Selectively sequentially pseudocompact group topologies on torsion and torsion-free Abelian groups

Abstract

A space X is selectively sequentially pseudocompact if for every family {Un:n∈N} of non-empty open subsets of X, one can choose a point xn∈Un for every n∈N in such a way that the sequence {xn:n∈N} has a convergent subsequence. Let G be a group from one of the following three classes: (i) V-free groups, where V is an arbitrary variety of Abelian groups; (ii) torsion Abelian groups; (iii) torsion-free Abelian groups. Under the Singular Cardinal Hypothesis SCH, we prove that if G admits a pseudocompact group topology, then it can also be equipped with a selectively sequentially pseudocompact group topology. Since selectively sequentially pseudocompact spaces are strongly pseudocompact in the sense of García-Ferreira and Ortiz-Castillo, this provides a strong positive (albeit partial) answer to a question of García-Ferreira and Tomita.