Locally pseudocompact topological groups
type: Item
instance of: scholarly article
title: Locally pseudocompact topological groups
zbMATH Open document ID: 0828.22003
DOI: 10.1016/0166-8641(94)00048-8
author: F. Javier Trigos-Arrieta
author: W. W. Comfort
published in: Topology and its Applications
publication date: 1996-01-02T00:00:00Z
review text: By a result of \textit{W. Comfort} and \textit{K. Ross} [Pac. J. Math. 16, 483- 496 (1966; Zbl 0214.285)], the following conditions are equivalent for a dense subgroup \(G\) of a compact topological group \(\overline {G}\): (a) \(G\) is pseudocompact; (b) \(G\) is \(C^*\)-embedded in \(\overline {G}\); (c) \(G\) is \(G_\delta\)-dense in \(\overline {G}\), that is, \(G\) intersects every non-empty \(G_\delta\)-subset of \(\overline {G}\). -- What happens if the ``enveloping'' group \(\overline {G}\) is locally pseudo-compact? Following A. Weil, the authors call a group \(G\) locally bounded if there exists a bounded neighborhood \(U\) of the identity in \(G\), that is, \(U\) can be covered by finitely many translations of any non-empty open subset of \(G\). The main result of the paper is the following theorem which answers the question above. Theorem. If \(G\) is a locally bounded group with Weil completion \(\overline {G}\), then the following conditions are equivalent: (a) \(G\) is locally pseudocompact; (b) \(G\) is \(C^*\)-embedded in \(\overline{G}\); (c) the set \(\text{cl}_G U\) is \(C^*\)-embedded in \(\text{cl}_{\overline {G}} U\) for some non-empty open subset \(U\) of \(G\); (d) the set \(\text{cl}_G U\) is \(C^*\)-embedded in \(\text{cl}_{\overline {G}} U\) for every bounded open subset \(U\) of \(G\). -- The paper also contains helpful discussions and an extensive bibliography.
Mathematics Subject Classification ID: 22A10
Mathematics Subject Classification ID: 54H11
Mathematics Subject Classification ID: 54C45
zbMATH DE Number: 762125
zbMATH Keywords: locally compact group
zbMATH Keywords: pseudocompact space
zbMATH Keywords: Weil completion
reviewed by: Mikhail Tkachenko
MaRDI profile type: Publication
full work available at URL: 0166-8641(94)00048-8
OpenAlex ID: W1984177542
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description: scientific article; zbMATH DE number 762125