Convergent sequences in minimal groups

Dmitri Shakhmatov

Published 2009-01-01Version 1

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).