A topology generated by selections
type: Item
instance of: scholarly article
title: A topology generated by selections
zbMATH Open document ID: 1089.54005
published in: Topology and its Applications
publication date: 2006-03-20T00:00:00Z
review text: In this paper the authors continue their investigations from the paper [Appl. Gen. Topol. 2, 205-218 (2001; Zbl 0993.54019)]. Let \((X, {\mathcal T})\) be a \(T_1\) topological space. Let \({\mathcal F}_2 (X)\) be the family of at most two-point subsets of \(X\) equipped with the Vietoris topology and let \(f: {\mathcal F}_2 (X) \rightarrow X\) be a selection. The selection \(f\) defines an order-like relation \(\preceq _f\) on \(X\) by \(x \preceq _f y\) if and only if \(f(\{x,y\})=x\). The ``open'' intervals \((x, \rightarrow )=\{y \in X: x \preceq _f y \) and \(y \neq x\}\) and \(( \leftarrow ,x)=\{y \in X: y \preceq _f x \) and \(y \neq x \}\) generate the topology \({\mathcal T}_f \). Properties of this topology are investigated. A number of interesting examples is presented, for instance, continuous selections \(f\) on various spaces \((X, {\mathcal T})\) such that \({\mathcal T}_f \neq {\mathcal T}\).
reviewed by: Władysław Makuchowski
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54C65
zbMATH DE Number: 5013562
zbMATH Keywords: Hyperspace topology
zbMATH Keywords: Vietoris topology
zbMATH Keywords: Continuous selection.
author: Valentin Gutev
author: T. Nogura
MaRDI profile type: Publication
full work available at URL: j.topol.2005.01.019
OpenAlex ID: W1976465856
cites work: Q3994935
cites work: Extreme selections for hyperspaces of topological spaces
cites work: Selections and order-like relations
cites work: Vietoris continuous selections and disconnectedness-like properties
cites work: Topologies on Spaces of Subsets
cites work: Characterizations of intervals via continuous selections
DOI: 10.1016/J.TOPOL.2005.01.019
description: scientific article; zbMATH DE number 5013562