Completeness properties of group topologies for \(\mathbb{R}\)
type: Item
instance of: scholarly article
title: Completeness properties of group topologies for \(\mathbb{R}\)
zbMATH Open document ID: 1322.22004
author: E. Martín-Peinador
author: T. Christine Stevens
published in: Topology and its Applications
publication date: 2015-08-19T00:00:00Z
review text: The authors study several complete Hausdorff group topologies on the real line. They recall that the character group of the discrete group \(\mathbb Q\) is a torsion-free divisible group of cardinality \(\mathfrak c\) and hence algebraically isomorphic to \(\mathbb R\) and show that this topology is not comparable with the usual topology \({\mathcal U}\) on \(\mathbb R\). Afterwards, they consider group topologies of the form \({\mathcal U}\times {\mathcal T}\) on \(\mathbb R\times \mathbb Z\) where \({\mathcal T}\) is some Hausdorff group topology on \(\mathbb Z\). They compute the completion of the quotient topology on \(\mathbb R\) induced by the mapping \((\mathbb R,{\mathcal U})\times (\mathbb Z,{\mathcal T})\to \mathbb R,\;(x,y)\mapsto x+y\) and apply the result to topologies \({\mathcal T}\) induced by \(T\)-sequences on \(\mathbb Z\) to obtain complete group topologies on \(\mathbb R\) which are strictly weaker than the usual topology.
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 43A40
zbMATH DE Number: 6473618
zbMATH Keywords: complete group topology
zbMATH Keywords: open mapping theorem
zbMATH Keywords: dual group
zbMATH Keywords: Pontryagin duality
zbMATH Keywords: metrizable group
reviewed by: Lydia Außenhofer
MaRDI profile type: Publication
full work available at URL: j.topol.2015.05.079
OpenAlex ID: W1628032554
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DOI: 10.1016/J.TOPOL.2015.05.079
description: scientific article; zbMATH DE number 6473618