Mathematics

Ya. L. Kreinin

On Perfect Compact Kernels of Sets Effectively Distinct from All \(\Phi\)-Sets

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

A number of fundamental problems of descriptive set theory are known, whose investigation has led to the boundaries of applicability of set-theoretic principles. Among these problems we shall point to the continuum problem and to the problem of the existence of a perfect compact kernel in projective sets.

With regard to the first of these problems, K. Gödel proved that Cantor’s continuum hypothesis does not contradict the system of axioms of set theory. As for the second problem, as P. S. Novikov showed, the assertion of the existence of a perfect compact kernel in projective sets is not derivable logically from set-theoretic principles. But since the question of the opposite assertions has not been resolved, N. N. Luzin’s supposition on the unsolvability of the above-mentioned problems of descriptive set theory has not yet been confirmed.

In this state of affairs it is of interest to consider the same problems of descriptive set theory in a formulation restricted by means of the notion of effective distinction, due to P. S. Novikov \((^{1-3})\). P. S. Novikov proved that the continuum problem in its effectively restricted formulation is resolved in the direction of confirming Cantor’s hypothesis. But, as shown in \((^3)\) and in the present paper, the proposition on the existence of a perfect compact kernel in the sets \(T_\Phi\), effectively distinct from projective sets, is derivable from set-theoretic principles.

Here we consider this result for a very general class of \(\delta s\)-operations \(\Phi\), which includes all operations yielding \(B\)-sets, \(C\)-sets, and projective sets of each given class (in item 1 beginning with \(F_\sigma\), and in item 2 beginning with \(F_{\sigma\delta}\)). At the same time it is not even required that \(T_\Phi\) be the complement of some \(\Phi\)-set. Moreover, the complements \(CT_\Phi\) also contain perfect compact kernels. The exposition is based on § 4 of \((^3)\).

\(1^\circ.\) By \(\Phi\) in this item is meant an arbitrary \(\delta s\)-operation possessing the following property \((\tau)\): whatever the number \(n_0\) may be, there exists such a (finite or infinite) chain \(\{n_1,n_2,\ldots\}\) of operations \(\Phi\) that
\[ n_0<n_1<n_2<\cdots . \]

\(R\) is an arbitrary metric space in which there exist sets effectively distinct from all \(\Phi\)-sets \((^3,\S 6)\).

Theorem 1. Let the set \(T\) \((T \subset R)\) be effectively distinct from all \(\Phi\)-sets of the space \(R\). Whatever the \(\Phi\)-set \(M\) may be, \(M \subset T\), there exists a discontinuum \(D\) which is contained in \(T\) and does not meet \(M\): \(D \subset T \cdot (R-M)\).

Proof. Let \(M \cdot Z=M_0=\Phi\{F_n^0\}\), where \(\{F_n^0\}\in \Pi^*(Z)\), and let
\[ \nu\{F_n^0\}=x_0. \]
Then
\[ x_0\in T\cdot CM_0+CT\cdot M_0=T\cdot CM_0. \]
Denoting

\(F_n^0+\langle x_0\rangle = F_n^1\) and \(\nu\{F_n^1\}=x_1\), we obtain
\(\Phi\{F_n^1\}=\Phi\{F_n^0\}+\langle x_0\rangle\subset T\),
\(x_1\in T\cdot C\Phi\{F_n^1\}\), \(x_0\ne x_1\).

Surround the points \(x_0\) and \(x_1\) by such \(\varepsilon\)-neighborhoods, respectively \(S_0\) and \(S_1\), that \(\varepsilon<1/2\) and \(\overline S_0\cdot \overline S_1=0\). From the continuity of \(\nu\) on \(\Pi t(Z)\) there follows the existence of a number \(q_1\) such that
\[ \nu\{F_1^{t_1},\ldots,F_{q_1}^{t_1},Q_1,Q_2,\ldots,Q_n,\ldots\}\in S_{t_1} \]
\((t_1=0,1)\), whatever the point \(\{Q_n\}\in\Pi t(Z)\). We next introduce the notation:
\[ F_n^{t_1 0}=F_n^0,\quad \nu\{F_1^{t_1},\ldots,F_{q_1}^{t_1},F_{q_1+1}^{t_1 0},F_{q_1+2}^{t_1 0},\ldots,F_n^{t_1 0},\ldots\}=x_{t_1 0}, \]
\[ F_n^{t_1 1}=F_n^{t_1 0}+\langle x_{t_1 0}\rangle,\quad \nu\{F_1^{t_1},\ldots,F_{q_1}^{t_1},F_{q_1+1}^{t_1 1},F_{q_1+2}^{t_1 1},\ldots,F_n^{t_1 1},\ldots\}=x_{t_1 1}. \]
Hence, by virtue of the property of the function \(\nu\) and the property \((\tau)\) of the operation \(\Phi\), we obtain:
\[ x_{t_1 1}\in T\cdot C\Phi\{F_1^{t_1},\ldots,F_{q_1}^{t_1},F_{q_1+1}^{t_1 1},F_{q_1+2}^{t_1 1},\ldots,F_n^{t_1 1},\ldots\}, \]
\[ x_{t_1 0}\in \Phi\{F_1^{t_1},\ldots,F_{q_1}^{t_1},F_{q_1+1}^{t_1 1},F_{q_1+2}^{t_1 1},\ldots,F_n^{t_1 1},\ldots\}, \]
\(x_{t_1 0}\ne x_{t_1 1}\). This makes it possible to surround \(x_{t_1 0}\) and \(x_{t_1 1}\) by such \(\varepsilon\)-neighborhoods, respectively \(S_{t_1 0}\) and \(S_{t_1 1}\), that
\[ S_{t_1 t_2}\subset S_{t_1},\quad \overline S_{t_1 0}\cdot \overline S_{t_1 1}=0,\quad \varepsilon<1/4. \]

