UDC 513.83

MATHEMATICS

M. M. CHOBAN

MULTIVALUED MAPPINGS

AND SPACES WITH A COUNTABLE NETWORK*

(Presented by Academician P. S. Aleksandrov on 1 XI 1968)

Let \(X\) be a topological space. Consider a certain system \(F_\pi(X)\) of closed subsets of the space \(X\). The system \(F_\pi(X)\) is complete if there exists a topological space \(\widetilde X \supseteq X\), complete in the sense of Čech, for which \(F_\pi(X) \subseteq F(\widetilde X)\). By \(F(\widetilde X)\) we denote the collection of all closed subsets of the space \(\widetilde X\).

Examples of complete systems:

1. The system \(C(X)\) of nonempty bicompact subsets of the space \(X\).

2. Let \((X,\rho)\) be a metric space. Then the system
   \[ F_\pi(X)=\{L\in F(X)\mid L \text{ is complete in the metric } \rho\} \]
   is complete.

In those cases when it is important to emphasize the complete system \(F_\pi(X)\) of the space \(X\), we shall use the notation \((X,F_\pi(X))\).

Let pairs \((X,F_\pi(X))\) and \((Y,F_\pi(Y))\) be given, where \(X\) and \(Y\) are completely normal spaces. The pairs \((X,F_\pi(X))\) and \((Y,F_\pi(Y))\) are \((\alpha,\beta)\)-homeomorphic if there exists a one-to-one mapping \(f:X\to Y\) such that:

1) the mapping \(f\) is \(B\)-measurable** of class \(\alpha\);

2) the mapping \(f^{-1}\) is \(B\)-measurable of class \(\beta\);

3) \(F_\pi(Y)=\{fL\mid L\in F_\pi(X)\}\).

Let \(Z\) be some set and let \(L_1(Z)\) be some algebra of subsets of the set \(Z\) (i.e., if \(A,B\in L_1(Z)\), then \(A\cup B\), \(A\cap B\), \(Z\setminus A\in L_1(Z)\)). Put
\[ S_1(Z)=\left\{\bigcup_{n=1}^{\infty} A_n \mid A_n\in L_1(Z),\ n=1,2,\ldots\right\}. \]
Denote by \(L_\beta(Z)\) the algebra of subsets generated by the system
\[ \bigcup\{S_\alpha(Z)\mid \alpha<\beta\}, \]
where \(\beta<\omega_1\). Put
\[ S_\beta(Z)=\left\{\bigcup_{n=1}^{\infty} A_n \mid A_n\in L_\beta(Z),\ n=1,2,\ldots\right\}. \]
The pair \((X,F_\pi(X))\) satisfies property \(l(\beta)\) (respectively, property \(u(\beta)\)) if, for every mapping \(\theta: Z\to F_\pi(X)\), where \(\theta^{-1}A\in S_1(Z)\) as soon as the set \(A\) is open (respectively, closed) in \(X\), there exists a one-to-one mapping*** \(\varphi:Z\to X\) such that:

1) \(\varphi z\in \theta z\) for every point \(z\in Z\);

2) for every open set \(U\) in \(X\) we have \(\varphi^{-1}U\in S_\beta(Z)\).

In [7] the following is proved:

Theorem 1 (Kuratowski, Ryll-Nardzewski). Let \(X\) be a separable metric space. Then, for any complete system \(F_\pi(X)\) of the space \(X\), the pair \((X,F_\pi(X))\) satisfies condition \(l(1)\).

The following lemmas are easily proved.

* A system \(S\) of subsets of the space \(X\) is called a network in \(X\) if, for any \(x\) and \(Ox\)—a neighborhood of it in \(X\)—there is a \(P\in S\) such that \(x\in P\subseteq Ox\). The concept of a network was introduced by A. V. Arhangel’skii and successfully used by him in works [1–3]. Note that all spaces are assumed in advance to be completely regular.

** For the definition and basic properties of \(B\)-measurable mappings, see [6], p. 382.

*** Such mappings are called selections.

Lemma 1. Let the pair \((X,F_\pi(X))\), where \(X\) is a perfectly normal space, satisfy property \(l(\beta)\). Then \((X,F_\pi(X))\) satisfies property \(u(\beta)\).

