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

TL;DR

The paper demonstrates that certain Abelian groups can be endowed with a selectively sequentially pseudocompact topology, strengthening previous pseudocompactness results under the Singular Cardinal Hypothesis.

Contribution

It proves that torsion, torsion-free, and V-free Abelian groups admit selectively sequentially pseudocompact topologies assuming SCH, advancing the understanding of group topologies with strong pseudocompactness.

Findings

Selective sequential pseudocompactness implies strong pseudocompactness.

Any group in the specified classes admits such a topology under SCH.

Provides partial positive answer to a question by García-Ferreira and Tomita.

Abstract

A space X is selectively sequentially pseudocompact if for every sequence (U_n) of non-empty open subsets of X, one can choose a point x_n in each U_n in such a way that the sequence (x_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 Garc\'ia-Ferreira and Ortiz-Castillo, this provides a strong positive (albeit partial) answer to a question of Garc\'ia-Ferreira and Tomita.