Quasi-uniform hyperspaces of compact subsets
type: Item
instance of: scholarly article
title: Quasi-uniform hyperspaces of compact subsets
zbMATH Open document ID: 0927.54008
DOI: 10.1016/S0166-8641(97)00133-8
author: Jiling Cao
author: Hans Peter Künzi
author: I. L. Reilly
author: Salvador Romaguera
published in: Topology and its Applications
publication date: 1999-12-12T00:00:00Z
review text: Suppose \((X,{\mathcal U})\) is a quasi-uniform space, \({\mathcal T}_V\) denotes the Vietoris topology and \({\mathcal U}^*\) denotes the Hausdorff-Bourbaki quasi-uniformity on \({\mathcal K}(X)\) which is the space of all non-empty compact subsets of \(X\). It is well known that if \({\mathcal U}\) is a uniformity, then \({\mathcal T}_V\) is compatible with \({\mathcal U}^*\) and \({\mathcal U}^*\) is complete if and only if \({\mathcal U}\) is complete. In this paper the authors show that \({\mathcal T}_V \subset {\mathcal T}({\mathcal U}^*)\) and investigate when they are equal. For this purpose they introduce the concepts ``compactly symmetric'' and ``small-set symmetric'' which strengthen the concept of ``local symmetry''. Main results: (1) If \((X,{\mathcal U})\) is compactly symmetric or small-set symmetric, then \({\mathcal T}_V\) is compatible with \({\mathcal U}^*\). (2) If \((X,{\mathcal U})\) is compactly symmetric, then \({\mathcal U}^*\) is complete if and only if \({\mathcal U}\) is complete. The result is not true for small-set symmetric spaces.
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 54E15
zbMATH DE Number: 1307962
zbMATH Keywords: Vietoris hyper-topology
zbMATH Keywords: compactly symmetric
zbMATH Keywords: small-set symmetric
zbMATH Keywords: Hausdorff-Bourbaki quasi-uniformity
zbMATH Keywords: complete
reviewed by: S. Naimpally
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1307962