On thin generating sets in topological groups

Abstract

If a discrete subset S of a topological group G with identity 1 generates a dense subgroup of G and S boolean OR {1} is closed in G, then S is called a suitable set for G. We show that every almost metrizable topological group has a suitable set, thus generalizing a similar result for metrizable groups obtained by Comfort, Morris, Robbie, Svetlichny and Tkacenko. It is also proved that the free topological groups of compact spaces from a class including all dyadic and all polyadic spaces have suitable sets.