MATHEMATICS

I. D. STUPINA

ON THE PROPERTIES OF SOME \(\delta s\)-OPERATIONS

(Presented by Academician P. S. Aleksandrov on 21 VI 1956)

In descriptive set theory a number of results of the following character have been obtained: the projection of a plane set is considered, and assertions are made about the descriptive nature of the set of points of the projection whose inverse images possess a certain special property. For example: if a plane \(A\)-set is projected, then the set of points whose inverse images contain at least two points is an \(A\)-set (N. N. Luzin); the set of points whose inverse images contain an uncountable set of points is likewise an \(A\)-set (W. Sierpiński).

A number of problems of the same character were solved by P. S. Novikov \(\left({}^{3}\right)\), V. Ya. Arsenin \(\left({}^{4,5}\right)\), C. Braun \(\left({}^{6}\right)\), K. Kunugui \(\left({}^{7}\right)\). Analogous theorems in the theory of operations on sets were established by A. A. Lyapunov \(\left({}^{8,9}\right)\) and Z. I. Kozlova \(\left({}^{10,11}\right)\).

In the present note some results in this direction are given*.

Let \(N\) be a rigid base** of some \(\delta s\)-operation. We shall call a point \(x\) a point of \(N\)-first type for the sequence of sets \(\{E_n\}\) if there exists an uncountable number of chains of the base \(N\) into whose kernels the point \(x\) enters. By \(\Phi_{N***}\) we denote the \(\delta s\)-operation which selects the points of \(N\)-first type. If each chain of an \(N\)-rigid base of a \(\delta s\)-operation is ordered in increasing order of the elements of the chain, then the resulting collection of chains will be called the rigid reduced base and denoted by \(\check N\).

A point \(x\) is determined by the chain \(\eta=\{n_i\}\) with respect to the sequence of sets \(\{E_n\}\), if

\[ x \in \prod_{n_i \in \eta} E_{n_i}. \]

Let \(M_x=\{\eta\}\) denote, for the given sequence of sets \(\{E_n\}\) and base \(N\), the set of all such chains \(\eta \in \check N\) that \(x\) is determined by the chain \(\eta\) with respect to the sequence of sets \(\{E_n\}\). We shall call a point \(x\), determined by the \(\delta s\)-operation \(\Phi_N\), a point of \(N\)-second type if the set \(M_x\) has non-compact closure. By \(\Phi_{\check N}\) we denote the \(\delta s\)-operation selecting the points of \(N\)-second type.

We shall call a point \(x\), determined by the \(\delta s\)-operation \(\Phi_N\), a point of \(N\)-third type of order \(\alpha\), if the set \(M_x\) is not a scattered set of index \(\leqslant \alpha\). By \(\Phi_{\check N\alpha}\) we denote the \(\delta s\)-operation selecting the points of \(N\)-third type of order \(\alpha\).

We shall consider linear sets \(E\) which are sums of a scattered family of sets whose closure is compact.

* We use the notation introduced in the works \(\left({}^{8-11}\right)\).

** In what follows we shall consider \(\delta s\)-operations \(\Phi_N\) with rigid bases \(N\) belonging to the Baire space \(J\).

Let \(E = E^{(0)}\). If \(\alpha\) is a transfinite number of the first kind, then \(E^{(\alpha)}\) denotes the set obtained from the set \(E^{(\alpha-1)}\) by removing all its isolated portions whose closures are compact; if \(\alpha\) is a transfinite number of the second kind, then, by definition,

\[ E^{(\alpha)}=\prod_{\alpha'<\alpha} E^{(\alpha')}. \]

The least number \(\beta\) such that \(E^{(\beta)}=0\) is called the index of the set \(E\).

We shall call a point \(x\), determined by the \(\delta s\)-operation \(\Phi_N\), a point of \(N\)-fourth type of order \(\alpha\), if the set \(M_x\) is not the sum of a dispersed family of sets whose closures are compact, of index \(\leqslant \alpha\). By \(\Phi_{\check N_*^{(\alpha)}}\) we shall denote the \(\delta s\)-operation selecting points of \(N\)-fourth type of order \(\alpha\).

Z. I. Kozlova showed \((^{11})\) that if \(N\) is a rigid base of an \(A\)-operation, then the operations \(\Phi_{N^{***}}, \Phi_{\check N_*}, \Phi_{\check N_\alpha}, \Phi_{\check N_*^{(\alpha)}}\) are no more powerful than \(A\)-operations.

Theorem 1. The operations \(\Phi_{N_c'***}, \Phi_{N_c'***n}\) are no more powerful than the \(\Gamma\)-operation, where \(N_c'\) is a rigid base of the \(\Gamma\)-operation.

Theorem 2. The operations \(\Phi_{N_c''***}, \Phi_{N_c''***n}\) are no more powerful than the \(CA_2\)-operation, where \(N_c''\) is a rigid base of the \(CA_2\)-operation.

