On the hyperspace of a non-separable
metric space
Author:
Camillo Costantini
Journal:
Proc. Amer. Math. Soc. 126 (1998), 3393-3396
MSC (1991):
Primary 54B20; Secondary 54E35, 54D35
DOI:
https://doi.org/10.1090/S0002-9939-98-04956-9
MathSciNet review:
1618729
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Abstract: We prove the following two results. 1) There exists a non-separable complete metric space whose Wijsman hypertopology is not Cech-complete. 2) There exist a non-separable metrizable space and two compatible metrics on it, such that the collections of the Borel sets generated by the relative Wijsman hypertopologies do not coincide.
- [Be1] G. Beer, Topologies on closed and closed convex sets, Kluwer Academic Publishers, Dordrecht, 1993. MR 95k:49001
- [Be2] G. Beer, A Polish topology for the closed subsets of a Polish space, Proc. Amer. Math. Soc. 113 (1991), 1123-1133. MR 92c:54009
- [CLZ] C. Costantini, S. Levi and J. Zieminska, Metrics that generate the same hyperspace convergence, Set-Valued Anal. 1 (1993), 141-157. MR 94h:54010
- [Co] C. Costantini, Every Wijsman topology relative to a Polish space is Polish, Proc. Amer. Math. Soc. 123 (1995), 2569-2574. MR 95j:54012
- [En] R. Engelking, General Topology, revised and completed edition, Heldermann Verlag, Berlin, 1989. MR 91c:54001
- [He] C. Hess, Loi de la probabilité et indépendence des ensembles aléatoires à valeurs fermées dans un espace de Banach, Séminaire d'Analyse Convexe de l'Université de Montpellier, Exposé 7, 1983. MR 85m:60020
- [Ke] A. S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, New York, 1995. MR 96e:03057
- [Zs] L. Zsilinszky, Baire spaces and hyperspace topologies, Proc. Amer. Math. Soc. 124 (1996), 2575-2584. MR 96j:54017
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Additional Information
Camillo Costantini
Affiliation:
Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy
Email:
costantini@dm.unito.it
DOI:
https://doi.org/10.1090/S0002-9939-98-04956-9
Keywords:
Non-separable metric space,
Wijsman hypertopology,
\v{C}ech-complete\-ness,
Borel sets
Received by editor(s):
January 17, 1996
Received by editor(s) in revised form:
November 18, 1996
Communicated by:
Franklin D. Tall
Article copyright:
© Copyright 1998
American Mathematical Society