Suppose now that for \(m\ge 2\): 1) integers \(q_0,q_1,\ldots,q_{m-1}\) have been constructed, with \(q_0=0<q_1<\cdots<q_{m-1}\); 2) for each of the tuples
\[ t_1,\ t_1t_2,\ldots,t_1t_2\ldots t_m\quad (t_1,t_2,\ldots,t_m=0;1) \]
closed sets
\[ F_n^{t_1},\ F_n^{t_1 t_2},\ldots,F_n^{t_1\ldots t_m}\quad (n=1,2,\ldots) \]
have been constructed, with \(F_n^0\subseteq F_n^{t_1\ldots t_m}\) and
\[ \Phi\{F_1^{t_1},\ldots,F_{q_1}^{t_1},F_{q_1+1}^{t_1 t_2},\ldots,F_{q_2}^{t_1 t_2},\ldots; F_{q_{m-2}+1}^{t_1\ldots t_{m-1}},\ldots,F_{q_{m-1}}^{t_1\ldots t_{m-1}}; F_{q_{m-1}+1}^{t_1\ldots t_m},\ldots,F_n^{t_1\ldots t_m},\ldots\}\subset T; \]
3) the points
\[ x_{t_1\ldots t_m} =\nu\{F_1^{t_1},\ldots,F_{q_1}^{t_1}; F_{q_1+1}^{t_1 t_2},\ldots,F_{q_2}^{t_1 t_2},\ldots; F_{q_{m-2}+1}^{t_1\ldots t_{m-1}},\ldots,F_{q_{m-1}}^{t_1\ldots t_{m-1}}; F_{q_{m-1}+1}^{t_1\ldots t_m},\ldots,F_n^{t_1\ldots t_m},\ldots\} \]
are distinct for distinct tuples \(t_1t_2\ldots t_m\); 4) the points \(x_{t_1t_2\ldots t_m}\) are surrounded by \(\varepsilon\)-neighborhoods \(S_{t_1t_2\ldots t_m}\), where \(\varepsilon<(1/2)^n\), and the closures of these neighborhoods are pairwise disjoint.

From the continuity of \(\nu\) on \(\Pi t(Z)\) there follows the existence of a number \(q_m,\ q_m>q_{m-1}\), such that
\[ \nu\{F_1^{t_1},\ldots,F_{q_1}^{t_1},\ldots; F_{q_{m-1}+1}^{t_1\ldots t_m},\ldots,F_{q_m}^{t_1\ldots t_m}; Q_1,Q_2,\ldots,Q_n,\ldots\}\in S_{t_1t_2\ldots t_m}, \]
whatever the point \(\{Q_n\}\in\Pi t(Z)\).

Introduce the notation:
\[ F_n^{t_1\ldots t_m0}=F_n^{t_1\ldots t_m-1 0}; \]
\[ \nu\{F_1^{t_1},\ldots,F_{q_1}^{t_1},\ldots; F_{q_{m-1}+1}^{t_1\ldots t_m},\ldots,F_{q_m}^{t_1\ldots t_m}; F_{q_m+1}^{t_1\ldots t_m0},\ldots,F_n^{t_1\ldots t_m0},\ldots\}=x_{t_1\ldots t_m0}; \]
\[ F_n^{t_1\ldots t_m1}=F_n^{t_1\ldots t_m0}+\langle x_{t_1\ldots t_m0}\rangle; \]
\[ \nu\{F_1^{t_1},\ldots,F_{q_1}^{t_1},\ldots; F_{q_{m-1}+1}^{t_1\ldots t_m},\ldots,F_{q_m}^{t_1\ldots t_m}; F_{q_m+1}^{t_1\ldots t_m1},\ldots,F_n^{t_1\ldots t_m1},\ldots\}=x_{t_1\ldots t_m1}. \]

