UDC 513.83

MATHEMATICS

V. A. GEILER

ON CONTINUOUS SELECTORS IN UNIFORM SPACES

(Presented by Academician P. S. Aleksandrov, May 11, 1970)

The present note is devoted to a generalization of some theorems of E. Michael on continuous selectors of multivalued mappings \((^{1,2})\) to the case where the multivalued mapping takes its values in a uniform space.

In what follows \(E\) will denote a certain separated uniform space, \(\mathcal U\) the filter of entourages of its uniformity, \(\mathcal U_0\) a certain filter of entourages of a weaker uniformity on \(E\) than the original one, having a countable base, and \(E_0\) the uniform space obtained by endowing \(E\) with the uniformity whose filter of entourages is \(\mathcal U_0\). By \(\mathfrak P(E)\), \(C(E)\), \(B(E)\) we shall denote, respectively, the set of all subsets, all complete subsets, and all bicompact subsets of \(E\).

Definition 1. A set \(\mathfrak A \subset \mathfrak P(E)\) is called uniformly metrizable by means of \(\mathcal U_0\) if the condition

\[ (*)\ \forall V \in \mathcal U\ \exists U \subset \mathcal U_0\ \forall A \in \mathfrak A:\ (A \times A)\cap U \subset V. \]

is satisfied.

If \(E\) is metrizable, then, obviously, every \(\mathfrak A \subset \mathfrak P(E)\) is uniformly metrizable by means of \(\mathcal U\). If \(E\) is a topological group endowed, say, with the right uniformity, and \(H\) is a metrizable subgroup of \(E\), then the set of right cosets modulo \(H\) is uniformly metrizable.

Theorem 1. Let \(X\) and \(Y\) be topological spaces, and let \(\mathfrak A\) be a subset of \(C(E)\) uniformly metrizable by means of \(\mathcal U_0\). Suppose a commutative diagram is given

\[ \begin{array}{ccc} X & \xrightarrow{\ \varphi\ } & \mathfrak A \\ {\scriptstyle h}\downarrow & & \downarrow{\scriptstyle i} \\ Y & \xrightarrow{\ \psi\ } & \mathfrak P(E_0) \end{array} \]

where \(h\) is continuous, \(\varphi\) and \(\psi\) are lower semicontinuous, and \(i\) is the inclusion. Then, if \(Y\) is perfectly normal*, there exist continuous mappings \(f:X\to E\) and \(g:Y\to E_0\) such that \(f(x)\in\varphi(x)\), \(g(y)\in\psi(y)\) \((x\in X,\ y\in Y)\), and the diagram

\[ \begin{array}{ccc} X & \xrightarrow{\ f\ } & E \\ {\scriptstyle h}\downarrow & & \downarrow{\scriptstyle 1} \\ Y & \xrightarrow{\ g\ } & E_0 \end{array} \]

is commutative, where \(1\) is the identity mapping of \(E\) onto \(E_0\).

Proof. Let \((U_n)_{n\ge 1}\) be a base of the filter \(\mathcal U_0\) such that

\[ U_n^{-1}=U_n,\quad U_{n+1}\subset U_n \]

and all \(U_n\) are open. We construct by induction sequences \((\varphi_n)_{n\ge 0}\) and \((\psi_n)_{n\ge 0}\) of lower semicontinuous mappings

\[ \varphi_n:X\to\mathfrak P(E),\qquad \psi_n:Y\to\mathfrak P(E_0) \]

such that 1) \(i\circ\varphi_n=\psi_n\circ h\), 2) \(\psi_{n+1}(y)\subset\)

\[ \text{* I.e., into every open cover of }Y\text{ one can inscribe an open cover consisting of pairwise disjoint sets.} \]

\(\subset \psi_n(y)\), 3) \(\psi_n(y)\) is of order of smallness \(U_n\). Put \(\varphi_0=\varphi\) and \(\psi_0=\psi\). Suppose that for \(k\leq n\) the \(\varphi_k\) and \(\psi_k\) have been constructed. For each \(a\in E\), let
\[ G(a)=\{y\in Y:\psi_n(y)\cap U_{n+1}(a)\ne\varnothing\}. \]
Then \((G(a))_{a\in E}\) is an open cover of \(Y\). Take an open refinement \((H_\lambda)_{\lambda\in L}\), inscribed in \((G(a))\), and for each \(\lambda\in L\) choose \(a_\lambda\in E\) so that \(H_\lambda\subset G(a_\lambda)\). Let \(g_n\) be the continuous mapping of \(Y\) into \(E\) defined by the relation \(g_n(y)=a_\lambda\) if \(y\in H_\lambda\), and let \(f_n=g_n\circ h\). Then one may put
\[ \varphi_{n+1}(x)=\varphi_n(x)\cap U_{n+1}(f_n(x)),\qquad \psi_{n+1}(y)=\psi_n(y)\cap U_{n+1}(g_n(y)). \]

