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

 

Pseudocompact topological group refinements of maximal weight


Authors: W. W. Comfort and Jorge Galindo
Journal: Proc. Amer. Math. Soc. 131 (2003), 1311-1320
MSC (2000): Primary 22A05, 54H11
Published electronically: September 5, 2002
MathSciNet review: 1948125
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Abstract: It is known that a compact metrizable group admits no proper pseudocompact topological group refinement. The authors show, in contrast, that every (Hausdorff) pseudocompact Abelian group $G=(G,{\mathcal T})$ of uncountable weight $\alpha$, satisfying any of the following conditions, admits a pseudocompact group refinement of maximal weight (that is, of weight $2^{\vert G\vert}$):

(i)
$G$ is compact;
(ii)
$G$ is torsion-free with $\alpha\leq\vert G\vert=\vert G\vert^\omega$;
(iii)
[GCH] $G$ is torsion-free.

Remark. (i) answers a question posed by Comfort and Remus [Math. Zeit- schrift 215 (1994), 337-346].


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Additional Information

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

Jorge Galindo
Affiliation: Departamento de Matemáticas, Universitat Jaume I, 8029-AP Castellón, Spain
Email: jgalindo@mat.uji.es

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06650-9
PII: S 0002-9939(02)06650-9
Keywords: Topological group, pseudocompact, refinement topology, maximal weight
Received by editor(s): August 21, 2000
Received by editor(s) in revised form: December 4, 2001
Published electronically: September 5, 2002
Additional Notes: This paper is based on work completed during the visit of the second-listed author to the Department of Mathematics of Wesleyan University, during the Fall Term of the academic year 1998-1999
The work of the second author was supported in part by Spanish DGES, grant number BFM 2000-0913. The second author acknowledges with thanks hospitality and support received from the Department of Mathematics of Wesleyan University
Communicated by: Alan Dow
Article copyright: © Copyright 2002 American Mathematical Society