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Infima of hyperspace topologies

Part of: Basic constructions

Published online by Cambridge University Press:  26 February 2010

C. Costantini ,
S. Levi  and
J. Pelant
C. Costantini
Affiliation:
Dipartimento di Matematica, Universitá di Milano, Via Saldini 50, 20133 Milano, Italy.
S. Levi
Affiliation:
Dipartimento di Matematica, Universitá di Milano, 20133 Milano, Italy.
J. Pelant
Affiliation:
Czech Academy of Sciences, Department of Mathematics, Žitná 25, 11567 Praha 1, Czech Republic.
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Abstract

We study infima of families of topologies on the hyperspace of a metrizable space. We prove that Kuratowski convergence is the infimum, in the lattice of convergences, of all Wijsman topologies and that the cocompact topology on a metric space which is complete for a metric d is the infimum of the upper Wijsman topologies arising from metrics that are uniformly equivalent to d.

MSC classification

Secondary: 54B20: Hyperspaces
Type
Research Article
Information
Mathematika , Volume 42 , Issue 1 , June 1995 , pp. 67 - 86
Copyright
Copyright © University College London 1995

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References

At.Atsuji, M.. Uniform continuity of continuous functions of metric spaces. Pacific J. Math., 8(1958), 11–16.CrossRefGoogle Scholar
Be 1.Beer, G.. Metric spaces with nice closed balls and distance functions for closed sets. Bull. Austral. Math. Soc, 35 (1987), 81–96.Google Scholar
Be 2.Beer, G.. Wijsman convergence of convex sets under renorming. Nonlinear Anal. To appear.Google Scholar
BLLN.Beer, G., Lechicki, A., Levi, S. and Naimpally, S.. Distance functional and suprema of hyperspace topologies. Ann. Mat. Pura Appl. (4), 162 (1992), 367–381.Google Scholar
Chr.Christensen, J. P. R.. Topology and Borel Structure. (North-Holland, Amsterdam, 1974).Google Scholar
CLZ.Costantini, C., Levi, S. and Zieminska, J.. Metrics that generate the same hyperspace convergence. Set-Valued Analysis, 1 (1993), 141–157.CrossRefGoogle Scholar
CV.Costantini, C. and Vitolo, P.. On the infimum of the Hausdorff metric topologies. Proc. London Math. Soc, 70 (1995), 414–480.Google Scholar
DG.Dolecki, S. and Greco, G.. Cyrtologies of convergences, I. Math. Nachr., 126 (1986), 327–348.CrossRefGoogle Scholar
DGL.Dolecki, S., Greco, G. and Lechicki, A.. When do the upper Kuratowski and co-compact topologies coincide. C.R. Acad. Sci. Paris, Ser I, 312 (1991), 923–926.Google Scholar
Du.Dugundji, J.. Topology (Allyn and Bacon, Boston 1966).Google Scholar
En.Engelking, R.. General Topology. Revised and completed edition (Heldermann Verlag, Berlin, 1989).Google Scholar
Fl.Flachsmeyer, J.. Verschiedene Topologisierungen im Raum der abgeschlossene Mengen. Math. Nach., 26 (1964), 321–337.Google Scholar
FLL.Francaviglia, S., A. Lechicki and S. Levi. Quasi-uniformization of hyperspaces and convergence of nets of semicontinuous multifunctions. J. Math. Anal. Appl., 112 (1985), 347–370.CrossRefGoogle Scholar
HL.Hola, L. and Lucchetti, R.. Comparison of hypertopologies. Set-Valued Analysis. To appear.Google Scholar
HP.Hille, E. and Phillips, R. S.. Functional Analysis and Semi-groups. American Mathematical Society Colloquium Publications (Vol. XXXI) (Providence, 1957).Google Scholar
Kl.Klee, V. L.. Some characterizations of compactness. Am. Math. Monthly, 58 (1951), 389–393.CrossRefGoogle Scholar
Ku.Kuratowski, K.. Topology, Vol. I (Academic Press, New York, 1966).Google Scholar
1.Lechicki, LL. A. and Levi, S.. Wijsman convergence in the hyperspace of a metric space. Boll. Un. Mat. Itat. (7), B.1 (1987), 439–451.Google Scholar
LLP.Levi, S., Lucchetti, R. and Pelant, J.. On the infimum of the Hausdorff and Vietoris topologies. Proc. Amer. Math. Soc, 118 (1993), 971–978.Google Scholar