By virtue of 3), \((\varphi_n(x))_{n\ge1}\) and \((\psi_n(y))_{n\ge1}\) are bases of Cauchy filters in \(\varphi(x)\) and \(\psi(y)\), respectively; let \(f(x)\) and \(g(x)\) be their limits. We shall show that \(f\) and \(g\) are continuous. Consider, for example, \(f\) (the proof for \(g\) is analogous). Let \(x_0\in X\), \(V\in\mathcal U\), and let \(V(f(x_0))\) be the corresponding neighborhood of \(f(x_0)\). Take an open \(W\in\mathcal U\) such that
\[ W^{-1}=W,\qquad \overset{2}{W}\subset V. \]
By virtue of \((*)\), there is an \(n\) such that
\[ (\varphi(x)\times\varphi(x))\cap \overset{3}{U}_n\subset W \]
for all \(x\in X\). Let
\[ O=\{x\in X:\varphi_n(x)\cap W(f(x_0))\ne\varnothing\}; \]
then \(O\) is open. We shall show that \(x\in O\) implies \(f(x)\in V(f(x_0))\). Choose
\[ a_0\in\varphi_n(x)\cap W(f(x_0)); \]
by virtue of 3), \(f(x)\in \overset{2}{U}_n(a_0)\). Hence \(f(x)\in W(a_0)\), and therefore
\[ f(x)\in W(f(x_0))\subset V(f(x_0)). \]
The theorem is proved.

Remark. If \(E\) is a locally convex space and the sets from \(\mathfrak A\) are convex, then it is enough to assume that \(Y\) is paracompact.

Corollary (P. Kenderov \((^3)\)). Let \(E\) be a locally convex space, and let \(F\) be its Fréchet subspace. Then the natural mapping \(p:E\to E/F\) has a continuous right inverse.

2. Below we shall give the definition of an almost metrizable uniform space, which is a generalization of the definition of an almost metrizable group introduced by B. A. Pasynkov \((^4)\).

Below, \(R\) will denote the equivalence relation in \(E\) whose graph is the set
\[ \bigcap\{U:U\in\mathcal U_0\}. \]

Definition \(2^*\). We shall call \(E\) almost metrizable with respect to \(\mathcal U_0\) if: 1) for every \(a\in E\), \(R(a)\) is a bicompactum; 2) for every \(V\in\mathcal U\) there exists \(U\in\mathcal U_0\) such that
\[ U\subset V\circ R. \]

Definition 3. A set \(\mathfrak A\subset\mathcal P(E)\) is called uniformly almost metrizable with respect to \(\mathcal U_0\) if: 1) for every \(A\in\mathfrak A\) and every \(a\in A\), \(R(a)\cap A\) is a bicompactum; 2) for every \(V\in\mathcal U\) there exists \(U\in\mathcal U_0\) such that, for every \(A\in\mathfrak A\),
\[ (A\times A)\cap U\subset V\circ R. \]

For example, if \(E\) is a topological group endowed with the right uniformity, and \(H\) is an almost metrizable subgroup of \(E\), then the set of right cosets modulo \(H\) is uniformly almost metrizable.

Theorem 2. Let \(X\) and \(Y\) be topological spaces, let \(\mathfrak A\) be a subset of \(C(E)\) uniformly almost metrizable with respect to \(\mathcal U_0\), and let the diagram
\[ \begin{array}{ccc} X & \xrightarrow{\ \varphi\ } & \mathfrak A\\ {\scriptstyle h}\downarrow & & \downarrow{\scriptstyle i}\\ Y & \xrightarrow{\ \psi\ } & \mathcal P(E_0) \end{array} \]
be commutative, where \(h,\varphi,\psi,i\) are as in Theorem 1. If \(Y\) is perfectly zero-dimensional, then there exist lower semicontinuous mappings \(\eta:X\to B(E)\) and \(\theta:Y\to B(E_0)\) such that \(\eta(x)\subset\varphi(x)\), \(\theta(y)\subset\psi(y)\) \((x\in X,\ y\in Y)\), and the diagram
\[ \begin{array}{ccc} X & \xrightarrow{\ \eta\ } & B(E)\\ {\scriptstyle h}\downarrow & & \downarrow{\scriptstyle j}\\ Y & \xrightarrow{\ \theta\ } & B(E_0) \end{array} \]
is commutative,

* This definition was proposed by P. Kenderov.

where \(j\) is an embedding. Here \(\eta(x)=R(f(x))\cap\varphi(x)\), and \(\theta(x)=R(g(x))\cap\psi(x)\), where \(f:X\to E\), \(g:Y\to E\) are certain single-valued mappings.