It is easy to see that
\[ x_{t_1\ldots t_m t_{m+1}}\in T\cdot S_{t_1\ldots t_m t_{m+1}},\quad F_n^0\subseteq F_n^{t_1\ldots t_m+1}; \]
\(x_{t_1\ldots t_m1}\in \overline E\), while
\[ x_{t_1\ldots t_m0}\in \Phi\{F_1^{t_1},\ldots,F_{q_1}^{t_1},\ldots; F_{q_{m-1}+1}^{t_1\ldots t_m},\ldots,F_{q_m}^{t_1\ldots t_m}; F_{q_m+1}^{t_1\ldots t_m1},\ldots,F_n^{t_1\ldots t_m1},\ldots\}; \]
\(x_{t_1\ldots t_m0}\ne x_{t_1\ldots t_m1}\).

We construct such \(\varepsilon\)-neighborhoods \(S_{t_1\ldots t_m0}\) and \(S_{t_1\ldots t_m1}\), \(\varepsilon<(1/2)^{m+1}\), respectively of the points \(x_{t_1\ldots t_m0}\) and \(x_{t_1\ldots t_m1}\), that
\[ S_{t_1\ldots t_m t_{m+1}}\subset S_{t_1\ldots t_m},\quad \overline S_{t_1\ldots t_m0}\cdot \overline S_{t_1\ldots t_m1}=0. \]

The inductively described process gives us, for any \(k\), points \(x_{t_1\ldots t_k}\), their \(\varepsilon\)-neighborhoods \(S_{t_1\ldots t_k}\), and closed sets \(F_n^{t_1\ldots t_k}\). Their properties show that for any sequence \(t_1,t_2,\ldots,t_k,\ldots\)
\[ \prod_{k=1}^{\infty}\overline S_{t_1\ldots t_k} = \nu\{F_1^{t_1},\ldots,F_{q_1}^{t_1},\ldots; F_{q_{k-1}+1}^{t_1\ldots t_k},\ldots,F_{q_k}^{t_1\ldots t_k}; F_{q_k+1}^{t_1\ldots t_{k+1}},\ldots,F_{q_{k+1}}^{t_1\ldots t_{k+1}},\ldots\} \]
and that, consequently, the set
\[ D=\sum_{t_1t_2\ldots t_k\ldots}\prod_{k=1}^{\infty}\overline S_{t_1\ldots t_k} \]
is a discontinuum, with \(D\subset T\cdot CM\). The theorem is proved.

\(2^\circ\). Here, by \(\Phi\) will be meant a \(\delta s\)-operation all of whose chains are infinite and which has the following property \((\omega)\): whatever sequences of sets
\(\{Q_{m1}\},\ \{Q_{m2}\},\ldots,\{Q_{mk}\},\ldots\) may be, one has
\[ \Phi\{\Phi_1\{Q_{m1}\},\ \Phi_1\{Q_{m2}\},\ldots,\Phi_1\{Q_{mk}\},\ldots\} = \Phi\{Q_1,Q_2,\ldots,Q_n,\ldots\}, \]
where the sequence \(\{Q_1,Q_2,\ldots,Q_n,\ldots\}\) is obtained as a result

arrangement of the sequences of sets \(\{Q_{m1}\}, \{Q_{m2}\}, \ldots, \{Q_{mk}\}, \ldots\) into one sequence of the same sets with the aid of some one-to-one mapping \(n=\varphi(mk)\) of the collection of all pairs \((mk)\) of natural numbers onto the natural series \(\{n\}\). By \(\Phi_1\) is denoted the operation of countable summation. We shall use the following notation:
\[ \Phi\{\{Q_{m1}\}, \{Q_{m2}\}, \ldots, \{Q_{mk}\}, \ldots\} = \Phi\{\Phi_1\{Q_{m1}\}, \Phi_1\{Q_{m2}\}, \ldots, \Phi_1\{Q_{mk}\}, \ldots\}; \]
if \(Q_{mk}\in F(Z)\), then
\[ \nu\{\{Q_{m1}\}, \ldots, \{Q_m\}, \ldots\} = \nu\{Q_1,Q_2,\ldots,Q_n,\ldots\}. \]

\(R\) will denote in this subsection a metric space in which the difference of two closed sets is an \(F_\sigma\)-set (for example, \(R\) is a space with a countable base). It follows from this that, subtracting from the \(F_\sigma\)-set \(\Phi_1\{F_n^0\}\) of the space \(R\) a closed set \(F\), we obtain an \(F_\sigma\)-set:
\[ \Phi_1\{F_n^0\}-F=\Phi_1\{F_n^1\}. \]

Theorem 2. If a set \(T\) of the space \(R\) is effectively distinct from all \(\Phi\)-sets of this space, then every \(\Phi\)-set \(N\) containing \(T\) contains within itself a discontinuum \(D\) which does not intersect \(T\): \(D\subset N\cdot(R-T)\).

