Convergent sequences in minimal 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. 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 Tkappa on the free group F(kappa) with kappa many generators having the following properties: (i) (F(kappa), Tkappa) is a minimal group; (ii) every subset of F(kappa) of size less than kappa is Tkappa-discrete (and thus also Tkappa-closed); (iii) there are no non-trivial proper Tkappa-closed normal subgroups of F(kappa). In particular, all compact subsets of (F(kappa), Tkappa) are finite, and every Hausdorff quotient group of (F(kappa), Tkappa) is minimal (that is, (F(kappa), Tkappa) is totally minimal).