Pseudocompact group topologies with no infinite compact subsets

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 ♯ ). Every pseudocompact Abelian group G with cardinality | G | ≤ 2 2 c satisfies this inequality and therefore admits a pseudocompact group topology with property ♯ . 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 ♯ . We also observe that pseudocompact Abelian groups with property ♯ contain no infinite compact subsets and are examples of Pontryagin reflexive precompact groups that are not compact.