Let \(T\cdot Z=Y,\; N\cdot Z=N_0=\Phi\{F_n^0\}\), where \(\{F_n^0\}\in \Pi_t(Z)\). The set \(Y\) is effectively distinct from all \(\Phi\)-sets of the space \(Z\) ((3), p. 136). With the aid of the mapping \(\varphi(mk)=n\), represent \(\Phi\{F_1^0,\ldots,F_n^0,\ldots\}\) in the form
\[ \Phi\{\{F_{m1}^0\},\ldots,\{F_{mk}^0\},\ldots\}; \]
introduce the notation:
\[ \nu\{\{F_{m1}^0\},\ldots,\{F_{mk}^0\},\ldots\}=x_0, \]
\[ \Phi_1\{F_{mk}^0\}-\langle x_0\rangle=\Phi_1\{F_{mk}^1\}; \qquad \nu\{\{F_{m1}^1\},\ldots,\{F_{mk}^1\},\ldots\}=x_1. \]

The further construction of the proof presents no difficulty and is carried out analogously to paragraph \(1^\circ\). The difference in the construction of the points \(x_{t_1\ldots t_p}\) consists only in the fact that, instead of the equality
\[ F_n^{t_1\ldots t_p{}_1}=F_n^{t_1\ldots t_p0}+\langle x_{t_1\ldots t_p0}\rangle \quad (p.1^\circ) \]
there is the equality
\[ \Phi_1\{F_{mk}^{t_1\ldots t_p1}\} = \Phi_1\{F_{mk}^{t_1\ldots t_p0}\} - \langle x_{t_1\ldots t_p0}\rangle. \]

At the end of the construction we arrive at the desired discontinuum
\[ D= \sum_{t_1,t_2\ldots t_p\ldots}^{\infty} \prod_{p=1}^{\infty}\overline{S}_{t_1\ldots t_p}, \]
all of whose points are points of the form
\[ \nu\{\{F_{m1}^{t_1}\},\ldots,\{F_{mq_1}^{t_1}\}; \{F_{mq_1+1}^{t_1t_2}\},\ldots,\{F_{mq_2}^{t_1t_2}\}; \ldots; \{F_{mq_{p-1}+1}^{t_1\ldots t_p}\},\ldots,\{F_{mq_p}^{t_1\ldots t_p}\}; \ldots\}. \]

\(3^\circ\). Considering that \(\Phi\) is an arbitrary \(\delta s\)-operation, suppose that \(T\) (\(T\subset R\)) is effectively distinct from all \(\Phi\)-sets of the space \(R\). Let \(K\) be the image of the set \(\Pi_t(Z)\) under the mapping \(\nu\), and suppose that it coincides with the image of the whole set \(\Pi_\Phi(R)\) under the same mapping.

It can be proved that the set \(T\cdot K+L\), where \(L\) is an arbitrary set of the space \(R\) not intersecting \(K\), is effectively distinct from all \(\Phi\)-sets of the space \(R\). From this follows the following proposition on the general type of properties generated by effective distinction. Denote by \(T\) a set effectively distinct from all \(\Phi\)-sets, by \(\Pi\) a universal proposition, and by \(\Sigma\) an existence proposition.

Theorem 3. In order that the property \(\alpha\) be inherent in every set effectively distinct from all \(\Phi\)-sets, it is necessary and sufficient that the condition
\[ \prod_T \sum_{K\subseteq R}\prod_{L\subseteq R-K} \bigl(T\cdot K+L\ \text{has property } \alpha\bigr) \]
be fulfilled. If \(R\) is a Euclidean space, then \(K\) is compact.

It follows from this theorem that measurability and Baire’s property do not belong to the general type of properties generated by effective distinction.

Crimean State Pedagogical Institute
named after M. V. Frunze

Received
3 VII 1957

CITED LITERATURE

\(^{1}\) P. S. Novikov, Izv. AN SSSR, ser. math., 3, No. 1, 35 (1939).
\(^{2}\) S. Saks, Matem. sborn., 7(49), 373 (1940).
\(^{3}\) Ya. L. Kreinin, Matem. sborn., 33(80), issue 2, 129 (1956).