Selections and hyperspaces
type: Item
instance of: scholarly article
zbMATH Open document ID: 1320.54002
publication date: 2014-08-12T00:00:00Z
Mathematics Subject Classification ID: 54-02
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C65
zbMATH DE Number: 6329563
zbMATH Keywords: hyperspace topologies
zbMATH Keywords: Vietoris topology
zbMATH Keywords: continuous selection
zbMATH Keywords: weak selection
zbMATH Keywords: (weak) orderability
author: Valentin Gutev
MaRDI profile type: Publication
title: Selections and hyperspaces
review text: Given a Hausdorff topological space \(X\), let \(\mathcal F(X)\) be the set of all nonempty closed subsets of \(X\), and denote by \(\tau_V\) the \textit{Vietoris topology} on \(\mathcal F(X)\) having as a base sets of the form \(\{S\in \mathcal F(X): S\subset \bigcup \mathcal V \;\text{and} \;S\cap V\neq \emptyset \;\text{whenever} \;V\in\mathcal V\}\), where \(\mathcal V\) runs through the finite families of nonempty open subsets of \(X\). For \(\mathcal D\subseteq \mathcal F(X)\), the map \(f:\mathcal D\to X\) is a \textit{selection} for \(\mathcal D\) provided \(f(S)\in S\) for each \(S\in \mathcal D\), and \(f\) is a \textit{\(\tau_V\)-continuous selection} for \(\mathcal D\), provided \(f\) is continuous with respect to the relative Vietoris topology on \(\mathcal D\); more generally, one can define the notion of \textit{\(\tau\)-continuous selections}, if \(\tau_V\) is replaced by other hyperspace topologies \(\tau\) on \(\mathcal D\).NEWLINENEWLINE In this survey paper the author reviews the status of the so-called \textit{hyperspace selection problem}: when is it possible to find a \(\tau\)-continuous selection for \(\mathcal D\), where \(\tau\) stands for various hypertopologies (such as the Vietoris, Fell, or Wijsman, respectively), and \(\mathcal D\) is \(\mathcal F(X)\), the nonempty compact subsets of \(X\) or, for a given \(n\), the up to \(n\)-element subsets of \(X\), respectively. The notable solutions are surveyed according to the properties of the spaces studied, namely, for connected-like, compact-like, disconnected-like, and metrizable spaces, respectively.NEWLINENEWLINEFor the entire collection see [Zbl 1282.54001].
reviewed by: Laszlo Zsilinszky
description: scientific article; zbMATH DE number 6329563