Suitable sets for topological groups
by
Michael G. Tkačenko
Universidad Autónoma Metropolitana, Iztapalapa
Coauthors: W. W. Comfort, S. Morris, D. Robbie, S. Svetlichny

The complete version of this work will appear under the same title in Topology and its Applications in 1997. In what follows all topological groups are assumed Hausdorff.

It is well known that every compact connected Abelian group of weight less than or equal to the continuum is monothetic, i. e., contains a dense cyclic subgroup. K. Hoffman and S. Morris proved the best possible analog of this fact in the non-Abelian case: Every compact connected group of weight at most the continuum contains a dense subgroup generated by two elements. These results suggest the following definition. A subset S of a topological group G is called suitable if S is discrete (in itself), S \cup {e_G} is closed in G and the subgroup <S> of G generated by S is dense in G. By a subtle unexpected theorem of Hofmann and Morris, every locally compact group has a suitable set. We examine, therefore, suitable sets in nonlocally compact groups.

Theorem 1. Every countable topological group G has a closed discrete subset S such that <S> = G. In particular, S is a suitable subset for G.

For a topological group G, let b(G) be the least cardinal \kappa such that for every neighborhood of the identity in G there exists a subset C subset or equal G with |C| < \kappa such that C·U=G. Note that b(G) <= d(G)⁺, where d(G) is the density of G.

Theorem 2. Let G be a topological group. If d(G) < b(G), then G has a closed suitable set.

In particular, every separable topological group which is not totally bounded has a suitable set. The case of totally bounded groups is more complicated.

Theorem 3. A separable totally bounded topological group of countable pseudocharacter has a suitable set.

Total boundedness of a group in Theorem 3 can be omitted because of Theorem 2. We do not know, however, if every totally bounded group of countable pseudocharacter has a suitable set. For metrizable groups, we have the following result.

Theorem 4. Every metrizable topological group G has a suitable set. If G is not compact, then it has a closed suitable set.

Date received: March 17, 1997

Copyright © 1997 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caal-19.