Categorically compact topological groups Original Research Article

We study the notion of a categorically compact topological group, suggested by the Kuratowski-Mrówka characterization of compact spaces. A topological group G is categorically compact, or C-compact, if for any topological group H the projection G × H → H sends closed subgroups to closed subgroups. We prove, among others, the following theorems: (1) any product of C-compact topological groups is C-compact; (2) separable C-compact groups are totally minimal; (3) C-compact soluble topological groups are compact.