Suitable sets for topological groups

Abstract

A subset S of a topological group G is said to be a suitable set if (a) it has the discrete topology, (b) it is a closed subset of G{1}, and (c) the subgroup generated by S is dense in G. K.H. Hoffmann and S.A. Morris proved that every locally compact group has a suitable set. In this paper it is proved that every metrizable topological group and every countable Hausdorff topological group has a suitable set. Examples of Hausdorff topological groups without suitable sets are produced. The free abelian topological group on the Stone-Čech compactification of any uncountable discrete space is one such example. Under the assumption of the Continuum Hypothesis or Martin's Axiom it is shown that examples exist of separable Hausdorff topological groups with no suitable set. It is not known if such examples exist in ZFC alone. An example is produced here of a compact connected abelian group with a one-element suitable set which has a dense σ-compact connected subgroup with no suitable set.