UDC 513.83

MATHEMATICS

M. M. ČOBAN

MULTIVALUED MAPPINGS AND BOREL SETS

(Presented by Academician P. S. Aleksandrov, 30 I 1968)

Let \(X\) be a topological space. Denote by \(F(X)\) the space of all nonempty closed subsets of the space \(X\) in the Vietoris topology. In the space \(F(X)\) let us single out the subspace
\[ C(X)=\{L\in F(X)\mid L\text{ is bicompact}\}. \]
A single-valued mapping \(\theta:Y\to F(X)\) from the space \(Y\) into the space \(F(X)\) is called a multivalued mapping. With multivalued mappings there is associated the problem of sections (see \((^4,^5)\)): for which multivalued mappings \(\theta:Y\to F(X)\) does there exist a “good” single-valued mapping \(f:Y\to X\) such that \(fy\in \theta y\) for every point \(y\in Y\). Recall that a mapping \(\theta:Y\to F(X)\) is called upper (lower) semicontinuous if for every open (closed) set \(A\subset X\) the set
\[ \{y\in Y\mid \theta y\subset A\} \]
is open (closed) in the space \(Y\). A mapping \(\theta:Y\to F(X)\) is continuous if it is at the same time upper and lower semicontinuous.

§ 1. On canonical representation. Let \(\theta:X\to F(Y)\) be a multivalued mapping. The set
\[ \Gamma(X,Y,\theta)=\bigcup\{(\{x\}\times \theta x)\mid x\in X\}\subset X\times Y \]
is called the graph of the mapping \(\theta\). Denote by \(\pi_X\) and \(\pi_Y\) the natural projections of the set \(\Gamma\) onto \(X\) and onto \(Y\), respectively. We have \(\theta=\pi_Y\circ \pi_X^{-1}\). We shall say that the mapping \(\theta\) is \(X\)-open (\(X\)-closed) if the mapping \(\pi_X\) is open (closed). If \(\pi_X\) is perfect, then \(\theta\) is called \(X\)-perfect.

Proposition 1. The following properties are equivalent: a) the mapping \(\theta:X\to F(Y)\) is \(X\)-perfect; b) the mapping \(\theta\) is upper semicontinuous and \(\theta x\in C(Y)\) for every point \(x\in Y\).*

Proposition 2. The following properties are equivalent: a) the mapping \(\theta:X\to F(Y)\) is \(X\)-open; b) the mapping \(\theta\) is lower semicontinuous.

These propositions establish a connection between single-valued and multivalued mappings. Let us note that there exists an upper semicontinuous mapping \(\theta:X\to F(Y)\) which is not \(X\)-closed.

§ 2. Multivalued continuous mappings.

Theorem 1. Let the mapping \(\theta:X\to F(Y)\) be continuous, where \(Y\) is a complete zero-dimensional metric space. Then there exists a single-valued continuous mapping \(f:X\to Y\) such that \(fx\in \theta x\) for every point \(x\in X\).

Corollary 1. Let \(\varphi:X\to Y\) be an open-and-closed single-valued continuous mapping of a complete zero-dimensional metric space \(X\) onto \(Y\). Then there exists a closed subset \(X_1\subset X\) such that \(\varphi X_1=Y\) and \(\varphi|X_1\) is a homeomorphism.

Theorem 2. Let the mapping \(\theta:X\to F(Y)\) be continuous, where \(Y\) is a complete metric space. Then there exists a single-valued mapping \(f:X\to Y\) such that \(fx\in \theta x\) for every point \(x\in X\), and \(f^{-1}U\) is an \(F_\sigma\)-set whenever \(U\) is open in \(Y\).

Theorem 3. Let \(\theta:X\to F(Y)\) be a continuous mapping of a normal space \(X\) into a complete metric space \(Y\). Then there exist mappings \(\varphi:X\to C(Y)\) and \(\psi:X\to C(Y)\) such that: a) \(\varphi x\subset\)

* This proposition is a strengthening of a lemma of Yu. M. Smirnov and V. I. Ponomarev on the canonical representation of perfect mappings (see \((^8)\)).

