un 2 00 5 Precompact abelian groups and topological annihilators ∗

For a compact Hausdorff abelian group K and its subgroup H ≤ K, one defines the g-closure g 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 } inˆK (the Pontryagin dual of K) that converges to 0 in the topology that H induces onˆK. 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.