Precompact Fréchet topologies on abelian groups
type: Item
instance of: scholarly article
title: Precompact Fréchet topologies on abelian groups
zbMATH Open document ID: 1260.22001
published in: Topology and its Applications
publication date: 2012-10-31T00:00:00Z
review text: The authors introduce, for every countable abelian group \(G\), the notion of \(\gamma_G\)-set and show that a necessary and sufficient condition for the existence of a precompact Fréchet non-metrizable topology on \(G\) is the existence of an uncountable \(\gamma_G\)-set which separates points of \(G\). Assuming the existence of an uncountable \(\gamma\)-set, the authors show that every countable abelian group admits a non-metrizable precompact Fréchet topology. With the assumption \({\mathfrak p}>\omega_1\), they also establish the existence of a non-metrizable Fréchet topology on every countable group which admits a non-discrete topology at all. Through further study of the notion of \(\gamma_G\)-set, they show that the minimal size of a subset of the dual group \(G^\ast\) which is not a \(\gamma_G\)-set is the pseudointersection number \({\mathfrak p}\) for any countable abelian group \(G\).
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 54A20
Mathematics Subject Classification ID: 54A35
zbMATH DE Number: 6100635
zbMATH Keywords: abelian Fréchet groups
zbMATH Keywords: precompact topologies
zbMATH Keywords: \(\gamma\)-set
zbMATH Keywords: pseudointersection number
author: Michael Hrušák
author: Ulises Ariet Ramos-García
reviewed by: Shing So
MaRDI profile type: Publication
full work available at URL: j.topol.2012.09.004
OpenAlex ID: W2153675952
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DOI: 10.1016/J.TOPOL.2012.09.004
description: scientific article; zbMATH DE number 6100635