Groups without convergent sequences
The aim of the paper is to describe the compact abelian topological groups that contain a dense (pseudocompact) subgroup without nontrivial convergent sequences. It is shown, for example, that if \(G\) is a precompact abelian topological group such that for some integer \(m\) the subgroup \(mG\) of \(G\) is infinite and \(w(mG) =\kappa\) is a strong limit cardinal, then every dense subgroup of \(G\) has convergent sequences of length \(cf(\kappa)\). By a result of Sirota, the group \(\mathbb{Z} (2)^\kappa\) has a dense pseudocompact subgroup without nontrivial convergent sequences for every \(\kappa>1\) satisfying \(\kappa^\omega= \kappa\). The authors generalize this result to products of compact groups of a given order \(n\) and weight \(\leq\lambda\) and present an example of a compact metric group no power of which has a dense subgroup without non-trivial convergent sequences. Finally, the authors make use of \(G\subset H\) to characterize compact abelian torsion groups every dense subgroup of which contains convergent sequences.
scientific article; zbMATH DE number 987930