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

 

Baire spaces, Tychonoff powers and the Vietoris topology


Authors: Jiling Cao and A. H. Tomita
Journal: Proc. Amer. Math. Soc. 135 (2007), 1565-1573
MSC (2000): Primary 54E52; Secondary 54B10, 54B20, 91A05
Published electronically: January 4, 2007
MathSciNet review: 2276668
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Abstract: In this paper, we show that if the Tychonoff power $ X^\omega$ of a quasi-regular space $ X$ is Baire, then its Vietoris hyperspace $ 2^X$ is also Baire. We also provide two examples to show (i) the converse of this result does not hold in general, and (ii) the Baireness of finite powers of a space is insufficient to guarantee the Baireness of its hyperspace.


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Additional Information

Jiling Cao
Affiliation: School of Mathematical Sciences, Auckland University of Technology, Private Bag 92006, Auckland 1020, New Zealand
Email: jiling.cao@aut.ac.nz

A. H. Tomita
Affiliation: Departamento de Matemática, Institute of Mathematics and Statistics, University of São Paulo, 05315 São Paulo, Brazil
Email: tomita@ime.usp.br

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08855-7
PII: S 0002-9939(07)08855-7
Keywords: Baire space, Choquet game, Tychonoff power, Vietoris topology.
Received by editor(s): February 13, 2006
Published electronically: January 4, 2007
Additional Notes: The first-named author acknowledges the support from a Marsden Fund grant, UOA0422, administrated by the Royal Society of New Zealand. The second-named author wishes to thank the financial support from the Foundation of Research, Science and Technology of New Zealand and the warm hospitality of his host during his visit to the University of Auckland in July 2005.
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.