Minimal pseudocompact group topologies on free abelian groups

TL;DR

This paper characterizes the structure of minimal abelian groups, explores conditions for pseudocompact topologies on free abelian groups, and establishes limitations on locally connected minimal topologies.

Contribution

It extends previous results by providing detailed conditions for minimal and pseudocompact topologies on free abelian groups, linking set-theoretic hypotheses to topological group properties.

Findings

Infinite minimal abelian groups have specific cardinality and weight relations.

Existence of pseudocompact topologies on free abelian groups depends on set-theoretic hypotheses.

No infinite torsion-free abelian group admits a locally connected minimal topology.

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 {\sigma_n : n\in N} of cardinals such that w(G) = sup {\sigma_n : n \in N} and sup {2^{\sigma_n} : n \in N} \leq |G| \leq 2^{w(G)}, where w(G) is the weight of G. If G is an infinite minimal abelian group, then either |G| = 2^\sigma for some cardinal \sigma, or w(G) = min {\sigma: |G| \leq 2^\sigma}; moreover, the equality |G| = 2^{w(G)} holds whenever cf (w(G)) > \omega. For a cardinal \kappa, we denote by F_\kappa the free abelian group with \kappa many generators. If F_\kappa admits a pseudocompact group topology, then \kappa \geq 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^{\omega_1} = c. For \kappa > c, we prove that F_\kappa admits a (zero-dimensional) minimal pseudocompact group topology if and only if F_\kappa has both a minimal group topology and a pseudocompact group topology. If \kappa > c, then F_\kappa admits a connected minimal pseudocompact group topology of weight \sigma if and only if \kappa = 2^\sigma. Finally, we establish that no infinite torsion-free abelian group can be equipped with a locally connected minimal group topology.