Precompact abelian groups and topological annihilators

  • Lukács G
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For a compact Hausdorff abelian group K and its subgroup H ≤ K, one defines the g-closuregK(H) of H in K as the subgroup consisting of χ ∈ K such that χ (an) {long rightwards arrow} 0 in T = R / Z for every sequence {an} in over(K, ̂) (the Pontryagin dual of K) that converges to 0 in the topology that H induces on over(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 Gδ-closure on compact Hausdorff abelian groups, and thus captures realcompactness and pseudocompactness of subgroups. © 2006 Elsevier Ltd. All rights reserved.

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  • Gábor Lukács

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