Theorem 3. If the class of sets \(\Xi\) and the rigid base \(N\) are in a completely regular relation, then the relations

\[ \Phi_{\check N_*}(\Xi)\subset \Phi_N(\Xi), \qquad \Phi_{\check N_*^n}(\Xi)\subset \Phi_N(\Xi) \]

hold.

Theorem 4. If the class of sets \(\Xi\) is invariant with respect to the \(\delta s\)-operations \(\Phi_N\), where the base \(N\) is an \(A_2\)-set, then

\[ \Phi_{\check N_\alpha''}(\Xi)\subset \Xi. \]

If, however, the class of sets \(\Xi^*\) is such that \(\Phi_{N''}(\Xi^*)\) belongs to the class of \(A_2\)-sets \(\Theta\), then

\[ \Phi_{\check N_\alpha''}(\Xi^*)\subset \Theta, \]

where \(N''\) is a rigid base of the \(A_2\)-operation.

Theorem 5. If the class of sets \(\Xi\) is invariant with respect to the \(\delta s\)-operations \(\Phi_N\), where the base \(N\) is an \(A_2\)-set, then

\[ \Phi_{\check N_*^{\prime\prime(\alpha)}}(\Xi)\subset \Xi. \]

If, however, the class of sets \(\Xi^*\) is such that \(\Phi_{N''}(\Xi^*)\) belongs to the class of \(A_2\)-sets \(\Theta\), then

\[ \Phi_{\check N_*^{\prime\prime(\alpha)}}(\Xi^*)\subset \Theta. \]

From the general theorem on covering sets of Z. I. Kozlova \(((^{11}),\) theorem 1) and the preceding results, the following propositions follow:

Theorem 6. For every sequence of \(CA_2\)-sets \(\{E_n\}\) such that no point of the set \(\Phi_N\{E_n\}\) is a point of \(N\)-first type, there exists a sequence of \(B_2\)-sets \(\{H_n\}\) such that \(H_n \supset E_n\) and no point of the set \(\Phi_N\{H_n\}\) is a point of \(N\)-first type, where \(N\) is a rigid base of a \(\Gamma\)-operation, of a \(CA_2\)-operation.

Theorem 7. For every sequence of \(CA_2\)-sets \(\{E_n\}\) such that no point of the set \(\Phi_N\{E_n\}\) is a point of the second \(N\)-type, there exists a sequence of \(B_2\)-sets \(\{H_n\}\) such that \(H_n \supset E_n\) and no point of the set \(\Phi_N\{H_n\}\) is a point of the second \(N\)-type, where \(N\) is a rigid base of the \(\Gamma\)-operation, a \(CA_2\)-operation.

By virtue of a theorem of P. S. Novikov \(^{13}\) stating that, in the system \(\Sigma\) of axioms of K. Gödel’s set theory \(^{12}\), the separation theorems for projective sets \(CA_n\), for sufficiently large \(n\), are not contradictory, one may, in the sense of consistency, formulate analogous propositions on covering \(CA_n\)-sets.

Received
5 VI 1956

CITED LITERATURE

\(^{1}\) N. N. Luzin, Lectures on Analytic Sets and Their Applications, Moscow, 1953.
\(^{2}\) S. Mazurkiewicz, W. Sierpiński, Fund. Math., 6, 161 (1924).
\(^{3}\) P. S. Novikov, DAN, 23, No. 9, 863 (1939).
\(^{4}\) V. Ya. Arsenin, Izv. AN SSSR, Ser. Mat., No. 2, 233 (1939).
\(^{5}\) V. Ya. Arsenin, Izv. AN SSSR, Ser. Mat., No. 4, 403 (1940).
\(^{6}\) S. Braun, Fund. Math., 20, 166 (1933).
\(^{7}\) K. Kunugui, J. Faculty Sci. Hokkaido Imp. Univ., Ser. I, Math., 7, No. 3–4, 187 (1939); 8, No. 1, 1 (1939).
\(^{8}\) A. A. Lyapunov, Izv. AN SSSR, Ser. Mat., 17, 563 (1953).
\(^{9}\) A. A. Lyapunov, Tr. Mosk. Matem. Obshch., 6, 195 (1957).
\(^{10}\) E. I. Kozlova, Izv. AN SSSR, Ser. Mat., 19, 125 (1955).
\(^{11}\) E. I. Kozlova, Izv. AN SSSR, Ser. Mat., No. 3, 349 (1957).
\(^{12}\) Yu. S. Yuchan, Matem. Sbornik, 10 (52), No. 3, 151 (1942).
\(^{13}\) P. S. Novikov, Tr. Matem. Inst. im. V. A. Steklova, 38, 279 (1951).
\(^{14}\) K. Gödel, Uspekhi Matem. Nauk, 3, issue 1 (23), 96 (1948).