Building Suitable Sets for Locally Compact Groups by Means of Continuous Selections *

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 [8] on the existence of suitable sets in locally compact groups. Our approach uses only elementary facts from (topological) group theory. All topological groups considered in this paper are assumed to be Hausdorff, and all topological spaces are assumed to be Tychonoff. 1 Motivating background Let G be a group. We use 1G to denote the identity element of G. If X is a subset of G, then 〈X〉 will denote the smallest subgroup of G containing X, and we say that X (algebraically) generates 〈X〉. Definition 1. [2, 13, 8] A subset X of a topological group G is called a suitable set for G provided that: (i) X is discrete, (ii) X ∪ {1G} is closed in G, (iii) 〈X〉 is dense in G. Suitable sets were considered first in the early sixties by Tate in the framework of Galois cohomology (see [2]). Tate proved1 that every profinite group has a suitable set. This result has later been proved also by Mel’nikov [13]. Later on, Hofmann and Morris discovered the following fundamental theorem: ∗MSC Subj. Class.: Primary: 22D05; Secondary: 22A05, 22C05, 54A25, 54B05, 54B35, 54C60, 54C65, 54D30, 54D45, 54H11.