On countably complete topological groups

Abstract

In this paper, we give a characterization of countably complete topological groups and study when a countably complete subgroup of a topological group is C-embedded. We mainly show that (1) a topological group G is countably complete (the notion introduced by M. Tkachenko in 2012) iff G contains a closed r-pseudocompact subgroup H such that the quotient space G/H is completely metrizable and the canonical quotient mapping π:G→G/H satisfies that π−1(F) is r-pseudocompact in G for each r-pseudocompact set F in G/H; (2) every countably complete weakly Ψω-factorizable and ω-balanced subgroup H of a topological group G is C-embedded; (3) every countably complete subgroup H of a pointwise pseudocompact topological group G is C-embedded; (4) every uniformly strongly countably complete and weakly Ψω-factorizable subgroup H of a topological group G is C-embedded. Further, an ω-narrow locally compact subgroup H of a topological group G is C-embedded.