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Dense subsets of maximally almost periodic groups. (English) Zbl 0955.22003

A topological group \((G,\mathcal{T})\) is said to be totally bounded if its topology is Hausdorff and, for every open subset \(U \in \mathcal{T}\), there is a finite subset \(F\) of \(G\) such that \(G=UF\).
Refining other related properties due to J. G. Ceder [Fundam. Math. 55, 87-93 (1964; Zbl 0139.40401)] and Malykhin, the authors of this paper propose and study the concept of strong extraresolvability of a topological space. A totally bounded group \((G,\mathcal{T})\) is strongly extraresolvable if there is a family \(\mathcal{D}\) of dense subsets of \((G,\mathcal{T})\) such that (a) \(|\mathcal{D} |> |G |\), (b) each \(D \in \mathcal{D}\) is dense in \((G,\mathcal{T})\) and (c) distinct \(D,E\in \mathcal{D}\) satisfy \(|D \cap E|<d(G)\), where \(d(G)\) denotes, as usual, the density character of \((G,\mathcal{T})\).
The main theorem of the paper proves that every totally bounded group \((G,\mathcal{T})\) satisfying the equality \(|G |=d(G)\) (i.e., containing no dense subset \(A\) with \(|A |< |G |\)) is strongly extraresolvable. It therefore follows that every totally bounded group has a dense extraresolvable subgroup.
Every group \(G\) admitting a totally bounded group topology (i.e., every group which, as a discrete group, is maximally almost periodic in the sense of von Neumann) must also admit a largest one. The authors apply the main theorem to this topology and show that every infinite maximally almost periodic group \(G\) has a subgroup \(H\) and a family \(\mathcal{D}\) of subsets of \(G\) such that: (i) \(H\) is dense in every totally bounded group topology on \(G\), (ii) the family \(\mathcal{D}\) makes \(H\) into a strongly extraresolvable group for every totally bounded topology \(\mathcal{T}\) on \(H\) for which the equality \(|H |=d(H,\mathcal{T})\) is satisfied, and (iii) \(H\) admits such a topology.
As the authors remark, the subgroup \(H\) obtained above must in some cases equal \(G\) (e.g., when \(G\) is Abelian) while in others the choice \(H=G\) is impossible (e.g., when \(G\) is a compact simple Lie group).

MSC:

22A05 Structure of general topological groups
54A05 Topological spaces and generalizations (closure spaces, etc.)
54H11 Topological groups (topological aspects)

Citations:

Zbl 0139.40401
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References:

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