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

Abstract

We describe the structure of totally disconnected minimal ω \omega - bounded abelian groups by reducing the description to the case of those of them which are subgroups of powers of the p p -adic integers Z p \mathbb {Z}_{p} . 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 \omega -bounded abelian groups are minimal. We prove that the positive answer to this question is equivalent to non-existence of measurable cardinals.