Pseudocompact totally dense subgroups

Abstract

It was shown by Dikranjan and Shakhmatov in 1992 that if a compact abelian group K K admits a proper totally dense pseudocompact subgroup, then K K cannot have a torsion closed G δ G_\delta -subgroup; moreover this condition was shown to be also sufficient under LH. We prove in ZFC that this condition actually ensures the existence of a proper totally dense subgroup H H of K K that contains an ω \omega -bounded dense subgroup of K K (such an H H is necessarily pseudocompact). This answers two questions posed by Dikranjan and Shakhmatov (Proc. Amer. Math. Soc. 114 (1992), 1119–1129).