\(\subset \psi x \subset \theta x\) for every point \(x \in X\); b) the mapping \(\psi\) is upper semicontinuous; c) the mapping \(\varphi\) is lower semicontinuous; d) if, moreover, \(\dim X = 0\), then the mapping \(\psi\) is single-valued, i.e., \(\varphi x\) and \(\psi x\) are singletons.

In the paper \((^7)\) E. Michael proved an analogous theorem for lower semicontinuous mappings under the assumption that the space \(X\) is paracompact. Let us note that the assertion converse to Michael’s theorem is true.

Theorem 4. Let \(X\) be a topological \(T_1\)-space. If for every lower semicontinuous mapping \(f: X \to F(Y)\), where \(Y\) is an arbitrary complete metric space, there exists an upper semicontinuous mapping \(\theta: X \to C(Y)\) such that \(\theta x \subset f x\) for every point \(x \in X\), then the space \(X\) is paracompact.

Proof. First, let us prove the normality of the space \(X\). Let \(A\) and \(B\) be arbitrary closed and disjoint subsets of the space \(X\). Let \(Y=\{a,b\}\) be an ordinary two-point set (with the discrete topology). We construct a mapping \(f: X \to F(Y)\), where

\[ f x = \begin{cases} a, & \text{if } x \in A,\\ b, & \text{if } x \in B,\\ Y, & \text{if } x \in X \setminus (A \cup B). \end{cases} \]

The mapping \(f\) is lower semicontinuous, since \(f^{-1} a = X \setminus B\) and \(f^{-1} b = X \setminus A\). By assumption there exists an upper semicontinuous mapping \(\theta: X \to C(Y)\) such that \(\theta x \subset f x\) for every point \(x \in X\). Put \(U=\{x \in X \mid \theta x \subset \{a\}\}\) and \(V=\{x \in X \mid \theta x \subset \{b\}\}\). By construction \(U \cap V=\varnothing\), \(U \supset A\), and \(V \supset B\). From the definition of upper semicontinuity it follows that the sets \(U\) and \(V\) are open in the space \(X\). The normality of \(X\) is proved.

Now let \(\omega=\{U_\alpha \mid \alpha \in Y\}\) be some open cover of the space \(X\). On the set \(Y\) define a metric in the following way: \(\rho(\alpha,\beta)=1\) if \(\alpha \ne \beta\), and \(\rho(\alpha,\alpha)=0\). Put \(f: X \to F(Y)\), where \(f x=\{\alpha \in Y \mid x \in U_\alpha\}\). The mapping \(f\) is lower semicontinuous, since \(f^{-1}\alpha=U_\alpha\) for every \(\alpha \in Y\). By assumption, there exists an upper semicontinuous mapping \(\theta: X \to C(Y)\) such that \(\theta x \subset f x\) for every point \(x \in X\).

Put \(\gamma=\{F_\alpha=\theta^{-1}\alpha=\{x \in X \mid \theta x \cap \{\alpha\}\ne \varnothing\}\mid \alpha \in Y\}\). The system \(\gamma\) covers the space \(X\). Since \(F_\alpha \subset U_\alpha\) for every \(\alpha \in Y\), the cover \(\gamma\) is inscribed in the cover \(\omega\). Let \(x_0 \in X\). Since the space \(Y\) is discrete, the set \(\theta x_0=\{\alpha_1(x),\ldots,\alpha_{n(x)}\}\) is finite and open. Consequently, the set \(O x_0=\{x \in X \mid \theta x \subset \theta x_0\}\) is also open in the space \(X\) and has nonempty intersection only with the sets \(F_\alpha\) where \(\alpha \in \theta x_0\). Since the set \(\theta x_0\) is finite and \(x_0 \in O x_0\), the cover \(\gamma\) is locally finite. On the basis of a known theorem of E. Michael from \((^{10})\), the space \(X\) is paracompact.

Theorem 3 and Theorem 3.2 from \((^5)\) allow us to obtain the following theorem.

Theorem 5. Let \(\theta: X \to F(Y)\) be a continuous mapping of a collectionwise normal space \(X\) into a Banach space \(Y\). Then there exists a single-valued continuous mapping \(f: X \to Y\) such that \(f x \in \operatorname{conv}(\theta x)\) for every point \(x \in X\).

