UDC 513.83

MATHEMATICS

M. M. CHOBAN

MULTIVALUED MAPPINGS

AND SOME RELATED QUESTIONS

(Presented by Academician P. S. Aleksandrov on 6 VI 1969)

Let \(X\) be a topological space. Denote by \(A(X)\) the collection of all nonempty subsets of the space \(X\). A single-valued mapping \(\theta: Y \to A(X)\) is called a multivalued mapping of the space \(Y\) into \(X\). A single-valued mapping \(f: Y \to X\) is called a section for the multivalued mapping \(\theta: Y \to A(X)\) if \(fx \in \theta x\) for every point \(x \in Y\) \((^{5-10})\). We note that, unless a separation axiom is explicitly named, the space should be regarded as completely regular.

§ 1. Continuous sections and \(\sigma\)-discrete sets. Put
\[ C(X)=\{L\in A(X)\mid L \text{ is bicompact}\}. \]

Theorem 1. Let \(\theta: Y \to A(X)\) be a lower semicontinuous* mapping of a paracompact space \(Y\) into a complete metric space \(X\). If the set
\[ Y_1=\{y\in Y\mid \theta y \text{ is not closed in } X\} \]
is \(\sigma\)-discrete in \(Y\), then there exist multivalued mappings \(\varphi: Y \to C(X)\) and \(\psi: Y \to C(X)\) such that: a) \(\varphi y \subseteq \psi y \subseteq \theta y\) for every point \(y\in Y\); b) the mapping \(\varphi\) is lower semicontinuous; c) the mapping \(\psi\) is upper semicontinuous.

Theorem 1 allows us to strengthen a number of theorems from \((^{5,6})\). In particular, we obtain:

Theorem 2. Let \(\theta: Y \to A(X)\) be a lower semicontinuous mapping of a paracompact space \(Y\) into a Banach space \(X\). If
\[ Y_1=\{y\in Y\mid \theta y \text{ is not closed in } X\} \]
is \(\sigma\)-discrete in \(Y\), then there exists a continuous single-valued mapping \(f: Y \to X\) such that: a) \(fy\in \operatorname{conv}(\theta y)\) for every point \(y\in Y\setminus Y_1\); b) \(fy\in \theta y\) for every point \(y\in Y_1\).

Theorem 3. Let \(\theta: Y \to A(X)\) be a lower semicontinuous mapping of a zero-dimensional paracompact space \(Y\) into a complete metric space \(X\). If
\[ Y_1=\{y\in Y\mid \theta y \text{ is not closed in } X\} \]
is \(\sigma\)-discrete in \(Y\), then there exists a continuous section \(f: Y \to X\) for the mapping \(\theta\).

Corollary 1. Let \(\theta: Y \to A(X)\) be a lower semicontinuous mapping of a \(\sigma\)-discrete paracompact space \(Y\) into a \(T_1\)-space \(X\) with the first axiom of countability. Then there exists a continuous section \(f: Y \to X\) for the mapping \(\theta\).

§ 2. \(\sigma\)-continuous sections and dimension. In \((^{3,4})\) A. V. Arhangel’skii obtained a number of results on preservation of dimension under open and closed mappings. In doing so, A. V. Arhangel’skii used, in implicit form, the method of sections for single-valued mappings.

* A mapping \(\theta: Y \to A(X)\) is lower (upper) semicontinuous if, for every open (closed) set \(A\) in \(X\), the set
\[ \theta^{-1}A=\{y\in Y\mid \theta y\cap A\ne \varnothing\} \]
is open (closed) in \(Y\). Following V. Hurewicz (and in fact also Cauchy), V. I. Ponomarev in his fundamental works on multivalued mappings \((^{12-14})\) calls mappings that are upper semicontinuous and, respectively, lower semicontinuous, continuous and, respectively, quasicontinuous.

A single-valued mapping \(f:X\to Y\) is \(\sigma\)-continuous if \(X=\bigcup_{i=1}^{\infty}X_i\) and \(f|X_i\) is continuous for every natural number \(i\). If, moreover, for every natural number \(i\) the set \(X_i\) is closed in \(X\), then the mapping \(f\) is called \(F_\sigma\)-continuous.

Theorem 4. Let \(\theta:Y\to A(X)\) be an upper semicontinuous isolated mapping* of the space \(Y\) into a regular space \(X\) with a countable net. Then there exists a \(\sigma\)-continuous selection \(f:Y\to X\) for the mapping \(\theta\).

Theorem 5. Let \(\theta:Y\to A(X)\) be a \(Y\)-closed mapping (see (7)) of a perfectly normal space \(Y\) into a metric space \(X\). Suppose, further, that \(\psi:Y\to A(X)\) is a lower semicontinuous isolated mapping such that \(\psi y\subset \theta y\) for every point \(y\in Y\). Then there exists an \(F_\sigma\)-continuous selection for the mapping \(\theta\).

