Infima and complements in the lattice of quasi-uniformities
type: Item
instance of: scholarly article
title: Infima and complements in the lattice of quasi-uniformities
zbMATH Open document ID: 1168.54010
published in: Topology and its Applications
publication date: 2007-06-26T00:00:00Z
review text: If \(({\mathcal U}_i)_{i\in I}\) is a family of quasi-uniformities on a set \(X\) such that every \({\mathcal U}_i\) has a certain property (P) it is quite natural to ask whether \(\inf({\mathcal U}_i)_{i\in I}\) and \(\sup({\mathcal U}_i)_{i\in I}\) also have property (P), where the infimum and the supremum is taken in the lattice \((q(X), \subseteq)\) of all quasi-uniformities on \(X\). It shown that if (P) is the property of being proximally symmetric, then \(\inf({\mathcal U}_i)_{i\in I}\) has (P). However, there exists a set \(X\) and proximally symmetric quasi-uniformities \({\mathcal U}_1\) and \({\mathcal U}_2\) on \(X\) such that \(\sup({\mathcal U}_1,{\mathcal U}_2)\) is not proximally symmetric. On the other hand, if (P) is the property of being transitive, then \(\sup({\mathcal U}_i)_{i\in I}\) has (P) while there exists a set \(X\) and transitive quasi-uniformities \({\mathcal U}_1\) and \({\mathcal U}_2\) on \(X\) such that \(\inf({\mathcal U}_1,{\mathcal U}_2)\) is not transitive. Additionally, it is shown that the Pervin quasi-uniformity of an irreducible \(T_1\)-space \(X\) does not have a complement in the lattice \((q(X), \subseteq)\) whereas the fine quasi-uniformity of each countable \(T_1\)-space \(X\) has a complement in \((q(X),\subseteq)\).
reviewed by: Harald Brandenburg
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 54E05
Mathematics Subject Classification ID: 06B05
zbMATH DE Number: 5167611
zbMATH Keywords: quasi uniform space
zbMATH Keywords: proximally symmetric quasi uniform space
zbMATH Keywords: transitive quasi uniform space
zbMATH Keywords: Pervin quasi-uniformity
zbMATH Keywords: fine quasi-uniformity
zbMATH Keywords: irreducible topological space
author: Eliza P. de Jager
author: Hans Peter Künzi
MaRDI profile type: Publication
full work available at URL: j.topol.2006.01.017
OpenAlex ID: W2052118283
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DOI: 10.1016/J.TOPOL.2006.01.017
description: scientific article; zbMATH DE number 5167611