Ordinals and set-valued zero-selections for hyperspaces
type: Item
instance of: scholarly article
title: Ordinals and set-valued zero-selections for hyperspaces
zbMATH Open document ID: 1211.54026
published in: Topology and its Applications
publication date: 2009-04-23T00:00:00Z
review text: 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.
reviewed by: Władysław Makuchowski
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C65
Mathematics Subject Classification ID: 54G12
zbMATH DE Number: 5546600
zbMATH Keywords: hyperspace
zbMATH Keywords: Vietoris topology
zbMATH Keywords: Fell topology
zbMATH Keywords: semi-continuous multi-selection
zbMATH Keywords: scattered space
author: Valentin Gutev
MaRDI profile type: Publication
full work available at URL: j.topol.2008.12.003
OpenAlex ID: W2021599057
cites work: Selectors and scattered spaces
cites work: Zero-selectors and GO spaces
cites work: Selections and suborderability
cites work: Q3994935
cites work: Characterizations of compact ordinal spaces via continuous selections
cites work: Weak factorizations of continuous set-valued mappings
cites work: Fell continuous selections and topologically well‐orderable spaces
cites work: Topologies on Spaces of Subsets
cites work: Q4882379
DOI: 10.1016/J.TOPOL.2008.12.003
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description: scientific article; zbMATH DE number 5546600