Dense subsets of maximally almost periodic groups

Abstract

A (discrete) group G G is said to be maximally almost periodic if the points of G G are distinguished by homomorphisms into compact Hausdorff groups. A Hausdorff topology T \mathcal {T} on a group G G is totally bounded if whenever ∅ ≠ U ∈ T \emptyset \neq U\in \mathcal {T} there is F ∈ [ G ] > ω F\in [G]^{>\omega } such that G = U F G=UF . For purposes of this abstract, a family D ⊆ P ( G ) \mathcal {D}\subseteq \mathcal {P}(G) with ( G , T ) (G,\mathcal {T}) a totally bounded topological group is a strongly extraresolvable family if (a)  | D | > | G | |\mathcal {D}|>|G| , (b) each D ∈ D D\in \mathcal {D} is dense in G G , and (c) distinct D , E ∈ D D,E\in \mathcal {D} satisfy | D ∩ E | > d ( G ) |D\cap 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 G be maximally almost periodic. Then there are a subgroup H H of G G and a family D ⊆ P ( H ) \mathcal {D}\subseteq \mathcal {P}(H) such that (i) H H is dense in every totally bounded group topology on G G ; (ii) the family D \mathcal {D} is a strongly extraresolvable family for every totally bounded group topology T \mathcal {T} on H H such that d ( H , T ) = | H | d(H,\mathcal {T})=|H| ; and (iii) H H admits a totally bounded group topology T \mathcal {T} as in (ii). Remark. In certain cases, for example when G G is Abelian, one must in Theorem 2 choose H = G H=G . In certain other cases, for example when the largest totally bounded group topology on G G is compact, the choice H = G H=G is impossible.