Precompact abelian groups and topological annihilators

Abstract

For a compact Hausdorff abelian group K and its subgroup H, one defines the g-closure g(H) of H in K as the subgroup consisting of $\chi \in K$ such that $\chi (a_n)⟶0$ in T=R/Z for every sequence {a_n} in $\hat{K}$ (the Pontryagin dual of K) that converges to 0 in the topology that H induces on $\hat{K}$. We prove that every countable subgroup of a compact Hausdorff group is g-closed, and thus give a positive answer to two problems of Dikranjan, Milan and Tonolo. We also show that every g-closed subgroup of a compact Hausdorff group is realcompact. The techniques developed in the paper are used to construct a close relative of the closure operator g that coincides with the $G_{\delta }$-closure on compact Hausdorff abelian groups, and thus captures realcompactness and pseudocompactness of subgroups.

Version 1.0 - submitted