Baire spaces, Tychonoff powers and the Vietoris topology
HTML articles powered by AMS MathViewer
- by Jiling Cao and A. H. Tomita PDF
- Proc. Amer. Math. Soc. 135 (2007), 1565-1573 Request permission
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.References
- J. Cao, S. García-Ferreira and V. Gutev, Baire spaces and Vietoris hyperspaces, Proc. Amer. Math. Soc., 135 (2007), 299–303.
- J. Chaber and R. Pol, On hereditarily Baire spaces, $\sigma$-fragmentability of mappings and Namioka property, Topology Appl. 151 (2005), no. 1-3, 132–143. MR 2139747, DOI 10.1016/j.topol.2004.04.011
- W. G. Fleissner and K. Kunen, Barely Baire spaces, Fund. Math. 101 (1978), no. 3, 229–240. MR 521125, DOI 10.4064/fm-101-3-229-240
- David Fremlin, Tomasz Natkaniec, and Ireneusz Recław, Universally Kuratowski-Ulam spaces, Fund. Math. 165 (2000), no. 3, 239–247. MR 1805426, DOI 10.4064/fm-165-3-239-247
- M. R. Krom, Cartesian products of metric Baire spaces, Proc. Amer. Math. Soc. 42 (1974), 588–594. MR 334138, DOI 10.1090/S0002-9939-1974-0334138-9
- Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam, 1983. An introduction to independence proofs; Reprint of the 1980 original. MR 756630
- Robert A. McCoy, Baire spaces and hyperspaces, Pacific J. Math. 58 (1975), no. 1, 133–142. MR 410689, DOI 10.2140/pjm.1975.58.133
- Ernest Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182. MR 42109, DOI 10.1090/S0002-9947-1951-0042109-4
- John C. Oxtoby, The Banach-Mazur game and Banach category theorem, Contributions to the theory of games, vol. 3, Annals of Mathematics Studies, no. 39, Princeton University Press, Princeton, N.J., 1957, pp. 159–163. MR 0093741
- John C. Oxtoby, Cartesian products of Baire spaces, Fund. Math. 49 (1960/61), 157–166. MR 140638, DOI 10.4064/fm-49-2-157-166
- Jean Saint-Raymond, Jeux topologiques et espaces de Namioka, Proc. Amer. Math. Soc. 87 (1983), no. 3, 499–504 (French, with English summary). MR 684646, DOI 10.1090/S0002-9939-1983-0684646-1
- Rastislav Telgársky, Topological games: on the 50th anniversary of the Banach-Mazur game, Rocky Mountain J. Math. 17 (1987), no. 2, 227–276. MR 892457, DOI 10.1216/RMJ-1987-17-2-227
- László Zsilinszky, Products of Baire spaces revisited, Fund. Math. 183 (2004), no. 2, 115–121. MR 2127961, DOI 10.4064/fm183-2-3
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
- 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
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 1565-1573
- MSC (2000): Primary 54E52; Secondary 54B10, 54B20, 91A05
- DOI: https://doi.org/10.1090/S0002-9939-07-08855-7
- MathSciNet review: 2276668