Pseudocompact group topologies with no infinite compact subsets

TL;DR

This paper demonstrates that under certain cardinal conditions, Abelian groups can be endowed with pseudocompact topologies where all countable subgroups are maximally totally bounded, and explores the structural properties of such groups.

Contribution

It introduces a new class of pseudocompact Abelian groups with property $ ext{h}$, showing their existence under mild cardinal inequalities and the SCH, and analyzes their algebraic and topological features.

Findings

Existence of pseudocompact topologies with property $ ext{h}$ for certain Abelian groups.

Such groups contain no infinite compact subsets.

They are examples of Pontryagin reflexive precompact groups that are not compact.

Abstract

We show that every Abelian group satisfying a mild cardinal inequality admits a pseudocompact group topology from which all countable subgroups inherit the maximal totally bounded topology (we say that such a topology satisfies property $\h$). Every pseudocompact Abelian group $G$ with cardinality $|G|\leq 2^{2^\cc}$ satisfies this inequality and therefore admits a pseudocompact group topology with property $\h$. Under the Singular Cardinal Hypothesis (SCH) this criterion can be combined with an analysis of the algebraic structure of pseudocompact groups to prove that every pseudocompact Abelian group admits a pseudocompact group topology with property $\h$. We also observe that pseudocompact Abelian groups with property $\h$ contain no infinite compact subsets and are examples of Pontryagin reflexive precompact groups that are not compact.