Lemma 2. Let \((X,C(X))\), where \(X\) is a perfectly normal space, satisfy property \(u(\beta)\). Then \((X,C(X))\) also satisfies property \(l(1+\beta)\).

Lemma 3. Let \(g:X\to Y\) be a continuous perfect mapping. If the pair \((X,C(X))\) satisfies property \(u(\beta)\), then the pair \((Y,C(Y))\) also satisfies property \(u(\beta)\).

The following establishes a curious connection.

Theorem 2. Let the pairs \((X,F_\pi(X))\) and \((Y,F_\pi(Y))\) be \((0,\alpha)\)-homeomorphic. If the pair \((Y,F_\pi(Y))\) satisfies property \(l(\beta)\) (respectively property \(u(\beta)\)), then the pair \((X,F_\pi(X))\) satisfies property \(l(\beta+\alpha)\) (respectively property \(u(\beta+\alpha)\)).

Proof. Let \(f:X\to Y\) be a \((0,\alpha)\)-homeomorphism for which
\[ F_\pi(Y)=\{fL\mid L\in F_\pi(X)\}. \]
Let, further, \(\theta:Z\to F_\pi(X)\) be such that \(\theta^{-1}U\in S_1(Z)\) whenever \(U\) is open (respectively closed) in \(X\). Consider \(\psi:Z\to F_\pi(Y)\), where \(\psi z=f(\theta z)\) for every point \(z\in Z\). By the continuity of the mapping \(f\), we have \(\psi^{-1}V\in S_1(Z)\) whenever the set \(V\) is open (respectively closed) in \(Y\). By hypothesis there exists a single-valued mapping \(\varphi:Z\to Y\) such that \(\varphi z\in\theta z\) and \(\varphi^{-1}G\in S_\beta(Z)\) whenever the set \(G\) is open in \(Y\). Put \(g:Z\to X\), where
\[ gz=f^{-1}(\varphi z). \]
Clearly \(gz\in\theta z\) for every point \(z\in Z\). Let \(U\) be an arbitrary open set of the space \(X\). Since \(fU\) is a Borel set of class \(\le \alpha\), it follows that
\[ \varphi^{-1}(fU)\in S_{\beta+\alpha}(Z). \]
In view of the equality
\[ g^{-1}U=\varphi^{-1}(fU), \]
we have
\[ g^{-1}U\in S_{\beta+\alpha}(Z). \]
This proves Theorem 2.

Theorem 3. Let \(X\) be a paracompact space with a \(\sigma\)-discrete net. For every complete system \(F_\pi(X)\) of subsets of the space \(X\) there exists a continuous one-to-one mapping \(f:X\to Y\), where \((Y,\rho)\) is a metrizable space, such that: 1) \(fU\) is an \(F_\sigma\)-set for every open set \(U\) in \(X\); 2) \(f|L\) is a homeomorphism for every set \(L\in F_\pi(X)\); 3) the set \(fL\) is complete with respect to the metric \(\rho\) for every set \(L\in F_\pi(X)\). Moreover, if \(X\) has a countable net, then \(Y\) has a countable base.

Theorem 3 is easily derived from the following two propositions.

Proposition 1. Let \(f:X\to Y\) be a one-to-one continuous mapping of a paracompact space \(X\) onto a metrizable space \(Y\). For every complete system \(F_\pi(X)\) there exists a one-to-one continuous mapping \(g:X\to Z\) of the space \(X\) onto a metrizable space \(Z\) such that: 1) \(g|L\) is a homeomorphism for every \(L\in F_\pi(X)\); 2)
\[ F_\pi(Z)=\{gA\mid A\in F_\pi(X)\} \]
is a complete system of closed subsets of the space \(Z\).

Proof. Let \(\widetilde X\) be a Čech-complete topological space for which \(X\subseteq\widetilde X\) and \(F_\pi(X)\subseteq F(X)\). Then there exists a paracompact Čech-complete space \(X'\) such that
\[ X\subseteq X'\subseteq\widetilde X. \]
This fact is proved by the same methods as Theorem 5.8 in (3).

