), \(\Pi_\Phi(E)\), \(F(E)\), \(Pt(E)\) we assign the same meaning as in \((^{3})\), pp. 129–130. The set \(\Pi_\Phi(E)\) is called the \(\Phi\)-base of the space \(E\). The definition itself of a set effectively different from \(\Phi\)-sets is as follows:

Definition. Let \(E\) be a metric space, \(\Phi\) an arbitrary \(\delta s\)-operation. We shall say that a set \(T\), \(T \subseteq E\), is effectively different from all \(\Phi\)-sets of the space \(E\), if there exist a compactum \(Z\), \(0 \subset Z \subseteq E\), and a mapping \(\nu\) of some \(\Phi\)-base \(\Pi_\Phi(E)\) into the set \(Z\), for which the following conditions are fulfilled: a) for every sequence of closed sets \(\{F_n\} \in \Pi_\Phi(E)\) we have
\[ \nu\{F_n\}\in T\cdot C\Phi\{F_n\}+CT\cdot \Phi\{F_n\}; \]
b) \(Pt(Z) \subseteq \Pi_\Phi(E)\); c) the mapping \(\nu\) is continuous on the metric space \(\Pi t(Z)\).

In this case we call \(\nu\) a distinguishing function for \(T\), and the compactum \(Z\) the metric base of the function \(\nu\).

Comparing this definition with the basic definition in \((^{3})\), p. 135, we note, first, that they differ only in that in \((^{3})\) it is assumed with respect to \(Z\) that it is a bounded closed subset of the space \(E\), whereas here we require that \(Z\) be a compactum. Secondly, for a Euclidean space \(E\) the two definitions coincide completely. Thirdly, both the existence theorem for a set \(T\) effectively different from all \(\Phi\)-sets of the space \(E\) \(( (^{3}), p. 146)\) and all properties of this set \(T\) found in \((^{3-6})\) are completely preserved if one assumes that \(Z\) is a compactum. In this case only the metric of the space \(Z\) is used, and not at all that of the space \(E\), so that everywhere one may regard \(E\) as a topological space containing a metric compactum as a subspace.

\(2^\circ\). In what follows, by \(\Phi\) we shall mean a \(\delta s\)-operation with the following property:

\((\tau)\). All chains of the operation \(\Phi\) are infinite, and for every \(n_0\) there exists a chain \(\{n_1,n_2,\ldots\}\) of the operation \(\Phi\) such that
\[ n_0<n_1<n_2<\cdots . \]

For an operation \(\Phi\) with this property, evidently,
\[ \Phi\{F_1,\ldots,F_{n_0}; M,\bar M,M,\ldots,M,\ldots\}=M. \]
All operations that are stronger than the operation of the lower limit, in particular the operations \(B_\alpha(\alpha\ge 2)\) and \(A\), possess property \((\tau)\).

Theorem 1. If a set \(T\), situated in the space \(E\), is effectively distinct from all \(\Phi\)-sets of this space, then there exists a perfect compact set \(Z_1\), \(0 \subset Z_1 \subset E\), such that the intersection \(T\cdot Z_1\) is a \(C\Phi\)-set of the space \(Z_1\), effectively distinct from all \(\Phi\)-sets of this space.

Proof. Let \(\nu\) be a distinguishing function for \(T\), and let \(Z\) be a metric basis of the function \(\nu\). From the compactness of \(Z\) it follows that \(\Pi t(Z)\) is also compact. Consequently, the image of the space \(\Pi t(Z)\) under the continuous mapping \(\nu\), which we denote by \(Z_1\), is also compact. According to Definition II.1°, \(Z_1 \subset Z\).

