Topological groups in which all countable subgroups are closed

Abstract

We study the class CC of topological Abelian groups G such that all countable subgroups of G are closed. It is shown that all countably compact subsets of a bounded torsion group in CC are finite, while in general countably compact subsets of any group in CC are countable and compact.It was proved by the author in 1992 that there exist arbitrarily big pseudocompact groups in CC; however all these groups did not contain non-trivial convergent sequences. For every infinite cardinal κ satisfying κω=κ, we construct here a pseudocompact Abelian group G∈CC of cardinality κ which contains non-trivial convergent sequences.We show, however, that all countably pseudocompact groups as well as all countably pracompact groups in the class CC are finite.