Dense subsets of maximally almost periodic groups

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 $\mathcal{T}$ on a group $G$ is totally bounded if whenever $\emptyset\neq U\in\mathcal{T}$ there is $F\in[G]^{<\omega}$ such that $G=UF$. For purposes of this abstract, a family $\mathcal{D}\subseteq\mathcal{P}(G)$with $(G,\mathcal{T})$ a totally bounded topological group is a strongly extraresolvable family if (a)  $\vert\mathcal{D}\vert>\vert G\vert$, (b) each $D\in\mathcal{D}$ is dense in $G$, and (c) distinct $D,E\in\mathcal{D}$ satisfy $\vert D\cap E\vert<d(G)$; 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 $\mathcal{D}\subseteq\mathcal{P}(H)$ such that

(i) $H$ is dense in every totally bounded group topology on $G$;

(ii) the family $\mathcal{D}$ is a strongly extraresolvable family for every totally bounded group topology $\mathcal{T}$ on $H$ such that $d(H,\mathcal{T})=\vert H\vert$; and

(iii) $H$ admits a totally bounded group topology $\mathcal{T}$ 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.