The sets \(Z_1\cdot T\) and \(Z_1\cdot(E-T)=Z_1\cdot CT\) are nonempty. Indeed,
\(\nu\{Z,Z,\ldots,Z,\ldots\}\in Z_1\cdot(T\cdot C\Phi\{Z\}+CT\cdot\Phi\{Z\})=Z_1\cdot CT\), since \(Z_1\cdot CT\ne0\). On the other hand,
\(\nu\{0,0,\ldots,0,\ldots\}\in Z\cdot(T\cdot C\Phi\{0\}+CT\cdot\Phi\{0\})=Z\cdot T\). Consequently, also \(Z\cdot T\ne0\). If now by \(F^0\) we denote any nonempty closed set contained in \(Z\cdot T\), then

\[ \nu\{F^0,F^0,\ldots,F^0,\ldots\}\in Z_1\cdot(T\cdot C\Phi\{F^0\}+CT\cdot\Phi\{F^0\})= \]

\[ =Z_1\cdot T\cdot C\Phi\{F^0\}, \tag{*} \]

which proves the nonemptiness of the set \(Z_1\cdot T\).

Next, let \(z=\nu\{F_1,F_2,\ldots,F_n,\ldots\}\) be an arbitrary point of the set \(Z_1\) \((\{F_n\}\in\Pi t(Z))\), and let \(U(z)\) be any neighborhood (relative to the space \(E\)) of this point. From the continuity of the function \(\nu\) at the point \(\{F_n\}\) it follows that, for some \(n_0\),
\(\nu\{F_1,\ldots,F_n;Q_1,Q_2,\ldots,Q_n,\ldots\}\in U(z)\), whatever the point \(\{Q_n\}\in\Pi t(Z)\) may be. At the same time, for the closed set \(F^0\), \(0\subset F^0\subset TZ\),
\(\nu\{F_1,\ldots,F_n;F^0,F^0,\ldots,F^0,\ldots\}\in Z_1\cdot T\cdot C\Phi\{F^0\}\) (see \((*)\), and also property \((\tau)\) of the operation \(\Phi\)), while
\(\nu\{F_1,\ldots,F_n;Z,Z,\ldots,Z,\ldots\}\in Z_1\cdot CT\). Thus, in an arbitrary neighborhood \(U(z)\) of any point \(z\in Z_1\) there are both points of \(T\cdot Z_1\) and points of \(CT\cdot Z_1\), and the compact set \(Z_1\) is perfect.

The set \(T\cdot Z_1\) is effectively distinct from all \(\Phi\)-sets of the space \(Z_1\). We construct a distinguishing function \(\nu_1\) for \(T\cdot Z_1\) as follows. As its domain we take the \(\Phi\)-basis \(\Pi_\Phi(Z_1)\), which is obtained by adjoining to the set \(\Pi t(Z_1)\) the single sequence \(\{0,0,\ldots,0,\ldots\}\) (see (3), p. 136, Remark 1), and as metric basis the compact set \(Z_1\) itself. Denoting by \(z_0\) some fixed point of the set \(Z_1\cdot T\), we put

\[ \nu_1\{F_n\}= \begin{cases} \nu\{F_n\}, & \text{for } \{F_n\}\in\Pi t(Z_1),\\ z_0, & \text{for } \{F_n\}=\{0,0,\ldots,0,\ldots\}. \end{cases} \]

It is easy to see that \(\nu_1\) is indeed a distinguishing function for \(T\cdot Z_1\) (relative to the space \(Z_1\)).

It remains to prove that \(CT\cdot Z_1\) is a \(\Phi\)-set of the space \(Z_1\). We turn to the continuous mapping \(\nu\) of the space \(\Pi t(Z)\) onto \(Z_1\) and define diagonal sets \(L_n\), \(L_n\subset Z_1\), for each natural number \(n\) in the following way. A point \(x\in Z_1\) belongs to \(L_n\) if and only if among all sequences
\(\{F_1^x,F_2^x,\ldots,F_n^x,\ldots\}\in\Pi t(Z)\) that are preimages of the point \(x\) under the mapping \(\nu\), there is at least one such sequence for which \(x\in F_n^x\).

The sets \(L_n\ne0\), for, obviously,
\(\nu\{Z,Z,\ldots,Z,\ldots\}\in\prod_{n=1}^{\infty}L_n\).

