Precompact abelian groups and topological annihilators

For a compact Hausdorff abelian group K and its subgroup H≤K, one defines the g-closureg_K(H) of H in K as the subgroup consisting of χ∈K such that χ(a_n)?0 in T=R/Z for every sequence {a_n} in (the Pontryagin dual of K) that converges to 0 in the topology that H induces on . 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_δ-closure on compact Hausdorff abelian groups, and thus captures realcompactness and pseudocompactness of subgroups.