On countably complete topological groups

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 $\pi :G\to G/H$ satisfies that ${\pi }^{-1}\left(F\right)$ is r-pseudocompact in G for each r-pseudocompact set F in $G/H$; (2) every countably complete weakly ${\Psi }_{\omega }$-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 ${\Psi }_{\omega }$-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.