A characterization of strongly countably complete topological groups

Abstract

We prove that a topological group G is strongly countably complete (the notion introduced by Z. Frolík in 1961) iff G contains a closed countably compact subgroup H such that the quotient space G/H is completely metrizable and the canonical mapping π:G→G/H is closed. We also show that every strongly countably complete group is sequentially complete, has countable Gδ-tightness, and its completion is a Čech-complete topological group. Further, a pseudocompact strongly countably complete group is countably compact. An example of a pseudocompact topological Abelian group H with the Fréchet–Urysohn property is presented such that H fails to be sequentially complete, thus answering a question posed by Dikranjan, Martín Peinador, and Tarieladze in [Appl. Categor. Struct. 15 (2007) 511–539].