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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Dense subsets of maximally almost periodic groups


Authors: W. W. Comfort and Salvador García-Ferreira
Journal: Proc. Amer. Math. Soc. 129 (2001), 593-599
MSC (1991): Primary 22A05, 54A05, 54H11
Published electronically: July 27, 2000
MathSciNet review: 1707513
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Abstract:

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.


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Additional Information

W. W. Comfort
Affiliation: Department of Mathematics, Wesleyan University, Middletown, Connecticut 06459
Email: wcomfort@wesleyan.edu

Salvador García-Ferreira
Affiliation: Instituto de Matemáticas, Ciudad Universitaria (UNAM), 04510 México D.F., México
Email: garcia@servidor.unam.mx, sgarcia@zeus.ccu.umich.mx

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05557-X
PII: S 0002-9939(00)05557-X
Received by editor(s): May 10, 1998
Received by editor(s) in revised form: April 23, 1999
Published electronically: July 27, 2000
Additional Notes: This work was written during the visit of the second-listed author to the Department of Mathematics of Wesleyan University, during the period September, 1997–March, 1998.
The second author acknowledges with thanks the generous hospitality and support received from the Department of Mathematics of Wesleyan University.
Communicated by: Alan Dow
Article copyright: © Copyright 2000 American Mathematical Society