Convergent sequences in minimal groups

Publication date: 1 January 2009

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).

Mathematics Subject Classification ID

Structure of general topological groups (22A05) Topological groups (topological aspects) (54H11) ``(P)-minimal and ``(P)-closed spaces (54D25) Several topologies on one set (change of topology, comparison of topologies, lattices of topologies) (54A10) Compact groups (22C05) Cardinality properties (cardinal functions and inequalities, discrete subsets) (54A25) Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.) (54A20)

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