Short notes on quasi-uniform spaces. I: Uniform local symmetry
type: Item
instance of: scholarly article
title: Short notes on quasi-uniform spaces. I: Uniform local symmetry
zbMATH Open document ID: 0841.54021
DOI: 10.1007/BF01874619
published in: Acta Mathematica Hungarica
publication date: 1996-05-13T00:00:00Z
review text: The author calls a quasi-uniformity \({\mathcal U}\) uniformly locally symmetric provided that for any \(U \in {\mathcal U}\) there is a \(V \in {\mathcal U}\) such that for any \(x \in X\) there is a \(W \in {\mathcal U}\) with \(W^{-1} (V(x)) \subseteq U(x)\). Clearly each uniformly locally symmetric quasi-uniformity is locally symmetric. He proves that each mixed-symmetric, uniformly regular quasi-uniform space is uniformly locally symmetric and that each uniformly locally symmetric quasi-uniformity is quiet. (A quasi-uniformity is called quiet provided that the Cauchy filter pairs are uniformly weakly concentrated.) Furthermore he shows that a quasi-uniform space is uniformly locally symmetric provided that it has a uniformly locally symmetric sup-dense subspace. Some counterexamples are also constructed, e.g. an open-symmetric, not point-symmetric \(T_2\)-quasi-uniformity. The paper is well motivated by numerous additional interesting remarks on symmetry properties often studied in quasi-uniform spaces.
Mathematics Subject Classification ID: 54E15
zbMATH DE Number: 878098
zbMATH Keywords: quiet quasi-uniformity
zbMATH Keywords: uniformly locally symmetric quasi-uniformity
zbMATH Keywords: mixed-symmetric
author: J. Deak
reviewed by: Hans Peter Künzi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 878098