Clearly,
\[ F_\pi(X)\subseteq F(X'). \]
In this case there exists a perfect mapping \(\varphi:X'\to S\), where \(S\) is a complete metrizable space (see (3, 8)). Consider \(g:X\to Z\subseteq S\times Y\), where
\[ gx=(\varphi x,fx) \]
for every point \(x\in X\). By Lemma 1.5 of (5), \(g|L\) is a homeomorphism for every set \(L\in F_\pi(X)\). Let \(\rho\) be an arbitrary metric on \(Y\) and \(d\) some complete metric on \(S\). In the space \(S\times Y\) consider the metric
\[ \mu((s,y),(s',y'))=d(s,s')+\rho(y,y'). \]
It is easily verified that the set \(gL\) is complete with respect to the metric \(\mu\) for every set \(L\in F_\pi(X)\). Proposition 1 is proved.

Proposition 2. Let \(X\) be a perfectly normal paracompact space. Let, further,
\[ \gamma=\{F_\alpha\mid \alpha\in A\} \]
be a discrete system

closed subsets of \(X\). Then there exists a continuous mapping \(f:X\to Y\), where \(Y\) is a metric space, such that the system \(\omega=\{fF_\alpha\mid \alpha\in A\}\) is closed and discrete in \(X\), and, moreover, the set \(F_\alpha\) is marked* for every \(\alpha\in A\).

Proof. By hypothesis there exists a discrete system \(\Omega=\{U_\alpha\mid \alpha\in A\}\) of open subsets of the space \(X\) such that \(F_\alpha\subseteq U_\alpha\) for every \(\alpha\in A\). By the perfectly normality of the space \(X\), for each \(\alpha\in A\) there exists a continuous function \(f_\alpha(x)\) such that

\[ f_\alpha(x)= \begin{cases} 0, & \text{if } x\in X\setminus U_\alpha,\\ 1, & \text{if } x\in F_\alpha,\\ 0<f_\alpha(x)<1, & \text{if } x\in U_\alpha\setminus F_\alpha. \end{cases} \]

Consider the mapping \(f:X\to S(A)\)**, where \(fx=\{f_\alpha(x)\}\in S(A)\). Put

\[ g_\alpha(\beta)= \begin{cases} 0, & \text{if } \beta\ne \alpha,\\ 1, & \text{if } \beta=\alpha. \end{cases} \]

It is obvious that \(fF_\alpha=g_\alpha\) and \(f^{-1}g_\alpha=F_\alpha\) for every \(\alpha\in A\). Moreover, the system of points \(\omega=\{g_\alpha=fF_\alpha\mid \alpha\in A\}\) is discrete. This proves Proposition 2.

Remark 1. In Proposition 2 it suffices to assume that \(X\) is collectively normal and that \(F_\alpha\) is a \(G_\delta\)-set for every \(\alpha\in A\). Theorem 3, together with Theorems 1, 2 and Lemma 1, permits the following conclusion.

Theorem 4. Let \(F_\pi(X)\) be a complete system of subsets of a space \(X\) with a countable network. Then the pair \((X,F_\pi(X))\) satisfies the conditions \(l(2), u(2)\).

A single-valued mapping \(\varphi:Y\to X\) is called an \(F_\alpha\)-section (respectively, a \(G_\alpha\)-section) for a multivalued mapping \(\theta:Y\to F_\pi(X)\), if: 1) \(\varphi y\in \theta y\) for every point \(y\in Y\); 2) for every open set \(U\) in \(X\), the set \(\varphi^{-1}U\) belongs to the class*** \(F_\alpha(Y)\) (respectively, to the class \(G_\alpha(Y)\)).

Theorem 5. Let the pairs \((X,F_\pi(X))\) and \((Y,F_\pi(Y))\) be \((0,\alpha)\)-homeomorphic. Further, let \(Z\) be some topological space. If for every continuous** mapping \(\theta:Z\to F_\pi(Y)\) there exists an \(F_\beta\)-section, then for every continuous mapping \(\psi:Z\to F_\pi(X)\) there exists:

a) a \(G_{\beta+\alpha-1}\)-section, if \(\beta\) is even and \(\alpha\) is odd and finite;
b) a \(G_{\beta+\alpha}\)-section, if \(\beta\) is odd and \(\alpha\) is odd and finite;
c) a \(G_{\beta+\alpha}\)-section, if \(\alpha\) is even and infinite;
d) an \(F_{\beta+\alpha-1}\)-section, if \(\beta\) is even and \(\alpha\) is even and finite;
e) an \(F_{\beta+\alpha}\)-section, if \(\beta\) is odd and \(\alpha\) is even and finite;
f) an \(F_{\beta+\alpha}\)-section, if \(\alpha\) is odd and infinite.

