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A (discrete) group G is said to be maximally almost periodic if the points of G are distinguished by homomorphisms into compact Hausdorff groups. A Hausdorff topology T on a group G is totally bounded if whenever ∅ 6= U ∈ T there is F ∈ [G] |G|, (b) each D ∈ D is dense in G, and (c) distinct D,E ∈ D satisfy |D ∩ E| < d(G); a totally bounded topological group with such a family is a strongly extraresolvable topological group. We give two theorems, the second generalizing the first. Theorem 1. Every infinite totally bounded group contains a dense strongly extraresolvable subgroup. Corollary. In its largest totally bounded group topology, every infinite Abelian group is strongly extraresolvable. Theorem 2. Let G be maximally almost periodic. Then there are a subgroup H of G and a family D ⊆ P(H) such that (i) H is dense in every totally bounded group topology on G; (ii) the family D is a strongly extraresolvable family for every totally bounded group topology T on H such that d(H, T ) = |H|; and (iii) H admits a totally bounded group topology T as in (ii). Remark. In certain cases, for example when G is Abelian, one must in Theorem 2 choose H = G. In certain other cases, for example when the largest totally bounded group topology on G is compact, the choice H = G is impossible.

Dense subsets of maximally almost periodic groups