Categorically compact topological groups
type: Item
instance of: scholarly article
title: Categorically compact topological groups
zbMATH Open document ID: 0887.22001
DOI: 10.1016/S0022-4049(96)00139-9
author: Dikran Dikranjan
author: Vladimir V. Uspenskij
published in: Journal of Pure and Applied Algebra
publication date: 1998-04-28T00:00:00Z
review text: Several notions, in various settings, have been called categorical compactness; the authors give numerous references, including \textit{D. Dikranjan} and \textit{E. Giuli} [Proc. Internat. Conf. Categorical Topology, Prague 1988, 284-296 (1989)] and \textit{D. Dikranjan} and \textit{W. Tholen} [Mathematics and its Applications, vol. 346 (Dordrecht 1995; Zbl 0853.18002)] and earliest, \textit{E. Manes} [General Topol. Appl. 4, 341-360 (1974; Zbl 0289.54003)]. The new version here seems to be the first one treated seriously in Hausdorff topological groups. Such a group \(G\) is called categorically compact or \(C\)-compact if for every such group \(H\) the projection \(G\times H\to H\) takes closed subgroups to closed subgroups. The authors do not know a \(C\)-compact Hausdorff group which is not compact, but they speculate that infinite discrete examples exist. A \(C\)-compact group \(G\) must be compact if it is solvable, or more generally if no nonzero subgroup of \(G\) has dense commutator subgroup. There are numerous further results and examples. For instance, all products of \(C\)-compact groups are \(C\)-compact; separable \(C\)-compact groups are totally minimal. The proof of the product theorem uses a characterization of \(C\)-compact groups by convergence of certain filters.
Mathematics Subject Classification ID: 22A05
Mathematics Subject Classification ID: 18A32
Mathematics Subject Classification ID: 22C05
Mathematics Subject Classification ID: 54B10
Mathematics Subject Classification ID: 54D25
Mathematics Subject Classification ID: 54D30
Mathematics Subject Classification ID: 54D65
zbMATH DE Number: 1121694
zbMATH Keywords: categorically compact
zbMATH Keywords: product
zbMATH Keywords: totally minimal
zbMATH Keywords: \(h\)-complete
zbMATH Keywords: Hausdorff topological groups
reviewed by: J. R. Isbell
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1121694