The relationship between the Vietoris topology and the Hausdorff quasi-uniformity
type: Item
instance of: scholarly article
title: The relationship between the Vietoris topology and the Hausdorff quasi-uniformity
zbMATH Open document ID: 1020.54017
DOI: 10.1016/S0166-8641(01)00252-8
author: Jesús Rodríguez-López
author: Salvador Romaguera
published in: Topology and its Applications
publication date: 2003-03-10T00:00:00Z
review text: One early result in the study of hyperspace quasi-uniformities is that the Hausdorff quasi-uniformity of a Pervin quasi-uniformity of a topological space \(X\) induces the Vietoris topology on the family of nonempty subsets of \(X\), see [\textit{N. Levine} and \textit{W. J. Stager jun.}, Math. J. Okayama Univ. 15, 101-106 (1972; Zbl 0246.54033)]. The authors extend this result to the other members of the Pervin quasi-proximity class: If \({\mathcal U}\) belongs to the Pervin quasi-proximity class of a topological space \(X\), the Hausdorff quasi-uniformity of \({\mathcal U}\) induces the Vietoris topology on the family of nonempty subsets of \(X\) if and only if \({\mathcal U}^{-1}\) is hereditarily precompact. (As the authors observe, it follows that the Hausdorff quasi-uniformity is compatible with the Vietoris topology for both the Pervin and well-monotone quasi-uniformities, since the inverse of these quasi-uniformities are hereditarily precompact.) The authors also consider the Vietoris topology on \(K_0(X)\), the nonempty compact subsets of \(X\), and on \({\mathcal C}{\mathcal L}_0(X)\), the nonempty closed subsets of \(X\). They establish that the Hausdorff quasi-uniformity of a quasi-uniform space \((X,{\mathcal U})\) induces the Vietoris topology on \(K_0(X)\) if and only if \({\mathcal U}^{-1}|K\) is precompact for each nonempty compact set \(K\) of \((X,{\mathcal I}({\mathcal U}))\). For a \(T_1\) quasi-uniform space \((X,{\mathcal U})\), they show that the Hausdorff quasi-uniformity induces the Vietoris topology on \({\mathcal C}{\mathcal L}_0(X)\) if and only if \({\mathcal U}\) is equinormal and \({\mathcal U}^{-1}\) is hereditarily precompact. Reviewer's remark: The authors' Lemma 1 can be used to show that the quasi-uniformities that satisfy the property QPC are exactly the quasi-uniformities that belong to the Pervin quasi-proximity class; the review incorporates this observation.
reviewed by: P. Fletcher
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54E35
zbMATH DE Number: 1879103
zbMATH Keywords: Hausdorff quasi-uniformity
zbMATH Keywords: Vietoris topology
zbMATH Keywords: proximal topology
zbMATH Keywords: hereditarily precompact
zbMATH Keywords: equinormal
zbMATH Keywords: Pervin quasi-uniformity
zbMATH Keywords: uqu space
zbMATH Keywords: quasi-metric
zbMATH Keywords: Pervin quasi-proximity class
MaRDI profile type: Publication
cites work: Distance functionals and suprema of hyperspace topologies
cites work: On Quasi-Uniformities in Hyperspaces
cites work: Q5289621
cites work: Quasi-uniform hyperspaces of compact subsets
cites work: Symmetry conditions in terms of open sets
cites work: C-complete quasi-uniform spaces
cites work: Topological Spaces with a Unique Compatible Quasi-Uniformity
cites work: Functorial admissible quasi-uniformities on topological spaces
cites work: Q4371050
cites work: Sobrification and bicompletion of totally bounded quasi-uniform spaces
cites work: Bicompleteness of the fine quasi-uniformity
cites work: Q4033051
cites work: Spaces of continuous functions and quasi-uniform convergence
cites work: Well-quasi-ordering and the Hausdorff quasi-uniformity
cites work: Bispaces admitting only bicomplete or only totally bounded quasi-metrics
cites work: Q3123097
cites work: Q5658811
cites work: Topologies on Spaces of Subsets
cites work: Cauchy sequences in quasi-pseudo-metric spaces
Recommended article: On the Infimum of the Hausdorff and Vietoris Topologies
Recommended article: Relations approximated by continuous functions in the Vietoris topology
Recommended article: Q5313952
Recommended article: Q4500925
Recommended article: Q5287708
Recommended article: Q3629286
Recommended article: Some compactness properties for Vietoris topological spaces
Recommended article: Continuity properties and Alexandroff theorem in Vietoris topology
Recommended article: The bounded Vietoris topology and applications
Recommended article: The Vietoris topology on rectifiable spaces
description: scientific article; zbMATH DE number 1879103