The impact of the Bohr topology on selective pseudocompactness

TL;DR

This paper investigates how the Bohr topology influences the property of selective pseudocompactness in Boolean groups, establishing conditions under which such groups fail to be selectively pseudocompact and constructing examples under certain set-theoretic assumptions.

Contribution

It proves that if the subgroup topology on every countable subgroup is finer than the Bohr topology, then the group is not selectively pseudocompact, and constructs pseudocompact reflexive groups that are not selectively pseudocompact.

Findings

Many pseudocompact Boolean groups are not selectively pseudocompact due to their subgroup topologies.

Under the Singular Cardinal Hypothesis, there exist pseudocompact reflexive groups that are not selectively pseudocompact.

Abstract

Recall that a space X is selectively pseudocompact if for every sequence (U_n) of non-empty open subsets of X one can choose a point x_n in U_n for all n such that the resulting sequence (x_n) has an accumulation point in X. This notion was introduced under the name strong pseudocompactness by Garc\'ia-Ferreira and Ortiz-Castillo, the present name is due to Dorantes-Aldama and the first author. In 2015, Garc\'ia-Ferreira and Tomita constructed a pseudocompact Boolean group that is not selectively pseudocompact. We prove that if the subgroup topology on every countable subgroup H of an infinite Boolean topological group G is finer than its maximal precompact topology (the so-called Bohr topology of H), then G is not selectively pseudocompact, and from this result we deduce that many known examples in the literature of pseudocompact Boolean groups automatically fail to be selectively pseudocompact. We also show that, under the Singular Cardinal Hypothesis, every infinite pseudocompact Boolean group admits a pseudocompact reflexive group topology which is not selectively pseudocompact.