Some topological groups with, and some without, proper dense subgroups (Q1183628)

\textit{M. Rajagopalan} and \textit{H. Subrahmanian} [Colloq. Math. 35, 289-292 (1976; Zbl 0331.22005)] constructed an example of a locally compact Abelian (LCA) group containing no proper dense subgroups. Subsequently \textit{M. I. Kabenyuk} [Sib. Mat. Zh. 21, 202-203 (1980; Zbl 0453.22003)] characterized all nondiscrete LCA groups having no proper dense subgroups. In this interesting paper, the authors continue earlier investigations into the question of the existence of a proper dense subgroup of a given topological group [the authors, Topology Appl. 33, 21-45 (1989; Zbl 0698.54003)]. It is shown that every infinite connected Abelian group, resp. every infinite pseudo-compact group contains a proper dense subgroup. Examples of totally bounded Abelian groups are given which possess (do not possess) a proper dense subgroup. The authors mention four unsolved problems. One of them is open for several years: Does every pseudocompact Abelian group of uncountable weight have a proper dense pseudocompact subgroup?