Convergent Sequences in Minimal Groups

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 κ we construct a Hausdorff group topology Tκ on the free group F (κ) with κ many generators having the following properties: (i) (F (κ),Tκ) is a minimal group; (ii) every subset of F (κ) of size less than κ is Tκ-discrete (and thus also Tκ-closed); (iii) there are no non-trivial proper Tκ-closed normal subgroups of F (κ). In particular, all compact subsets of (F (κ),Tκ) are finite, and every Hausdorff quotient group of (F (κ),Tκ) is minimal (that is, (F (κ),Tκ) is totally minimal). We denote by N the set the set of natural numbers. Let X be a topological space. A convergent sequence in X is a sequence S = {xn : n ∈ N} of points of X such that there exists a point x ∈ X (called the limit of S) so that S \ U is finite for every open subset U of X containing x. (We also say that S converges to x.) A sequence S is non-trivial provided that the set S is infinite. The identity element of a group G is denoted by 1. When G is abelian, the additive notation is used, and so 1 is replaced by the zero element 0 of G.