On s-reflexive spaces and continuous selections (Q2340481)

In this paper, the authors study s-reflexive spaces introduced by \textit{Z. Yang} and \textit{D. Zhao} [Fundam. Math. 192, No. 2, 111--120 (2006; Zbl 1111.47007)] in terms of continuous selections of carriers (set-valued mappings). Let \(X\) be a topological space and \(2^X\) the family of all subsets of \(X\). A family \({\mathcal A}\) of closed subsets of \(X\) is said to be reflexive if there exists a set \({\mathcal B}\) of continuous selfmaps on \(X\) such that \({\mathcal A}=\{A \in 2^X : A \text{ is closed and } f(A) \subset A \text{ for every } f \in {\mathcal B}\} \). Note that if \({\mathcal A}\) is a reflexive closed family, then \({\mathcal A}\) satisfies (a) \(X, \emptyset \in {\mathcal A}\), (b) \({\mathcal B} \subset {\mathcal A}\) implies \(\bigcap {\mathcal B} \in {\mathcal A}\), and (c) \({\mathcal B} \subset {\mathcal A}\) implies \(\overline{\bigcup {\mathcal B}} \in {\mathcal A}\). A topological space \(X\) is said to be s-reflexive if every closed family with the conditions (a)--(c) above is reflexive. In Section 2, a characterization of s-reflexive spaces in terms of continuous selections of l.s.c.\ carriers is given. Applying the characterization, the authors prove, for example, that every s-reflexive Hausdorff space is zero-dimensional. In Section 3, the notion of self-selective space is introduced and discussed. A space \(X\) is said to be self-selective if every lower semicontinuous carrier \(\Phi : X \to 2^Y\) with nonempty closed values has a continuous selection. It is proved that every self-selective \(T_1\)-space is s-reflexive and countably paracompact. In Section 4, it is proved that every strongly zero-dimensional completely metrizable space is s-reflexive by applying \textit{E. Michael}'s zero-dimensional selection theorem [Ann.\ Math.\ (2) 64, 562--580 (1956; Zbl 0073.17702)]. In Section 5, the s-reflexivity of several examples is discussed.