Concerning connected, pseudocompact Abelian groups
To avoid repetition, all groups in the sequel are assumed to be completely regular topological abelian groups. It is known [\textit{W. W. Comfort} and \textit{T. Soundararajan}, Pac. J. Math. 100, 61-84 (1982; Zbl 0451.22002)] that for every infinite cardinal \(\alpha\), there is an \(\omega\)-bounded (i.e., every countable subset has compact closure) group G whose weight w(G) is \(\alpha\), and such that no proper dense subgroup of G is countably compact (i.e., every countable open cover has a finite subcover). The group G can be made connected or 0-dimensional, the constructions are similar. Clearly every \(\omega\)-bounded space is countably compact; weaker, but closely related, is pseudocompactness (i.e., every continuous real-valued function is bounded). So it is natural to conjecture that for each infinite \(\alpha\) there is a pseudocompact group G with \(w(G)=\alpha\), and such that no proper dense subgroup of G is pseudocompact. When \(\alpha =\omega\), every pseudocompact group of weight \(\alpha\) is metrizable, hence compact. Thus there can never be a proper dense pseudocompact subgroup. When \(\alpha >\omega\), however, the conjecture is quite dramatically false. In \textit{W. W. Comfort} and \textit{L. C. Roberson} [Diss. Math. 272, 1-42 (1988)] it is shown that every 0-dimensional pseudocompact group of uncountable weight has a proper dense pseudocompact subgroup. In the present paper, the connected case is investigated, but the picture is much less complete. If G is any connected pseudocompact group of uncountable weight \(\alpha\), then G has a proper dense pseudocompact subgroup if any of the following conditions holds: w(G)\(\leq c\); \(| G| \geq \alpha^{\omega}\); \(\alpha\) is a strong limit cardinal of uncountable cofinality; \(| tor(G)| >c\); or G is nondivisible.
scientific article; zbMATH DE number 4144907
scientific article