Incomplete infima of complete uniformities related to topological groups and topological vector spaces
type: Item
instance of: scholarly article
title: Incomplete infima of complete uniformities related to topological groups and topological vector spaces
zbMATH Open document ID: 0772.54023
DOI: 10.1016/0022-4049(91)90015-T
published in: Journal of Pure and Applied Algebra
publication date: 1992-06-25T00:00:00Z
review text: We consider uniformities in terms of vicinities. Completeness is meant in the sense that all Cauchy filters (or, equivalently, all Cauchy nets) converge. A topological group \(X\) is called complete if its left uniformity \({\mathfrak L}\) (or, equivalently its right uniformity \({\mathfrak R}\)) is complete. This terminology applies as usual to topological vector spaces, in which case the left and right uniformities coincide. If \({\mathfrak V}\), \({\mathfrak W}\) are two complete uniformities on a set \(X\) then either infimum \({\mathfrak V}\wedge {\mathfrak W}\) (i.e., the finest uniformity on \(X\) which is coarser than \({\mathfrak V}\) and \({\mathfrak W}\)) need not be complete. The purpose of this article is to exhibit examples for this phenomenon under various circumstances, starting in a general uniform setting and then proceeding to more restricted situations in topological groups and topological vector spaces.
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 54H11
Mathematics Subject Classification ID: 46A19
Mathematics Subject Classification ID: 54B15
Mathematics Subject Classification ID: 57N17
zbMATH DE Number: 9460
zbMATH Keywords: completeness
zbMATH Keywords: Cauchy filters
zbMATH Keywords: Cauchy nets
zbMATH Keywords: uniformity
zbMATH Keywords: complete uniformities
author: Walter Roelcke
MaRDI profile type: Publication
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full work available at URL: 0022-4049(91)90015-t
OpenAlex ID: W2038763288
description: scientific article; zbMATH DE number 9460