Convergent sequences in minimal groups

TL;DR

This paper proves that all infinite minimal abelian groups contain non-trivial convergent sequences, extending known results from compact groups, and constructs specific minimal group topologies on free groups with unique properties.

Contribution

It extends the known convergence result from compact to minimal abelian groups and constructs new minimal topologies on free groups with special features.

Findings

Infinite minimal abelian groups contain non-trivial convergent sequences.

Constructed minimal topologies on free groups with no non-trivial proper closed normal subgroups.

All compact subsets in these topologies are finite.

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. Clearly, every compact Hausdorff group is minimal. It is well known that every infinite compact Hausdorff group contains a non-trivial convergent sequence. We extend this result to minimal abelian groups by proving that every infinite minimal abelian group contains a non-trivial convergent sequence. Furthermore, we show that "abelian" is essential and cannot be dropped. Indeed, for every uncountable regular cardinal kappa we construct a Hausdorff group topology T_kappa on the free group F(kappa) with kappa many generators having the following properties: (i) (F(kappa), T_kappa) is a minimal group; (ii) every subset of F(kappa) of size less than kappa is T_kappa-discrete (and thus also T_kappa-closed); (iii) there are no non-trivial proper T_kappa-closed normal subgroups of F(kappa). In particular, all compact subsets of (F(kappa), T_kappa) are finite, and every Hausdorff quotient group of (F(kappa), T_kappa) is minimal (that is, (F(kappa), T_kappa) is totally minimal).