Moscow spaces and selection theory (Q924282)

A Tychonoff space \(X\) is called a Moscow space if every regular closed set in \(X\) is the union of \(G_\delta\)-sets in \(X\). This notion was introduced by A. V. Arhangel'sklĭ in his research of topological groups. In this paper, the author studies selection theoretic aspects of Moscow spaces and proves the two main results (1) and (2) below. For a space \(Y\), let \({\mathcal C}(Y)\) be the set of all nonempty closed sets in \(Y\), and \({\mathcal S}(Y)\) (resp. \({\mathcal K}(Y)\)) the subset of \({\mathcal C}(Y)\) consisting of members which are separable (resp. compact). The first result (1) generalizes a remarkable property of Moscow spaces, namely that every \(G_\delta\)-dense subset is \(C\)-embedded. Theorem: (1) For a \(G_\delta\)-dense subset of a Moscow space \(X\), the following conditions (i)--(iv) are equivalent: {\parindent=8mm \begin{itemize}\item[(i)] For each metric space \(Y\), if \(\varphi:X\to{\mathcal S}(Y)\) is a continuous carrier, then every selection for \(\varphi|_S\) can be extended to a selection for \(\varphi\); \item[(ii)] same as (i), but with \({\mathcal S}(Y)\) replaced by \({\mathcal K}(Y)\); \item[(iii)] if \(\varphi:X\to{\mathcal C}({\mathbb R})\) is a continuous carrier, then every selection for \(\varphi|_S\) can be extended to a selection for \(\varphi\); and \item[(iv)] same as (iii), but with \({\mathcal C}({\mathbb R})\) replaced by \({\mathcal K}({\mathbb R})\). \end{itemize}} The second result (2) gives necessary and sufficient conditions for an open precompact subset \(S\) of a uniform space \((X,{\mathcal U})\) whose completion \(({\widehat X},{\widehat {\mathcal U})}\) is a Moscow space to be bounded in \(X\) (i.e., every real-valued continuous function on \(X\) is bounded on \(S\)). For a normed space \(Y\), let \({\mathcal F}(Y)\) be the set of all nonempty closed convex sets in \(Y\). Theorem: (2) If \(({\widehat X},{\widehat {\mathcal U})}\) is a Moscow space, then the following conditions (i)--(v) are equivalent for an open precompact subset \(S\) of \((X,{\mathcal U})\): {\parindent=8mm \begin{itemize}\item[(i)] \(S\) is bounded in \(X\); \item[(ii)] for each metric space \(Y\), if \(\varphi:\text{cl}_{\widehat X}S\to{\mathcal S}(Y)\) is a continuous carrier, then every selection for \(\varphi|_{\text{cl}_XS}\) can be extended to a selection for \(\varphi\); \item[(iii)] if \(\varphi:\text{cl}_{\widehat X}S\to{\mathcal C}({\mathbb R})\) is a continuous carrier, then every selection for \(\varphi|_{\text{cl}_XS}\) can be extended to a selection for \(\varphi\); \item[(iv)] if \(Y\) is a separable Banach space, every continuous carrier \(\varphi:\text{cl}_XS\to{\mathcal F}(Y)\) admits a uniformly continuous selection; and \item[(v)] every continuous carrier \(\varphi:\text{cl}_XS\to{\mathcal F}({\mathbb R})\) admits a uniformly continuous selection. \end{itemize}} In case \(S=X\), (2) characterizes pseudocompactness of a uniform space whose completion is a Moscow space.