Ya. L. Kreinin

ON A PROPERTY OF SETS EFFECTIVELY DISTINCT FROM ALL \(\Phi\)-SETS

(Presented by Academician S. L. Sobolev on 5 VII 1957)

In the paper \((^4)\) (p. 135), by analogy with the concept of effective uncountability belonging to P. S. Novikov \((^{1,2})\), the concept of a set effectively distinct from all \(\Phi\)-sets was introduced. There it was also proved that every set (in particular, a \(CA\)-set) effectively distinct from all \(A\)-sets contains a perfect compact nucleus. In the present note one of the further properties of sets effectively distinct from all \(\Phi\)-sets is considered. We use here the definitions adopted in \((^4)\).

1°. Definition. We shall say that a \(\delta s\)-operation \(\widetilde{\Phi}\) embraces a \(\delta s\)-operation \(\Phi\) (or say: \(\widetilde{\Phi}\) is embracing relative to \(\Phi\)) if there exists a mapping \(\tau(k)\) of the natural number sequence onto itself such that, whatever the space \(R\), for every sequence \(\{F_1,F_2,\ldots,F_n\}\) of sets \(F_n\) of the space \(R\) one can choose such a sequence \(\{F'_1,F'_2,\ldots,F'_k,\ldots\}\) of sets \(F'_k\) which satisfies the following requirements: a) for every \(k\), \(F'_k = F_{\tau(k)}\); b)
\[ \widetilde{\Phi}\{F'_1,F'_2,\ldots,F'_k,\ldots\} = \Phi\{F_1,F_2,\ldots,F_n,\ldots\}. \]

The concept of an embracing operation obviously has the property of transitivity.

We note that the \(A\)-operation embraces the operations of lower and upper limits.

Theorem 1. If the operation \(\widetilde{\Phi}\) embraces the operation \(\Phi\), and \(T\) is a set of the metric space \(R\) effectively distinct from all \(\widetilde{\Phi}\)-sets of the space \(R\), then \(T\) is effectively distinct from all \(\Phi\)-sets of this space.

2°. The proof of Theorem II § 4 of the paper \((^4)\), as well as Theorem 1 of the present note, make it possible to prove the following theorem.

Theorem 2. Let \(R\) be a metric space and let \(\Phi\) be a \(\delta s\)-operation embracing the operation of lower limit \(\underline{\Phi}\) and the operation of upper limit \(\overline{\Phi}\). If \(T\) \((T \subset R)\) is effectively distinct from all \(\Phi\)-sets of the space \(R\), then \(T\) and \(R-T\) contain, respectively, sets \(E_1\) and \(E_2\) possessing the following properties: a) there exist such discontinua \(D_1\) and \(D_2\) that \(E_1 \subset D_1\), \(E_2 \subset D_2\); b) \(E_1\) and \(E_2\) are absolute \(G_\delta\)’s; c) \(E_1\) is not separable from \(R-T\) by any absolute \(F_\sigma\)-set, and \(E_2\) is not separable from \(T\) by any absolute \(F_\sigma\)-set.

Proof. By virtue of Theorem 1, \(T\) is effectively distinct from all \(\underline{\Phi}\)-sets, and also effectively distinct from all \(\overline{\Phi}\)-sets.

We note that Theorem II § 4 \((^4)\) and its proof, given by us for the operation \(\underline{\Phi}\), remain valid also for \(\overline{\Phi}\), as well as for every

\(\delta s\)-operation \(\Phi\), possessing the following property: let a sequence of sets \(\{M_1, M_2, \ldots, M_n, \ldots\}\) and a set \(M\) be given; if for almost all \(n\), \(M_n \subset M\), then \(\Phi\{M_n\}\subset M\); while if for almost all \(n\), \(M_n \supset M\), then \(\Phi\{M_n\}\supset M\).

Without reproducing all the notation and arguments of § 4 of paper \((^4)\), let us recall only that we used two sequences \(\{F_n^0\}\) and \(\{F_n^1\}\) of closed sets of the space \(R\), satisfying the conditions:
\(0\subset F_n^0\subset Y_\Phi\subset F_n^1\subset Z_\Phi\) for every \(n\). At the end of the proof just mentioned we obtained a discontinuum \(D\), which is the aggregate of all points of the form
\(\nu\{F_1^{t_1},\ldots,F_{q_1}^{t_1}; F_1^{t_2},\ldots,F_{q_2}^{t_2};\ldots; F_1^{t_k},\ldots,F_{q_k}^{t_k}; F_1^{t_{k+1}},\ldots,\)
\(\ldots,F_{q_{k+1}}^{t_{k+1}},\ldots\}\), where \(\nu\) is a function ensuring the effective distinction of \(T\) from \(\Phi\)-sets; \(t_1,t_2,\ldots,t_k,t_{k+1},\ldots\) independently take the two values 0 and 1, and \(q_1,q_2,\ldots,q_k,q_{k+1},\ldots\) are natural numbers chosen in a definite manner.

Let \(\Phi\) be a lower-limit operation. Denote by \(D_1\) the discontinuum \(D\) constructed for this case in \((^4)\). By virtue of the properties of the function \(\nu\), the set \(R-T\) contains precisely those points
\(\nu\{F_1^{t_1},\ldots,F_{q_1}^{t_1};\ldots;F_1^{t_k},\ldots,F_{q_k}^{t_k};\ldots\}\)
of the discontinuum \(D_1\) for which almost all upper indices \(t_1,t_2,\ldots,t_k,\ldots\) take the value 1. It is easy to verify, in view of this, that \(D_1(R-T)\) is a countable everywhere dense subset of the discontinuum \(D_1\). Consequently, the set \(E_1=D_1\cdot T\) is an absolute \(G_\delta\), distinct from all absolute \(F_\sigma\)-sets. Whatever the set \(M\) such that \(E_1\subset M\subset T\), we have \(M\cdot D_1=E_1\). It follows from this that \(E_1\) is an absolute \(G_\delta\) not separable from \(R-T\) by any absolute \(F_\sigma\)-set.

Assuming, further, that \(\Phi\) is an upper-limit operation and denoting in this case the discontinuum \(D\) by \(D_2\), we see, by virtue of the properties of the function \(\nu\), that the set \(T\) contains precisely those points
\(\nu\{F_1^{t_1},\ldots,F_{q_1}^{t_1};\ldots;F_1^{t_k},\ldots,F_{q_k}^{t_k};\ldots\}\)
of the discontinuum \(D_2\) for which almost all indices \(t_1,t_2,\ldots,t_k,\ldots\) take the value 0. From this, analogously to the preceding, we conclude that the set \(E_2=(R-T)\cdot D_2\) is an absolute \(G_\delta\), not separable from \(T\) by any absolute \(F_\sigma\)-set. The theorem is proved.

Let us note the special case of this theorem when \(\Phi\) is an \(A\)-operation and \(T\) is a \(CA\)-set. This special case of Theorem 2 may be regarded as a certain supplement to Gurevich’s theorem (\((^3)\), p. 51).

Crimean State Pedagogical Institute
named after M. V. Frunze

Received
3 VII 1957

REFERENCES

1. P. S. Novikov, Izv. Akad. Nauk SSSR, Ser. Mat., 3, No. 1, 35 (1939).
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3. V. Ya. Arsenin, A. A. Lyapunov, Uspekhi matem. nauk, 5, issue 5 (39), 45 (1950).
4. Ya. L. Kreinin, Matem. sbornik, 38 (80), issue 2, 129 (1956).