Corollary 1. Suppose that in the notation of Theorem 2 \(X\) is perfectly zero-dimensional, \(E\) is a topological group, and \(\mathfrak A\) consists of all left cosets with respect to some almost metrizable Weil-complete subgroup \(H\subset E\). Then \(\varphi\) has a single-valued continuous selector.

Corollary 2. Let \(S\) be an open equivalence relation in \(E\), and let \(p:E\to E/S\) be the natural mapping. If the set of cosets with respect to \(S\) is uniformly almost metrizable and is contained in \(C(E)\), and the equivalence relation \(R\cap S\) is closed, then for every paracompact subset \(X\subset E/S\) (where \(E/S\) is endowed with the quotient topology) there exists a paracompact subset \(Y\subset E\) such that \(p(Y)=X\) and \(p|_Y\) is perfect.

Indeed, let \(\varphi:X\to\mathfrak P(E)\) be given by the relation \(\varphi(x)=p^{-1}(x)\); then \(\varphi\) is lower semicontinuous. By a known result of V. I. Ponomarev \((^5)\), there exists a perfectly zero-dimensional space \(\dot X\) (an absolution of \(X\)) and a perfect mapping \(\pi\) of \(\dot X\) onto \(X\). The mapping \(\varphi\circ\pi\) is lower semicontinuous; therefore, by Theorem 2, there is a single-valued mapping \(f:\dot X\to E\) such that the mapping \(\theta:x\to R(f(x))\cap S(f(x))\) is lower semicontinuous. But since \(R\cap S\) is closed, \(\theta\) is upper semicontinuous; consequently, one may put \(Y=\theta(X)\).

In exactly the same way, from Corollary 1 one can obtain Theorem 2 of B. A. Pasynkov’s paper \((^6)\).

Corollary 3. Let \(S\) be an equivalence relation, open and regular* both in \(E\) and in \(E_0\). If the set of cosets with respect to \(S\) is uniformly almost metrizable by means of the \(\mathfrak U_0\)-subset \(C(E)\), and the equivalence relation \(R\cap S\) is closed in \(E\), then there exists an \(X\subset E\) such that the restriction to \(X\) of the natural mapping \(p:E\to E/S\) is perfect, and \(p(X)=E/S\).

From Corollary 3 there follows the following theorem of P. Kenderov \((^8)\):

Let \(G\) be a topological group, \(H\) a Weil-complete, almost metrizable, normal subgroup of \(G\), and \(p:G\to G/H\) the natural mapping. Then there exists \(X\subset G\) such that \(p(X)=G\) and \(p|_X\) is perfect.

I express my deep gratitude to Prof. D. A. Raikov, who drew my attention to this topic, and to P. Kenderov and M. Choban for discussions and valuable comments.

Mordovian
State University
Saransk

Received
13 III 1970

REFERENCES

\({}^1\) E. Michael, Ann. math., 63, 361 (1956).
\({}^2\) E. Michael, Duke math. J., 26, No. 4, 647 (1959).
\({}^3\) P. Kenderov, Vestn. Moscow Univ., Math. Mech. (in press).
\({}^4\) B. A. Pasynkov, DAN, 161, No. 2, 281 (1965).
\({}^5\) V. I. Ponomarev, DAN, 143, No. 1, 46 (1962).
\({}^6\) B. A. Pasynkov, DAN, 188, No. 2, 286 (1969).
\({}^7\) V. L. Levin, D. A. Raikov, DAN, 150, No. 5, 981 (1963).
\({}^8\) P. Kenderov, DAN, 194, No. 4 (1970).

* V. L. Levin and D. A. Raikov call an equivalence relation \(S\) in \(E\) regular if \(\forall V\in\mathfrak U\ \exists U\subset\mathfrak U:\ S\circ U\subset V\circ S\).