Some cases of preservation of the Pontryagin dual by taking dense subgroups
type: Item
instance of: scholarly article
title: Some cases of preservation of the Pontryagin dual by taking dense subgroups
zbMATH Open document ID: 1235.22004
published in: Topology and its Applications
publication date: 2011-09-22T00:00:00Z
review text: For an abelian topological group \(G\) let \(G^\wedge\) denote the group of continuous characters of \(G\), endowed with the compact-open topology. A dense subgroup \(H\) of an abelian topological group \(G\) is said to \textit{determine} \(G\) if the restriction operator \(G^\wedge\to H^\wedge\) is a topological isomorphism. An abelian topological group is \textit{determined} if each dense subgroup of \(G\) determines \(G\). The authors detect some operations preserving determined groups and present several representative examples of determined and non-determined groups. One of the main results is Theorem 14 saying that each compact abelian group \(G\) of weight \(w(G)\geq \mathfrak c\) contains a dense pseudocompact subgroup which does not determine \(G\).
reviewed by: Taras Banakh
Mathematics Subject Classification ID: 22A05
zbMATH DE Number: 5949216
zbMATH Keywords: topological abelian group
zbMATH Keywords: character group
zbMATH Keywords: dense subgroup
zbMATH Keywords: determined group
zbMATH Keywords: determining subgroup
zbMATH Keywords: pseudocompact group
author: M. J. Chasco
author: X. Domínguez
author: F. Javier Trigos-Arrieta
MaRDI profile type: Publication
full work available at URL: j.topol.2011.06.018
OpenAlex ID: W2016192069
cites work: Precompact noncompact reflexive Abelian groups
cites work: Q5553887
cites work: Q4484934
cites work: On the arc component of a locally compact Abelian group
cites work: Additive subgroups of topological vector spaces
cites work: Open subgroups and Pontryagin duality
cites work: Pontryagin duality in the class of precompact abelian groups and the Baire property
cites work: Pontryagin duality for metrizable groups
cites work: The Dual Group of a Dense Subgroup
cites work: Quasi-convex density and determining subgroups of compact Abelian groups
cites work: Sequential completeness of quotient groups
cites work: Dual topologies on non-abelian groups
cites work: Q5609457
cites work: Pseudocompact group topologies with no infinite compact subsets
cites work: Uncountable products of determined groups need not be determined
cites work: Q4843641
cites work: Extensions of the Pontrjagin duality. I: Infinite products
cites work: Q4097467
cites work: Q5605131
DOI: 10.1016/J.TOPOL.2011.06.018
description: scientific article; zbMATH DE number 5949216