Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 54B20, 54E35, 54D35

Retrieve articles in all journals with MSC (1991): 54B20, 54E35, 54D35


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

American Mathematical Society