Ordinals and set-valued zero-selections for hyperspaces
A topological space \(X\) is said to be scattered if every nonempty closed subset of \(X\) has an isolated point. Let \({\mathcal F}(X)\) be the hyperspace of all nonempty closed subsets of \(X\) equipped with the Vietoris topology. A map \(f: {\mathcal F}(X) \rightarrow X\) is said to be a selection for \({\mathcal F}(X)\) if \(f(S) \in S\) for every \(S \in {\mathcal F}(X)\). If \(f(S)\) is an isolated point of \(S\) for every \(S \in {\mathcal F}(X)\) then \(f\) is called zero-selection. A multi-selection for \({\mathcal F}(X)\) is a set-valued mapping \(\theta : {\mathcal F}(X) \rightarrow {\mathcal F}(X)\) such that \(\theta(S) \subset S\) for every \(S \in {\mathcal F}(X)\). A mapping \(\theta : {\mathcal F}(X) \rightarrow {\mathcal F}(X)\) is called a zero multi-selection if \(\theta(S)\) consists of isolated points of \(S\) for each \(S \in {\mathcal F}(X)\). \textit{S. Fujii} and \textit{T. Nogura} have proved in [Topology Appl. 91, 65--69 (1999; Zbl 0926.54011)] that a compact scattered space \(X\) is homeomorphic to an ordinal space if and only if there exists a continuous zero-selection for \({\mathcal F}(X)\). In this paper it is proved that a compact scattered space \(X\) has an upper semi-continuous zero multi-selection for \({\mathcal F}(X)\) if and only if \(X\) can be mapped onto an ordinal space by a continuous finite-to-one surjection. Also, the space \({\mathcal F}(X)\) with the Fell topology is investigated and a similar characterization is obtained.
scientific article; zbMATH DE number 5546600