Theorem 6. Let \(\theta:Y\to A(X)\) be a lower semicontinuous mapping of a perfectly normal paracompact space \(Y\) into a complete metric space \(X\). If
\[ Y_1=\{y\in Y\mid \theta y \text{ is not closed in } X \text{ and } \operatorname{card}\theta y>\aleph_0\} \]
is \(\sigma\)-discrete in \(Y\), then there exists an \(F_\sigma\)-continuous selection for the mapping \(\theta\).

A system \(F_\pi(X)\) of subsets of a space \(X\) is called complete if there exists a space \(\tilde X\supset X\), complete in the sense of Čech, such that \([L]_{\tilde X}=L\) for every \(L\in F_\pi(X)\) (see \((8,9)\)).

A topological space \(X\) is called a \(G_\delta\)-space if \(\{(x,x)\mid x\in X\}\) is a \(G_\delta\)-set in \(X\times X\).

Theorem 7. Let \(\theta:Y\to F_\pi(X)\) be a \(Y\)-closed mapping of a perfectly normal space \(Y\) into a complete system \(F_\pi(X)\) of a paracompact \(G_\delta\)-space \(X\). Suppose, further, that \(\psi:Y\to A(X)\) is a lower semicontinuous isolated mapping such that \(\psi y\subset \theta y\) for every point \(y\in Y\). Then there exists an \(F_\sigma\)-continuous selection for the mapping \(\psi\).

Theorem 8. Let \(\theta:Y\to A(X)\) be a \(Y\)-closed lower semicontinuous isolated mapping of a perfectly normal space \(Y\) into a space \(X\) with a \(\sigma\)-discrete net. Then there exists an \(F_\sigma\)-continuous selection for the mapping \(\theta\).

We note that all symmetrizable spaces with the first axiom of countability and all spaces with a \(\sigma\)-discrete net are \(G_\delta\)-spaces.

From Theorem 8 it follows that

Corollary 2. Let \(f:X\to Y\) be an open-closed continuous and isolated mapping of a space \(X\) with a \(\sigma\)-discrete net onto \(Y\). Then \(\dim Y\le \dim X\) and \(\operatorname{Ind}Y\le \operatorname{Ind}X\). Moreover, if \(X\) is weakly finite-dimensional or countably dimensional, then so is \(Y\).

Theorem 7 allows us to obtain

Corollary 3. Let \(f:X\to Y\) be a closed, inductively open and isolated mapping of a paracompact \(G_\delta\)-space \(X\) onto a perfectly normal space \(Y\). Then \(\dim Y\le \dim X\) and \(\operatorname{Ind}Y\le \operatorname{Ind}X\). Moreover, if \(X\) is weakly finite-dimensional or countably dimensional, then so is \(Y\).

Theorem 9. Let the space \(X\) satisfy the weak first axiom of countability (see (1), p. 148). If \(X\) is the union of a countable number of closed symmetrizable subspaces, then \(X\) is symmetrizable.

Theorem 10. Let \(\{U_\alpha\mid \alpha\in A\}\) be an open point-finite cover of a \(T_1\)-space \(X\). If each \(U_\alpha\) is symmetrizable, then so is \(X\).

Theorem 11. Let \(\{F_\alpha\mid \alpha\in A\}\) be a closed locally finite cover of a space \(X\) by symmetrizable subspaces. Then \(X\) is symmetrizable.

Theorems 9 and 11 allow us to obtain

Corollary 4. Let \(f:X\to Y\) be a closed, inductively open

* The mapping \(\theta:Y\to A(X)\) is isolated if for every point \(y\in Y\) there is an isolated point in \(\theta y\) (relative to \(\theta y\)).

and an isolated mapping of a paracompact symmetrizable space \(X\) with the first axiom of countability. Then \(Y\) is symmetrizable.

§ 3. Generalized sections. A \(k\)-metric space is a pair \((X,d)\), where \(X\) is a topological space and \(d\) is a pseudometric on \(X\) satisfying the following conditions: a) \(H(x,d)=\{y\in X\mid d(y,x)=0\}\) is a bicompactum; b) if \(x\in [M]\), then \(d(x,M)=0\); c) let \(\Phi\) be a bicompactum and \(F\) a closed subset of \(X\). If \(d(x,y)>0\) for any \(x\in\Phi\) and \(y\in F\), then also \(d(\Phi,F)>0\).

For any \(L\subseteq X\) put \(H(L,d)=\{x\in X\mid d(x,L)=0\}\).

Theorem 12. Let \(\theta:X\to A(Y)\) be a lower semicontinuous mapping of a paracompact space \(X\) into a \(k\)-metric space \((Y,d)\). Suppose, furthermore, that \(F_\pi(Y)=\{\theta x\mid x\in X\setminus X_1\}\), where \(X_1\) is a \(\sigma\)-discrete subset of \(X\), is a complete system of the space \(Y\). Then there exists an upper semicontinuous mapping \(\psi:X\to C(Y)\) such that \(\psi x\subseteq H(\psi x,d)\) and \(\theta x\cap\psi x\ne\varnothing\) for every point \(x\in X\).

