Selection topologies
type: Item
instance of: scholarly article
title: Selection topologies
zbMATH Open document ID: 1335.54032
published in: Topology and its Applications
publication date: 2015-12-18T00:00:00Z
review text: For a topological space \(X\), let \(\mathcal{F}_2(X)\) denote the set of all non-empty subsets of \(X\) with at most two points. A map \(\sigma : \mathcal{F}_2(X) \to X\) is called a weak selection on \(X\) if \(\sigma(F) \in F\) for each \(F \in \mathcal{F}_2(X)\). A weak selection on \(X\) is said to be continuous if it is continuous with respect to the Vietoris topology on \(\mathcal{F}_2(X)\). For a weak selection \(\sigma\) on \(X\) and \(x \in X\), put \((\leftarrow , x)_\sigma = \{ y \in X : \sigma(\{x,y \}) =y \text{ and } x \neq y \}\) and \((x, \rightarrow )_\sigma = \{y \in X : \sigma(\{x,y \}) =x \text{ and } x \neq y \}\). The topology \(\tau_\sigma\) induced by the subbase consisting of all intervals \((\leftarrow , x)_\sigma\) and \((x, \rightarrow )_\sigma\), \(x \in X\), is called the selection topology induced by \(\sigma\), which was introduced by the author and \textit{T. Nogura} [Appl. Gen. Topol. 2, No. 2, 205--218 (2001; Zbl 0993.54019)]. A weak selection \(\sigma\) on a space \((X, \tau)\) is said to be separately continuous if \(\tau_\sigma \subset \tau\). Every continuous weak selection is separately continuous. In [\textit{S. García-Ferreira} et al., Houston J. Math. 39, No. 4, 1385--1399 (2013; Zbl 1287.54003)], S. García-Ferreira, K. Miyazaki, T. Nogura and A. H. Tomita introduced the notions of a weak selection space and a continuous weak selection space as generalizations of a topological space with a selection topology. A space \((X,\tau)\) is called a weak selection space (resp., continuous weak selection space) if there is a set \(\Sigma\) of weak selections (resp., continuous weak selections) on \(X\) such that \(\tau\) is equal to the smallest topology on \(X\) containing \(\tau_\sigma\) for all \(\sigma \in \Sigma\). In this paper, the author prove that a space with a separately continuous (resp., continuous) weak selection is a weak selection space (resp., continuous weak selection space) if and only if each of its components i…
reviewed by: Takamitsu Yamauchi
Mathematics Subject Classification ID: 54F05
Mathematics Subject Classification ID: 54C65
Mathematics Subject Classification ID: 54B20
zbMATH DE Number: 6522004
zbMATH Keywords: weak selection
zbMATH Keywords: weakly orderable space
zbMATH Keywords: suborderable space
author: Valentin Gutev
MaRDI profile type: Publication
full work available at URL: j.topol.2015.05.023
OpenAlex ID: W4206628781
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DOI: 10.1016/J.TOPOL.2015.05.023
description: scientific article; zbMATH DE number 6522004