Building suitable sets for locally compact groups by means of continuous selections

Abstract

If a discrete subset S of a topological group G with the identity 1 generates a dense subgroup of G and S ∪ { 1 } is closed in G, then S is called a suitable set for G. We apply Michael's selection theorem to offer a direct, self-contained, purely topological proof of the result of Hofmann and Morris [K.-H. Hofmann, S.A. Morris, Weight and c, J. Pure Appl. Algebra 68 (1–2) (1990) 181–194] on the existence of suitable sets in locally compact groups. Our approach uses only elementary facts from (topological) group theory.