Minimal pseudocompact group topologies on free abelian groups

Abstract

A Hausdorff topological group G is minimal if every continuous isomorphism f : G → H between G and a Hausdorff topological group H is open. Significantly strengthening a 1981 result of Stoyanov, we prove the following theorem: For every infinite minimal abelian group G there exists a sequence { σ n : n ∈ N } of cardinals such that w ( G ) = sup { σ n : n ∈ N } and sup { 2 σ n : n ∈ N } ⩽ | G | ⩽ 2 w ( G ) , where w ( G ) is the weight of G. If G is an infinite minimal abelian group, then either | G | = 2 σ for some cardinal σ, or w ( G ) = min { σ : | G | ⩽ 2 σ } ; moreover, the equality | G | = 2 w ( G ) holds whenever cf ( w ( G ) ) > ω . For a cardinal κ, we denote by F κ the free abelian group with κ many generators. If F κ admits a pseudocompact group topology, then κ ⩾ c , where c is the cardinality of the continuum. We show that the existence of a minimal pseudocompact group topology on F c is equivalent to the Lusin's Hypothesis 2 ω 1 = c . For κ > c , we prove that F κ admits a (zero-dimensional) minimal pseudocompact group topology if and only if F κ has both a minimal group topology and a pseudocompact group topology. If κ > c , then F κ admits a connected minimal pseudocompact group topology of weight σ if and only if κ = 2 σ . Finally, we establish that no infinite torsion-free abelian group can be equipped with a locally connected minimal group topology.