Nonstandard methods of completing quasi-uniform spaces
type: Item
instance of: scholarly article
title: Nonstandard methods of completing quasi-uniform spaces
zbMATH Open document ID: 0816.54021
DOI: 10.1016/0166-8641(94)00052-5
author: Hermann Render
published in: Topology and its Applications
publication date: 1995-07-19T00:00:00Z
review text: A quasi-uniform space is called complete provided that each Cauchy filter in the sense of Pervin and Sieber has a cluster point. A completion of a quasi-uniform space \((X,{\mathcal V})\) is a complete quasi-uniform space that has a dense subspace quasi-isomorphic to \((X,{\mathcal V})\). Using the tools of nonstandard analysis, a method to construct \(T_ 2\)-completions of a quasi-uniform space is presented. It is used to show that each quasi- uniformity that contains a compatible uniformity has a \(T_ 2\)- completion. Furthermore the following necessary condition in order that a quasi-uniform space \((X,{\mathcal V})\) has a \(T_ 2\)-completion is obtained: If \(\mathcal F\) is a Cauchy filter, \(U \in {\mathcal V}\) and \(x \in X\), then there are \(F \in {\mathcal F}\), \(V, W \in {\mathcal V}\), \(V\) symmetric, such that \(\{z \in F:\) There is \(y \in W(F)\) such that \((x,y)\), \((y,z) \in V\} \subseteq U(x)\). For locally compact spaces the method given can be modified to obtain quasi-uniform \(T_ 2\)-compactifications. In the final section of the paper various problems in the theory of Doitchinov completeness are discussed from a nonstandard point of view.
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 54J05
Mathematics Subject Classification ID: 03H05
zbMATH DE Number: 740455
zbMATH Keywords: quasi-uniform space
zbMATH Keywords: completion
zbMATH Keywords: quasi-uniformity
zbMATH Keywords: completeness
reviewed by: Hans Peter Künzi
MaRDI profile type: Publication
full work available at URL: 0166-8641(94)00052-5
OpenAlex ID: W1991157430
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description: scientific article; zbMATH DE number 740455