Chains of pseudocompact group topologies
type: Item
instance of: scholarly article
title: Chains of pseudocompact group topologies
zbMATH Open document ID: 0909.22002
DOI: 10.1016/S0022-4049(96)00110-7
author: Dikran Dikranjan
published in: Journal of Pure and Applied Algebra
publication date: 1998-02-25T00:00:00Z
review text: For an infinite cardinal \(\sigma\) and a group \(G\) let \({\mathcal P}_\sigma (G)\) be the poset of all pseudocompact group topologies of weight \(\sigma\) on \(G\). \({\mathcal P} (\sigma)\) denotes the class of groups admitting a pseudocompact group topology of weight \(\sigma\), and let \({\mathbf P} (\sigma)\) be the power set of \(\sigma\). Then the main theorem of this interesting paper is the following (cf. Theorems 2.3, 4.1, 5.1): If \(G\in {\mathcal P} (\sigma)\), then the poset \({\mathcal P}_\sigma (G)\) contains a copy of the Boolean algebra \({\mathbf P} (\sigma)\) in the following cases: (i) \(G\) is relatively free; (ii) \(G\) is Abelian and torsion; (iii) \(G\) is Abelian with \(\delta (\log| G|) \leq r(G)\) (in particular, \(G\) is Abelian and torsion-free); (iv) \(G\) is Abelian and divisible. \((r(G)\) is the torsion-free rank of \(G\); for a cardinal \(\alpha\), \(\delta (\alpha)\) denotes the \(G_\delta\)-density of \(\{0;1\}^\alpha)\).
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 22C05
Mathematics Subject Classification ID: 54A10
Mathematics Subject Classification ID: 54A35
Mathematics Subject Classification ID: 54D05
zbMATH DE Number: 1121647
zbMATH Keywords: chain
zbMATH Keywords: poset
zbMATH Keywords: pseudocompact group topologies
reviewed by: Dieter Remus
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1121647