On Hewitt groups and finer locally compact group topologies
type: Item
instance of: scholarly article
title: On Hewitt groups and finer locally compact group topologies
zbMATH Open document ID: 1347.22002
author: Daniele Impieri
author: Dikran Dikranjan
published in: Topology and its Applications
publication date: 2015-08-19T00:00:00Z
review text: The authors study the class \(\mathcal{H}\) of all Hewitt abelian topological groups. A topological group \((G,\tau)\) is called a Hewitt group if the only strictly finer than \(\tau\) locally compact group topology on \(G\) is the discrete topology. Hewitt showed that the reals \(\mathbb R\) and the circle group \(\mathbb T\) belong to \(\mathcal{H}\). \textit{N. W. Rickert} [Trans. Am. Math. Soc. 126, 225--235 (1967; Zbl 0189.32404)] and \textit{M. Rajagopalan} [Math. Ann. 176, 169--180 (1968; Zbl 0174.05601)] characterized the non-discrete abelian locally compact groups in the class \(\mathcal{H}\) as those that contain either \(\mathbb R\), \(\mathbb T\), or the compact group \(\mathbb J_p\) of \(p\)-adic integers as an open subgroup. The authors prove that a non-discrete topological abelian group \(H\) is a Hewitt group precisely when it satisfies one of the following conditions: (1) if \(H\) is non-locally compact, then it does not have infinite compact subgroups and pseudolines (i.e., subgroups isomorphic to a continuous injective image of \(\mathbb R\)); (2) if \(H\) is locally compact, then it is either topologically isomorphic to \(\mathbb R\times D\) for a discrete group \(D\) or contains an open subgroup isomorphic to either \(\mathbb J_p\) or \(\mathbb T\).
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 54H05
Mathematics Subject Classification ID: 22B05
zbMATH DE Number: 6473607
zbMATH Keywords: Hewitt group
zbMATH Keywords: locally compact abelian group
zbMATH Keywords: finer topology
zbMATH Keywords: polishable subgroup
zbMATH Keywords: circle group
zbMATH Keywords: reals
zbMATH Keywords: \(p\)-adic integers
reviewed by: Saak S. Gabriyelyan
MaRDI profile type: Publication
full work available at URL: j.topol.2015.05.072
OpenAlex ID: W659265658
cites work: Good approximation and characterization of subgroups of R = Z
cites work: On characterized subgroups of compact abelian groups
cites work: Questions on the Borel Complexity of Characterized Subgroups
cites work: On the Borel complexity of characterized subgroups
cites work: A characterization of the maximally almost periodic abelian groups
cites work: On T-sequences and characterized subgroups
cites work: Characterizable groups: Some results and open questions
cites work: A remark on characters of locally compact Abelian groups
cites work: The structure of compact groups. A primer for the student -- a handbook for the expert
cites work: The classification of hypersmooth Borel equivalence relations
cites work: Polish LCA groups are strongly characterizable
cites work: Q4085968
cites work: Topologies in locally compact groups
cites work: Locally Compact Topologies for Groups
DOI: 10.1016/J.TOPOL.2015.05.072
description: scientific article; zbMATH DE number 6473607