ar X iv : m at h / 05 02 22 3 v 1 [ m at h . G N ] 1 0 Fe b 20 05 Notes on the g - closure Gábor Lukács

In this note, we prove that H is a g-closed subgroup of the compact group K if and only if every sequentially continuous character of the dual of H is continuous-in other words, if the dual of H is a so-called sk-group. We conclude that every countable subgroup of a compact group is g-closed, and thus give a positive answer to a problem of Dikranjan, Milan and Tonolo. We also show that the relationship between precompact sk-groups and precompact groups is similar to that between (Hausdorff) k-spaces and (Hausdorff) topological spaces. A topological group G is precompact if for every neighborhood U of the identity in G, there exists a finite subset F ⊂ G such that G = F U. Throughout this note, K is a fixed abelian compact Hausdorff topological group, and A = ˆ K is its Pontryagin dual. It is a well-known result of Comfort and Ross [6] that every precompact Hausdorff group topology τ on A coincides with the initial topology on A with respect to characters χ in a dense subgroup H ≤ K. In other words, one has τ = τ H , where τ H is the coarsest (group) topology on A such that each χ : A → T in H is continuous. In fact, H ≤ K separates the points of A if and only if H is dense in K, and H = (A, τ) (cf. [6]).