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Infima of hyperspace topologies

  • C. Costantini (a1), S. Levi (a2) and J. Pelant (a3)


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.



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At.Atsuji, M.. Uniform continuity of continuous functions of metric spaces. Pacific J. Math., 8(1958), 1116.
Be 1.Beer, G.. Metric spaces with nice closed balls and distance functions for closed sets. Bull. Austral. Math. Soc, 35 (1987), 8196.
Be 2.Beer, G.. Wijsman convergence of convex sets under renorming. Nonlinear Anal. To appear.
BLLN.Beer, G., Lechicki, A., Levi, S. and Naimpally, S.. Distance functional and suprema of hyperspace topologies. Ann. Mat. Pura Appl. (4), 162 (1992), 367381.
Chr.Christensen, J. P. R.. Topology and Borel Structure. (North-Holland, Amsterdam, 1974).
CLZ.Costantini, C., Levi, S. and Zieminska, J.. Metrics that generate the same hyperspace convergence. Set-Valued Analysis, 1 (1993), 141157.
CV.Costantini, C. and Vitolo, P.. On the infimum of the Hausdorff metric topologies. Proc. London Math. Soc, 70 (1995), 414480.
DG.Dolecki, S. and Greco, G.. Cyrtologies of convergences, I. Math. Nachr., 126 (1986), 327348.
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), 923926.
Du.Dugundji, J.. Topology (Allyn and Bacon, Boston 1966).
En.Engelking, R.. General Topology. Revised and completed edition (Heldermann Verlag, Berlin, 1989).
Fl.Flachsmeyer, J.. Verschiedene Topologisierungen im Raum der abgeschlossene Mengen. Math. Nach., 26 (1964), 321337.
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), 347370.
HL.Hola, L. and Lucchetti, R.. Comparison of hypertopologies. Set-Valued Analysis. To appear.
HP.Hille, E. and Phillips, R. S.. Functional Analysis and Semi-groups. American Mathematical Society Colloquium Publications (Vol. XXXI) (Providence, 1957).
Kl.Klee, V. L.. Some characterizations of compactness. Am. Math. Monthly, 58 (1951), 389393.
Ku.Kuratowski, K.. Topology, Vol. I (Academic Press, New York, 1966).
1.Lechicki, LL. A. and Levi, S.. Wijsman convergence in the hyperspace of a metric space. Boll. Un. Mat. Itat. (7), B.1 (1987), 439451.
LLP.Levi, S., Lucchetti, R. and Pelant, J.. On the infimum of the Hausdorff and Vietoris topologies. Proc. Amer. Math. Soc, 118 (1993), 971978.
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  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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