Categorically compact topological groups
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.
scientific article; zbMATH DE number 1121694
scientific article