Amsterdam properties of Wijsman hyperspaces
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- by Jiling Cao and Heikki J. K. Junnila PDF
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Abstract:
In this paper we show the following results: (i) there exists a separable metric space of the first category whose Wijsman hyperspace is almost countably subcompact; (ii) there exists a $\sigma$-discrete crowded metric space whose Wijsman hyperspace is countably base-compact. Neither of these can occur with Vietoris hyperspaces.References
- J. M. Aarts and D. J. Lutzer, Pseudo-completeness and the product of Baire spaces, Pacific J. Math. 48 (1973), 1–10. MR 326666
- J. M. Aarts and D. J. Lutzer, Completeness properties designed for recognizing Baire spaces, Dissertationes Math. (Rozprawy Mat.) 116 (1974), 48. MR 380745
- Gerald Beer, A Polish topology for the closed subsets of a Polish space, Proc. Amer. Math. Soc. 113 (1991), no. 4, 1123–1133. MR 1065940, DOI 10.1090/S0002-9939-1991-1065940-6
- Gerald Beer, Topologies on closed and closed convex sets, Mathematics and its Applications, vol. 268, Kluwer Academic Publishers Group, Dordrecht, 1993. MR 1269778, DOI 10.1007/978-94-015-8149-3
- J. Cao and A. H. Tomita, The Wijsman hyperspace of a metric hereditarily Baire space is Baire, Topology Appl., to appear.
- J. Chaber and R. Pol, Note on the Wijsman hyperspaces of completely metrizable spaces, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 5 (2002), no. 3, 827–832 (English, with English and Italian summaries). MR 1934383
- C. Costantini, Every Wijsman topology relative to a Polish space is Polish, Proc. Amer. Math. Soc. 123 (1995), no. 8, 2569–2574. MR 1273484, DOI 10.1090/S0002-9939-1995-1273484-5
- Camillo Costantini, On the hyperspace of a non-separable metric space, Proc. Amer. Math. Soc. 126 (1998), no. 11, 3393–3396. MR 1618729, DOI 10.1090/S0002-9939-98-04956-9
- Edward G. Effros, Convergence of closed subsets in a topological space, Proc. Amer. Math. Soc. 16 (1965), 929–931. MR 181983, DOI 10.1090/S0002-9939-1965-0181983-3
- 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
- Zdeněk Frolík, Remarks concerning the invariance of Baire spaces under mappings, Czechoslovak Math. J. 11(86) (1961), 381–385 (English, with Russian summary). MR 133098
- J. de Groot, Subcompactness and the Baire category theorem, Nederl. Akad. Wetensch. Proc. Ser. A 66=Indag. Math. 25 (1963), 761–767. MR 0159303
- Yoshito Ikeda, Čech-completeness and countably subcompactness, Topology Proc. 14 (1989), no. 1, 75–87. MR 1081121
- A. Lechicki and S. Levi, Wijsman convergence in the hyperspace of a metric space, Boll. Un. Mat. Ital. B (7) 1 (1987), no. 2, 439–451 (English, with Italian summary). MR 896334
- D. J. Lutzer, J. van Mill, and V. V. Tkachuk, Amsterdam properties of $C_p(X)$ imply discreteness of $X$, Canad. Math. Bull. 51 (2008), no. 4, 570–578. MR 2462461, DOI 10.4153/CMB-2008-056-9
- Robert A. McCoy, Baire spaces and hyperspaces, Pacific J. Math. 58 (1975), no. 1, 133–142. MR 410689
- 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
- R. A. Wijsman, Convergence of sequences of convex sets, cones and functions. II, Trans. Amer. Math. Soc. 123 (1966), 32–45. MR 196599, DOI 10.1090/S0002-9947-1966-0196599-8
- László Zsilinszky, Baire spaces and hyperspace topologies, Proc. Amer. Math. Soc. 124 (1996), no. 8, 2575–2584. MR 1343733, DOI 10.1090/S0002-9939-96-03528-9
- László Zsilinszky, Polishness of the Wijsman topology revisited, Proc. Amer. Math. Soc. 126 (1998), no. 12, 3763–3765. MR 1458275, DOI 10.1090/S0002-9939-98-04526-2
- L. Zsilinszky, On Baireness of the Wijsman hyperspace, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 10 (2007), 1071-1079.
Additional Information
- Jiling Cao
- Affiliation: School of Computing and Mathematical Sciences, Auckland University of Technology, Private Bag 92006, Auckland 1142, New Zealand
- Email: jiling.cao@aut.ac.nz
- Heikki J. K. Junnila
- Affiliation: Department of Mathematics and Statistics, The University of Helsinki, P. O. Box 68, FI-00014, Helsinki, Finland
- Email: heikki.junnila@helsinki.fi
- Received by editor(s): February 3, 2009
- Received by editor(s) in revised form: June 8, 2009
- Published electronically: September 9, 2009
- Additional Notes: Research for this paper was partially conducted during the first author’s visit to the University of Helsinki in July 2008. He would like to acknowledge financial support from the Magnus Ehrnrooth Foundation, administered by the Finnish Society of Sciences and Letters, and he also thanks the Department of Mathematics and Statistics for hospitability.
The second author’s research was partially supported by Natural Science Foundation of China grant 10671173. - Communicated by: Alexander Dranishnikov
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 769-776
- MSC (2000): Primary 54E52; Secondary 54B10, 54B20
- DOI: https://doi.org/10.1090/S0002-9939-09-10072-2
- MathSciNet review: 2557194