On the minimality of powers of minimal 𝜔-bounded abelian groups

Abstract

Abstract. We describe the structure of totally disconnected minimal ω-bounded abelian groups by reducing the description to the case of those of them which are subgroups of powers of the p-adic integers Zp. In this case the description is obtained by means of a functorial correspondence, based on Pontryagin duality, between topological and linearly topologized groups introduced by Tonolo. As an application we answer the question (posed in Pseudocompact and countably compact abelian groups: Cartesian products and minimality, Trans. Amer. Math. Soc. 335 (1993), 775-790) when arbitrary powers of minimal ω-bounded abelian groups are minimal. We prove that the positive answer to this question is equivalent to non-existence of measurable cardinals.

Dense minimal pseudocompact subgroups of compact Abelian groups

“…On the other hand there exist compact abelian groups with proper dense essential countably compact subgroups [14,18]. Moreover the problem of the existence of connected compact abelian groups with proper dense essential countably compact subgroups is not decidable in ZFC, yet it is equivalent to that of the existence of measurable cardinals -see [17] and [9,Theorem 5.7].…”