Let \(x^0=\lim_{k\to\infty}x_k\), where \(x_k\in L_n\) for all \(k\) (\(n\) fixed).

From the definition of \(L_n\), by virtue of the compactness of \(\Pi t(Z)\), there follows the existence of a sequence
\(x_k^1,x_k^2,\ldots,x_k^p,\ldots\), for which:

1) \(\{x_k^1, x_k^2,\ldots,x_k^p,\ldots\}\) is a subsequence of the sequence \(\{x_1,\ldots,x_k,\ldots\}\);

2) in the space \(\operatorname{Pt}(Z)\) one can select a convergent sequence of its points \(\{F_m^1\}, \{F_m^2\},\ldots,\{F_m^p\},\ldots\), such that \(\nu\{F_m^p\}=x_k^p\) and \(x_k^p \in F_n^p\) \((p=1,2,\ldots)\). Denoting \(\lim_{p\to} \{F_m^p\}=\{F_m^0\}\), by virtue of the continuity of \(\nu\) at the point \(\{F_m^0\}\) we have \(\nu\{F_m^0\}=x^0\). Hence, in the space \(F(Z)\),

\[ \lim_{p\to\infty} F_m^p = F_m^0 \quad (m=1,2,\ldots). \]

In particular, for the \(n\) fixed above, \(\lim_{p\to\infty} F_n^p=F_n^0\). This, as well as the fact that \(x_k^p \in F_n^p\) for every \(p\) and that \(\lim_p x_k^p=x^0\), leads to the conclusion \(x^0 \in F_n^0\), consequently, \(x^0 \in L_n\). Thus \(L_n\) is closed for every \(n\).

Let us show that \(\Phi\{L_n\}=CT\cdot Z_1\). Take an arbitrary point \(x \in Z_1\). We denote every point of its complete preimage \(\nu^{-1}(x)\) under the mapping \(\nu\) of \(\operatorname{Pt}(Z)\) onto \(Z_1\) by \(\{F_1^x,F_2^x,\ldots,F_n^x,\ldots\}\). By the symbol \(\{F_1(x),F_2(x),\ldots,F_n(x),\ldots\}\) we denote any of those points \(\{F_1^x,F_2^x,\ldots,F_n^x,\ldots\}\) which are constructed in the following way: if, for a given \(n\) (and for the fixed \(x\) above), for every \(F_n^x\) we have \(x \notin F_n^x\), then as \(F_n(x)\) we choose any \(F_n^x\); if, however, there exist \(F_n^x\) for which \(x \in F_n^x\), then any of precisely these \(F_n^x\) is denoted by \(F_n(x)\). Thus such a part of the set \(\nu^{-1}(x)\), all points \(\{F_1(x),F_2(x),\ldots,F_n(x),\ldots\}\) of which possess the following property, has been singled out: for every \(n\), either \(x \in F_n(x)\) for all \(F_n(x)\), or for every \(F_n^x\) one has \(x \notin F_n^x\). From this property of the \(\{F_n(x)\}\) and from the definition of the diagonal sets \(L_n\) it follows that

\[ x \in L_n\cdot F_n(x)+(E-L_n)\cdot(E-F_n(x)). \]

Hence, by virtue of the definition of the \(\delta s\)-operation \(\Phi\),

\[ x \in \Phi\{L_n\}\cdot \Phi\{F_n(x)\}+C\Phi\{L_n\}\cdot C\Phi\{F_n(x)\}. \tag{1} \]

On the other hand,

\[ x=\nu\{F_n(x)\}\in Z_1\cdot CT\cdot \Phi\{F_n(x)\}+Z_1\cdot T\cdot C\Phi\{F_n(x)\}. \tag{2} \]

From comparing assertions (1) and (2), valid for any point \(x \in Z_1\), we easily infer that \(Z_1\cdot CT=\Phi\{L_n\}\), \(Z_1\cdot T=Z_1-\Phi\{L_n\}\). The theorem is proved.