Theorem 13. Let \(f:X\to Y\) be an open continuous mapping of a \(p\)-space \(X\) onto a paracompact \(Y\). If the system \(\{f^{-1}y\mid y\in Y\setminus Y_1\}\), where \(Y_1\) is a \(\sigma\)-discrete system in \(Y\), is complete in \(X\), then there exists such a closed subset \(X_1\subseteq X\) that \(fX_1=Y\) and \(f|X_1\) is a perfect mapping. In particular, \(Y\) is a \(p\)-space.

Corollary 5. Let \(f:X\to Y\) be an open bicompact mapping of a \(p\)-space \(X\) onto \(Y\). Then \(f\) is a \(k\)-covering mapping.

§ 4. The case of topological groups. Let \(\gamma\) be some uniform structure on \(X\). Denote by \(Z_\gamma(X)\) the collection of all \(\gamma\)-null-sets of the space \(X\). Consider the system \(A_\gamma(X)\) obtained as a result of applying the \(A\)-operation to the system \(Z_\gamma(X)\). The sets of the system \(A_\gamma(X)\) will be called \(\gamma\)-\(A\)-sets of the space \(X\).

Let \(G\) be a topological group. In what follows, by \(\gamma\) we shall denote the left uniform structure of the group \(G\).

Theorem 14. Let \(f:G\to X\) be an open mapping of an almost metrizable* group \(G\) onto a separable Fréchet–Urysohn space \(X\). If \(\{f^{-1}x\mid x\in X\}\subseteq A_\gamma(G)\), then there are a bicompact subgroup \(H\) of countable character in \(G\) and an open mapping \(\varphi:G/H\to X\) such that \(f=\varphi\circ\pi\), where \(\pi:G\to G/H\) is the natural mapping of \(G\) onto the quotient space \(G/H\). If, moreover, \(f^{-1}x\) is a finally compact subspace for every point \(x\in X\), then the space \(X\) has a countable base and \(G\) is finally compact.

Corollary 6. Let \(f:G\to X\) be an open bicompact mapping of an almost metrizable group \(G\) onto a regular separable space \(X\) with the first axiom of countability. Then \(X\) is metrizable.

Theorem 15. Let \(f:G\to X\) be an open continuous mapping of an almost metrizable group \(G\) onto a locally separable metrizable space \(X\). If \(F_\pi(G)=\{f^{-1}x\mid x\in X\}\subseteq A_\gamma(G)\) and the system \(F_\pi(G)\) is complete in \(G\), then for every zero-dimensional subspace \(X_1\subseteq X\) there exists a subset \(G_1\subseteq G\) such that \(fG_1=X_1\) and \(f|G_1\) is a homeomorphism.

Theorem 16. Let \(f:G\to X\) be an open continuous mapping of a locally bicompact group \(G\) onto a metrizable space \(X\). Then for every zero-dimensional subspace \(X_1\subseteq X\) there exists a subset \(G_1\subseteq G\) such that \(fG_1=X_1\) and \(f|G_1\) is a homeomorphism.

§ 5. An approximation theorem for upper semicontinuous mappings.

Theorem 17. Let \(\theta:X\to A(Y)\) be an upper semicontinuous mapping of a paracompact space \(X\) into a linearly topological space \(Y\). Suppose, furthermore, that \(V_0\) is some convex neighborhood of zero in \(Y\). If for every point \(x\in X\) there is a point \(a(x)\in Y\) for which

* A group is almost metrizable if it contains a bicompactum of countable character.

\(a(x)+V_0 \ni \theta x\), then there exists such a continuous single-valued mapping \(f:X\to Y\) that \(fx+V_0\ni \theta x\) for every point \(x\in X\).

§ 6. Theorems on the nonexistence of sections. Let a continuous mapping \(f:X\to Y\) be given. It is said that there exists a continuous section over a set \(Y_1\subseteq Y\) if there is a subset \(X_1\subseteq X\) such that \(fX_1=Y_1\) and \(f|X_1\) is a homeomorphism. The nonexistence of continuous sections is connected either with the absence of zero-dimensionality, or with the absence of completeness of the inverse images of points.

Theorem 18. Let \(X\) be a compact space without isolated points. Then there exists a space \(Y\) with a countable base, which is the union of a countable number of compacta, and an open continuous finite-to-one mapping \(f:Y\to X\) such that for every nonempty open subset \(U\subseteq X\) there is no continuous section.

Theorem 19. Let \(X\) be a complete metrizable space without isolated points. Then there exist a metrizable space \(Y\), which is the union of a countable number of closed and complete subspaces, and an open continuous finite-to-one mapping \(f:Y\to X\) such that for every open nonempty subset \(U\subseteq X\) there is no continuous section.

Mechanics and Mathematics Faculty
of Moscow State University
named after M. V. Lomonosov

Received
30 V 1969

CITED LITERATURE

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