Dense subsets of maximally almost periodic groups
Authors:
W. W. Comfort and Salvador GarcíaFerreira
Journal:
Proc. Amer. Math. Soc. 129 (2001), 593599
MSC (1991):
Primary 22A05, 54A05, 54H11
Published electronically:
July 27, 2000
MathSciNet review:
1707513
Fulltext PDF Free Access
Abstract 
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Abstract: A (discrete) group is said to be maximally almost periodic if the points of are distinguished by homomorphisms into compact Hausdorff groups. A Hausdorff topology on a group is totally bounded if whenever there is such that . For purposes of this abstract, a family with a totally bounded topological group is a strongly extraresolvable family if (a) , (b) each is dense in , and (c) distinct satisfy ; 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 be maximally almost periodic. Then there are a subgroup of and a family such that (i) is dense in every totally bounded group topology on ; (ii) the family is a strongly extraresolvable family for every totally bounded group topology on such that ; and (iii) admits a totally bounded group topology as in (ii). Remark. In certain cases, for example when is Abelian, one must in Theorem 2 choose . In certain other cases, for example when the largest totally bounded group topology on is compact, the choice 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 GarcíaFerreira
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/S000299390005557X
PII:
S 00029939(00)05557X
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 secondlisted 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
