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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
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]
Gerald
Beer, Topologies on closed and closed convex sets, Mathematics
and its Applications, vol. 268, Kluwer Academic Publishers Group,
Dordrecht, 1993. MR 1269778
(95k:49001)
- [Be2]
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
(92c:54009), http://dx.doi.org/10.1090/S0002-9939-1991-1065940-6
- [CLZ]
C.
Costantini, S.
Levi, and J.
Ziemińska, Metrics that generate the same hyperspace
convergence, Set-Valued Anal. 1 (1993), no. 2,
141–157. MR 1239401
(94h:54010), http://dx.doi.org/10.1007/BF01027689
- [Co]
C.
Costantini, Every Wijsman topology relative to a
Polish space is Polish, Proc. Amer. Math.
Soc. 123 (1995), no. 8, 2569–2574. MR 1273484
(95j:54012), http://dx.doi.org/10.1090/S0002-9939-1995-1273484-5
- [En]
Ryszard
Engelking, General topology, 2nd ed., Sigma Series in Pure
Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from
the Polish by the author. MR 1039321
(91c:54001)
- [He]
Christian
Hess, Loi de probabilité et indépendance des
ensembles aléatoires à valeurs fermées dans un espace
de Banach, Sém. Anal. Convexe 13 (1983), exp.
no. 7, 40 (French). MR 746116
(85m:60020)
- [Ke]
Alexander
S. Kechris, Classical descriptive set theory, Graduate Texts
in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597
(96e:03057)
- [Zs]
László
Zsilinszky, Baire spaces and hyperspace
topologies, Proc. Amer. Math. Soc.
124 (1996), no. 8,
2575–2584. MR 1343733
(96j:54017), http://dx.doi.org/10.1090/S0002-9939-96-03528-9
- [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:
http://dx.doi.org/10.1090/S0002-9939-98-04956-9
PII:
S 0002-9939(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
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