Abstract :
For a compact Hausdorff abelian group K and its subgroup H≤K, one defines the image-closure image of H in K as the subgroup consisting of χset membership, variantK such that χ(an)long right arrow0 in image for every sequence {an} in image (the Pontryagin dual of K) that converges to 0 in the topology that H induces on image. We prove that every countable subgroup of a compact Hausdorff group is image-closed, and thus give a positive answer to two problems of Dikranjan, Milan and Tonolo. We also show that every image-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 operatorimage that coincides with the Gδ-closure on compact Hausdorff abelian groups, and thus captures realcompactness and pseudocompactness of subgroups.
