Uncountable products of determined groups need not be determined
type: Item
instance of: scholarly article
title: Uncountable products of determined groups need not be determined
zbMATH Open document ID: 1156.22002
author: Salvador Hernández
author: Sergio Macario
author: F. Javier Trigos-Arrieta
published in: Journal of Mathematical Analysis and Applications
publication date: 2008-10-22T00:00:00Z
review text: The main object of the present paper are determined groups which are defined as follows: If \(H\) is a dense subgroup of the abelian topological group \(G\) then \(H\) determines \(G\) if their character groups endowed with the compact-open topology are topologically isomorphic. An abelian topological group is called determined if it is determined by all its dense subgroups. Every abelian metrizable group is determined. It is shown that any countable sum of determined groups determines the product. Conversely, if the sum of the real line (the integers or a non-trivial locally convex vector space) determines the product, the product must be countable. The main result is the following: If a compact abelian group is determined, then it must be metrizable. This was proved before by \textit{W. W. Comfort, S. U. Raczkowski} and \textit{F. J. Trigos-Arrieta} [Czech. Math. J. 54, No.~2, 509--533 (2004; Zbl 1080.22500)] under the assumption of the continuum hypothesis.
reviewed by: Lydia Außenhofer
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 28C10
Mathematics Subject Classification ID: 54D30
zbMATH DE Number: 5355978
zbMATH Keywords: dual group
zbMATH Keywords: determined group
zbMATH Keywords: metrizable groups, compact abelian group
zbMATH Keywords: Rosenthal compact
zbMATH Keywords: direct sum
zbMATH Keywords: direct product
zbMATH Keywords: reflexive group
zbMATH Keywords: dense subgroup
zbMATH Keywords: Aussenhofer-Chasco theorem
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.07.065
OpenAlex ID: W2018160612
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DOI: 10.1016/J.JMAA.2008.07.065
description: scientific article; zbMATH DE number 5355978