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

DOI:
https://doi.org/10.1090/S0002-9939-00-05557-X

Published electronically:
July 27, 2000

MathSciNet review:
1707513

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Abstract | References | Similar Articles | Additional Information

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 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:
https://doi.org/10.1090/S0002-9939-00-05557-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