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

 

A compact group which is not Valdivia compact


Authors: Wieslaw Kubis and Vladimir Uspenskij
Journal: Proc. Amer. Math. Soc. 133 (2005), 2483-2487
MSC (2000): Primary 54D30; Secondary 54C15, 22C05
Published electronically: February 25, 2005
MathSciNet review: 2138892
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Abstract: A compact space $K$ is Valdivia compact if it can be embedded in a Tikhonov cube $I^A$ in such a way that the intersection $K\cap\Sigma$ is dense in $K$, where $\Sigma$ is the sigma-product ($=$ the set of points with countably many non-zero coordinates). We show that there exists a compact connected Abelian group of weight $\omega_1$ which is not Valdivia compact, and deduce that Valdivia compact spaces are not preserved by open maps.


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Additional Information

Wieslaw Kubis
Affiliation: Institute of Mathematics, University of Silesia, Bankowa 14, 40-007 Katowice, Poland
Email: kubis@ux2.math.us.edu.pl

Vladimir Uspenskij
Affiliation: Department of Mathematics, 321 Morton Hall, Ohio University, Athens, Ohio 45701
Email: uspensk@math.ohiou.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-05-07797-X
PII: S 0002-9939(05)07797-X
Keywords: Valdivia compact space, open map, retract, indecomposable group
Received by editor(s): November 1, 2003
Received by editor(s) in revised form: April 11, 2004
Published electronically: February 25, 2005
Communicated by: Alan Dow
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.