§ 3. Lower semicontinuous mappings. Let \(X\) be a topological space. Put
\[ B_1(X)=\{A \subset X \mid A=L\setminus C;\ L,C \in F(X)\} \]
and
\[ B(X)=\{K \subset X \mid K=\bigcup_{n=1}^{\infty} A_n,\ A_n \in B_1(X)\ \text{and } n=1,2,\ldots\}. \]

It is easy to see that, in the case of perfectly normal spaces, the system \(B(X)\) coincides with the system of sets of type \(F_\sigma\).

Theorem 6. Let \(\theta: X \to C(Y)\) be a lower semicontinuous mapping, where \(Y\) is a metrizable space. Then there exists a single-valued mapping \(f: X \to Y\) such that \(f x \in \theta x\) for every point \(x \in X\), and \(f^{-1}U \in B(X)\) for every open set \(U\) in \(Y\).

Theorem 7. For every regular space \(X\) the following conditions are equivalent: a) the space \(X\) is weakly paracompact; b) for every lower semicontinuous mapping \(\theta: X \to F(Y)\), where \(Y\) is an arbitrary complete metric space, there exists a lower semicontinuous mapping \(\psi: X \to C(Y)\) such that \(\psi x \subset \theta x\) for every point \(x \in X\).

From Theorems 6 and 7 we obtain:

Theorem 8. Let \(\theta: X \to F(Y)\) be a lower semicontinuous mapping of a weakly paracompact regular space \(X\) into a complete metric space \(Y\). Then there exists a single-valued mapping \(f: X \to Y\) such that \(fx \in \theta x\) for every point \(x \in X\), and \(f^{-1}U \in B(X)\) for every open set \(U\) in \(Y\).

Theorem 7 allows us to definitively generalize a theorem of E. Michael from \((^7)\).

Theorem 9. Let \(f: X \to Y\) be an inductively open single-valued mapping* of a metric space \(X\) onto a regular weakly paracompact space \(Y\), with complete point-preimages (in the metric given on \(X\)). Then there exists a subspace \(X_1 \subset X\) such that \(fX_1 = Y\) and the mapping \(f|X_1\) is open and bicompact.

Theorem 10. Let the mapping \(\theta: X \to F(Y)\), where \(Y\) is a complete metric space, be lower semicontinuous. If the space \(X\) satisfies one of the following conditions: a) \(X\) is paracompact; b) it is perfectly normal and weakly paracompact; c) \(X\) is symmetrizable with the first axiom of countability; d) \(X\) has a \(\sigma\)-discrete net; e) \(X\) is perfectly normal and into every open cover one can inscribe a closed \(\sigma\)-discrete cover, then there exists a single-valued mapping \(f: X \to Y\) such that \(fx \in \theta x\) for every point \(x \in X\), and \(f^{-1}U\) is an \(F_\sigma\)-set for every open set \(U\) in \(Y\).

§ 4. \(X\)-closed mappings and projections of Borel sets.

Theorem 11. Let the mapping \(\theta: X \to F(Y)\) be \(X\)-closed, where \(Y\) is a complete metric space. If the space \(X\) is perfectly normal, then there exists a single-valued mapping \(f: X \to Y\) such that \(fx \in \theta x\) for every point \(x \in X\), and \(f^{-1}U\) is an \(F_\sigma\)-set for every open set \(U\) in \(Y\).

From Proposition 1 and Theorem 11 it follows:

Corollary 2. Let \(\theta: X \to C(Y)\) be an upper semicontinuous mapping of a perfectly normal space \(X\) into a metric space \(Y\). Then there exists a single-valued mapping \(f: X \to Y\) such that \(fx \in \theta x\) and \(f^{-1}U\) is an \(F_\sigma\)-set for every open set \(U\) in \(Y\).

Theorem 11 together with the theorem of I. A. Vainshtein (see \((^3)\), Theorem 5) makes it possible to extend one result of A. D. Taimanov to arbitrary metric spaces:

Theorem 12. Let \(f: X \to Y\) be a perfect mapping of a metric space \(X\) onto a space \(Y\). If the space \(X\) is an absolute Borel set of class \(\leq \alpha\), then the space \(Y\) is also an absolute Borel set of class \(\leq 1+\alpha\) for \(\alpha < \omega_0\) and \(\leq \alpha\) if \(\alpha \geq \omega_0\).

