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Baire spaces and Vietoris hyperspaces

Authors: Jiling Cao, Salvador García-Ferreira and Valentin Gutev
Journal: Proc. Amer. Math. Soc. 135 (2007), 299-303
MSC (2000): Primary 54E52; Secondary 26A21, 46A30, 54B10, 54B20
Published electronically: August 16, 2006
Erratum: Proc. Amer. Math. Soc. 136 (2008), 3729-3731
MathSciNet review: 2280197
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Abstract: We prove that if the Vietoris hyperspace $ CL(X)$ of all nonempty closed subsets of a space $ X$ is Baire, then all finite powers of $ X$ must be Baire spaces. In particular, there exists a metrizable Baire space $ X$ whose Vietoris hyperspace $ CL(X)$ is not Baire. This settles an open problem of R. A. McCoy stated in 1975.

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Additional Information

Jiling Cao
Affiliation: Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1, New Zealand
Address at time of publication: School of Mathematical Sciences, Auckland University of Technology, Private Bag 92006, Auckland 1020, New Zealand

Salvador García-Ferreira
Affiliation: Instituto de Matematicas (UNAM), Apartado Postal 61-3, Xangari 58089, Morelia, Michoacan, Mexico

Valentin Gutev
Affiliation: School of Mathematical Sciences, Faculty of Science, University of KwaZulu-Natal, King George V Avenue, Durban 4041, South Africa

Keywords: Baire space, product space, hyperspace, Vietoris topology, Volterra space
Received by editor(s): September 2, 2004
Published electronically: August 16, 2006
Additional Notes: The first author’s research was supported by the Foundation for Research, Science and Technology of New Zealand under project number UOAX0240.
The third author’s research was partially supported by the National Research Foundation of South Africa under grant number 2053735.
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2006 American Mathematical Society