Precompact abelian groups and topological annihilators

For a compact Hausdorff abelian group KK and its subgroup H≤KH≤K, one defines the gg-closure  gK(H)gK(H) of HH in KK as the subgroup consisting of χ∈Kχ∈K such that χ(an)⟶0χ(an)⟶0 in T=R/ZT=R/Z for every sequence {an}{an} in Kˆ (the Pontryagin dual of KK) that converges to 0 in the topology that HH induces on Kˆ. We prove that every countable subgroup of a compact Hausdorff group is gg-closed, and thus give a positive answer to two problems of Dikranjan, Milan and Tonolo. We also show that every gg-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 gg that coincides with the GδGδ-closure on compact Hausdorff abelian groups, and thus captures realcompactness and pseudocompactness of subgroups.