However, Michael’s theorem (see \((^7)\), Theorem 1.1) and Theorem 12 allow us to assert:

Theorem 13. Let \(f: X \to Y\) be an inductively open mapping of a metric space \(X\) onto a metric space \(Y\), with complete point-preimages (in the metric given on \(X\)). If the space \(X\) is an absolute Borel set of class \(\leq \alpha\), then the space \(Y\) is also an absolute Borel set of class \(\leq 1+\alpha\) for \(\alpha < \omega_0\) and \(\leq \alpha\) if \(\alpha \geq \omega_0\).

* A mapping \(f: X \to Y\) is inductively open if there exists a subspace \(X_0 \subset X\) such that \(fX_0 = Y\) and the mapping \(f_1 = f|X_0\) is open (on \(X_0\)) (see \((^1)\)).

From a preprint kindly sent by R. Engelking I have learned the following result of his: let \(f: X \to F(Y)\) be an upper semicontinuous mapping of a perfectly normal paracompact space \(X\) into a complete metric space \(Y\). If \(fx\) is separable for every point \(x \in X\), then there exists a single-valued mapping \(g: X \to Y\) such that \(gx \in fx\) for every point \(x \in X\), and \(g^{-1}U\) is an \(F_\sigma\)-set for every set \(U\) open in \(Y\). This prompted me to prove the following assertion.

Theorem 14. Let \(\theta: X \to F(Y)\) be an upper semicontinuous mapping of a perfectly normal and weakly paracompact space \(X\) into a complete metric space \(Y\), where \(\theta x\) is separable for every point \(x \in X\). Then there exists a single-valued mapping \(f: X \to Y\) such that \(fx \in \theta x\) for every point \(x \in X\), and \(f^{-1}U\) is an \(F_\sigma\)-set for every set \(U\) open in \(Y\).

§ 5. Remarks on \(\rho\)-continuous mappings.

A mapping \(\theta: X \to F(Y)\), where \((Y,\rho)\) is a metric space, is \(\rho\)-continuous if for every point \(x \in X\) and every \(\varepsilon > 0\) there exists \(Ox\) such that

\(\min \{r \mid O(\theta x,r) \supset \theta z,\ O(\theta z,r) \supset \theta x\} < \varepsilon\)

for every point \(z \in Ox\). From Theorem 10 it follows:

Theorem 15. Let \(\theta: X \to F(Y)\) be a \(\rho\)-continuous mapping, where \(Y\) is a complete metric space. Then there exists a single-valued mapping \(f: X \to Y\) such that \(fx \in \theta x\) for every point \(x \in X\), and \(f^{-1}U\) is an \(F_\sigma\)-set for every set \(U\) open in \(Y\).

Two theorems of E. Michael (see Theorem 1.1 of \((^7)\) and Theorem 2 of \((^6)\)) and the factorization theorem from \((^2)\) make it possible to obtain the theorem:

Theorem 16. Let \(\theta: X \to F(Y)\) be a \(\rho\)-continuous mapping of a normal space \(X\) into a complete metric space \(Y\). Then there exist mappings \(\varphi: X \to C(Y)\) and \(\psi: X \to C(Y)\) such that:
a) \(\varphi x \subset \psi x \subset \theta x\) for every point \(x \in X\);
b) \(\varphi\) is lower semicontinuous;
c) \(\psi\) is upper semicontinuous;
d) if \(\dim X = 0\), then the sets \(\varphi x\) and \(\psi x\) are singletons for every point \(x \in X\).

References

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2. A. V. Arkhangel’skii, DAN, 174, 1243 (1967).
3. I. A. Vainshtein, Uch. zap. Mosk. univ., v. 155, 5, 3 (1952).
4. K. Kuratowski, C. Kyl-Nardzewski, Bull. Acad. Polon. Sci., 13, 397 (1965).
5. E. Michael, Ann. Math., 63, 361 (1956).
6. E. Michael, Am. Math. Montly, 63, 233 (1956).
7. E. Michael, Duke Math. J., 26, 647 (1959).
8. V. I. Ponomarev, Matem. sborn., 51 (93), 515 (1960).
9. A. D. Taimanov, Matem. sborn., 52 (94), 557 (1960).
10. E. Michael, Proc. Am. Math. Soc., 8, 822 (1957).