Locally compact Abelian groups with totally suitable sets (Q2705011)

A subset \(S \subset G\setminus \{1_G\}\) in a topological group \(G\) is said to be suitable if it is relatively discrete, \(\overline S \subset S\cup \{1_G\}\) and the subgroup of \(G\) generated by \(S\) is dense in \(G\). This notion was introduced by \textit{K. H. Hofmann} and \textit{S. A. Morris} [J. Pure Appl. Algebra 68, 181-194 (1990; Zbl 0728.22006)]. Recently the second author of this paper and F.J. Trigos-Arrieta introduced the notion of totally suitable subsets of a topological group \(G\): a suitable subset \(S\) in \(G\) is totally suitable if the subgroup of \(G\) generated by \(S\) is totally dense in \(G\), i.e. its intersection with \(H\) is dense in \(H\) for every closed normal subgroup \(H\) of \(G\). They described countably compact Abelian groups having totally suitable sets and proved that all such groups are metrizable. They also posed the problem to describe locally compact Abelian groups that admit a totally suitable set. Let us note that according to a result by K. H. Hofmann and S. A. Morris from the mentioned paper every locally compact group has a suitable set. The paper under review contains the following results: (1) a characterization of locally compact Abelian groups with a totally suitable subset; (2) such groups need not be metrizable (recall that countably compact groups with totally suitable sets are metrizable); (3) the groups \(\mathbb Q_p\) of \(p\)-adic numbers are the only locally compact Abelian groups of density less than \(2^\omega\) in which each suitable set is totally suitable; (4) a locally compact Abelian group \(G\) has a totally suitable set iff the group \(G\) with the Bohr topology has such a set.