* A set \(F\) is marked if \(f^{-1}fF=F\). For properties of marked sets see [5].

** By \(S(A)\) we denote the totality of all real functions defined on \(A\) with norm

\[ \|g\|=\sum_{\beta\in A}|g(\beta)|<\infty \]

for every \(g\in S(A)\).

*** Let \(Y\) be a topological space. \(F_0(Y)\) is the family of closed \(G_\delta\)-sets. The elements of the family \(F_\alpha(Y)\) are intersections or unions of countable sequences of sets belonging to \(\bigcup\{F_\beta(Y)\mid \beta<\alpha\}\), depending on whether the number \(\alpha\) is even or odd. Put

\[ G_\alpha(Y)=\{A=Y\setminus L\mid L\in F_\alpha(Y)\}. \]

**** A mapping \(\theta:Z\to F_\pi(Y)\) is lower (upper) semicontinuous if, for every open (closed) set \(A\subseteq Y\), the set \(\theta^{-1}A=\{z\in Z\mid \theta z\cap A\ne\varnothing\}\) is open (closed) in the space \(Z\). A mapping \(\theta\) is continuous if it is simultaneously lower and upper semicontinuous (see [6,9]).

Proof. Let \(f:X\to Y\) be a \((0,\alpha)\)-homeomorphic mapping for which
\[ F_\pi(Y)=\{fL\mid L\in F_\pi(X)\}. \]
Let, further, \(\psi:Z\to F_\pi(X)\) be a continuous mapping. Consider \(\theta:Z\to F_\pi(Y)\), where \(\theta z=f(\psi z)\) for every point \(z\in Z\). By the continuity of the mapping \(f\), the mapping \(\theta\) is continuous. Consequently, there exists an \(F_\beta\)-section \(\varphi:Z\to Y\). Consider \(g:Z\to X\), where \(gz=f^{-1}(\varphi z)\) for every point \(z\in Z\). It is obvious that \(g\) is a section for the mapping \(\psi\). By elementary computations one can establish that the mapping \(g\) is the desired one.

The following theorem is proved analogously:

Theorem 6. Let the pairs \((X,F_\pi(X))\) and \((Y,F_\pi(Y))\) be \((0,\alpha)\)-homeomorphic. Let, further, \(Z\) be some topological space. If for every lower semicontinuous (respectively, upper semicontinuous) mapping \(\theta:Z\to F_\pi(Y)\) there exists an \(F_\beta\)-section, then also for every lower semicontinuous (respectively, upper semicontinuous) mapping \(\psi:Z\to F_\pi(X)\) there exists a section satisfying conditions a)—f) of Theorem 5.

Remark 2. If, under the hypotheses of Theorem 5 or 6, for the mappings \(\theta:Z\to F_\pi(Y)\) there exist \(G_\beta\)-sections, then for the mappings \(\psi:Z\to F_\pi(X)\) there exist: a) \(G_{\beta+\alpha}\)-sections when \(\alpha\) is even; b) \(F_{\beta+\alpha}\)-sections when \(\alpha\) is odd.

In paper (9) a number of cases were established in which, for multivalued mappings in metric spaces, \(F_1\)-sections exist. Theorems 3, 5 and 6 and the corresponding theorems from (9) make it possible to construct \(G_2\)-sections for multivalued mappings in paracompact spaces with a \(\sigma\)-discrete network. For example, from the above-mentioned theorems and Theorem 2 from (9) there follows

Theorem 7. Let \(F_\pi(X)\) be a complete system of a paracompact space \(X\) with a \(\sigma\)-discrete network. Then for every continuous mapping \(\theta:Y\to F_\pi(X)\), where \(Y\) is a perfectly normal space, there exists a \(G_2\)-section.

In the same way, Theorems 6, 10, 14 and Corollary 2 from (9) carry over to paracompact spaces with a \(\sigma\)-discrete network. We note that Theorem 11 from (9) also carries over, but with greater difficulties.

REFERENCES

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