Pseudocompact totally dense subgroups

Dikranjan, Dikran; Bruno, Anna

doi:10.1090/S0002-9939-07-09099-5

Pseudocompact totally dense subgroups
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by Dikran Dikranjan and Anna Giordano Bruno PDF

Proc. Amer. Math. Soc. 136 (2008), 1093-1103 Request permission

Abstract:

It was shown by Dikranjan and Shakhmatov in 1992 that if a compact abelian group $K$ admits a proper totally dense pseudocompact subgroup, then $K$ cannot have a torsion closed $G_\delta$-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 $H$ of $K$ that contains an $\omega$-bounded dense subgroup of $K$ (such an $H$ is necessarily pseudocompact). This answers two questions posed by Dikranjan and Shakhmatov (Proc. Amer. Math. Soc. 114 (1992), 1119–1129).

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