Remark 1. If in the statement of the theorem just proved the word “perfect” is omitted, then it will already be true for any \(\delta s\)-operation \(\Phi\), and not only for an operation with property \((\tau)\).

Remark 2. All discontinuums constructed by us in \((^3\text{–}^5)\) (see the beginning of the present article) are subsets of the compactum \(Z_1\), which is the image of \(\operatorname{Pt}(Z)\) under the mapping \(\nu\).

\(3^\circ\). In \((^6)\), p. 298, it is proved that if the \(\delta s\)-operation \(\Phi\) is stronger than the \(\delta s\)-operation \(\Psi\) and the set \(T\), \(T\subset E\), is effectively distinct from all \(\Phi\)-sets of the space \(E\), then \(T\) is also effectively distinct from all \(\Psi\)-sets of this space. In view of this, from Theorem 1 for \(\delta s\)-operations \(\Phi\) and \(\Psi\) with property \((\tau)\) there follows the following

Theorem 2. If the set \(T\), \(T\subset E\), is effectively distinct from all \(\Phi\)-sets of the space \(E\) and the operation \(\Phi\) is stronger than the operation \(\Psi\), then there exists a perfect compactum \(Z_1^\Psi\), \(0\subset Z_1^\Psi \subseteq E\), such that the intersection \(T\cdot Z_1^\Psi\) is a \(C\Psi\)-set of the space \(Z_1^\Psi\), effectively distinct from all \(\Psi\)-sets of this space.

Let us note that, as is seen from the proof of Theorem 1 and from the proof mentioned in \((^6)\), p. 298, the compacta \(Z_1\) and \(Z_1^\Psi\) in Theorems 1 and 2 respectively can be chosen so that \(Z_1^\Psi \subseteq Z_1\).

We now apply Theorem 2 to sets effectively distinct from \(A\)-sets. Using the known properties of Borel sets, as well as the fact that the \(A\)-operation is stronger than the operation \(B_\alpha\), which yields \(B\)-sets

of class $\alpha$, we obtain, as a consequence of Theorem 2, the following proposition.

Theorem 3. If a set (in particular, a $CA$-set) $T$, $T \subset E$, is effectively distinct from all $A$-sets of the space $E$, then for every ordinal $\alpha$, $2 \leq \alpha < \Omega$, there exists a perfect compactum $Z_1^\alpha$, $0 \subset Z_1^\alpha \subseteq E$, such that the intersection $Z_1^\alpha \cdot T$ is a $CB_\alpha$-set (and consequently a $B_{\alpha+1}$-set) of the space $Z_1^\alpha$, effectively distinct from all $B_\alpha$-sets of this space $Z_1^\alpha$.

Remark to $2^0$ remains valid also for Theorems 2 and 3. Omitting the word “perfect” from the formulation of the theorem, we may assume that it is also true for $\alpha = 0; 1$.

From Theorems 1 and 3 there follows the following proposition.

Corollary. Under the hypotheses of Theorem 3, both the set $T$ and its complement $E - T$, for every $\alpha < \Omega$, contain absolute $B_\alpha$-sets not separable respectively from $E - T$ and $T$ by any absolute $B_\beta$-sets, where $\beta < \alpha$. Moreover, $T$ contains an absolute $CA$-set not separable from $E - T$ by any absolute $A$-set, while $E - T$ contains an absolute $A$-set not separable from $T$ by any absolute $B$-set.

Received
12 IV 1965

CITED LITERATURE

$^1$ P. S. Novikov, Izv. AN SSSR, Ser. Matem., 3, No. 1, 35 (1939).
$^2$ C. Sacks, Matem. sborn., 7 (49), 343 (1940).
$^3$ Ya. L. Kreinin, Matem. sborn., 38 (80), no. 2, 129 (1956).
$^4$ Ya. L. Kreinin, DAN, 118, No. 3, 436 (1958).
$^5$ Ya. L. Kreinin, DAN, 118, No. 2, 237 (1958).
$^6$ F. V. Anufriev, Izv. Krymsk. ped. inst., 25, 295 (1961).