Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-19T23:46:37.614Z Has data issue: false hasContentIssue false
  • English
  • Français

Metric spaces with nice closed balls and distance functions for closed sets

Published online by Cambridge University Press:  17 April 2009

Gerald Beer
Gerald Beer
Affiliation:
Department of Mathematics, California State University, Los Angeles, Los Angeles, California 90032, United States of America.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A metric space 〈X,d〉 is said to have nice closed balls if each closed ball in X is either compact or the entire space. This class of spaces includes the metric spaces in which closed and bounded sets are compact and those for which the distance function is the zero-one metric. We show that these are the spaces in which the relation F = Lim Fn for sequences of closed sets is equivalent to the pointwise convergence of 〈d (.,Fn)〉 to d (.,F). We also reconcile these modes of convergence with three other closely related ones.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 35 , Issue 1 , February 1987 , pp. 81 - 96
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Atsuji, M., “Uniform continuity of continuous functions of metric spaces”, Pacific J. Math. 8 (1958), 11–16.CrossRefGoogle Scholar
[2]Aubin, J.P., Applied abstract analysis, (Wiley, New York, 1977).Google Scholar
[3]Bear, G., “On convergence of closed sets in a metric space and distance functions”, Bull. Australian Math. Soc. 31 (1985), 421–432.CrossRefGoogle Scholar
[4]Beer, G., “Metric spaces on which continuous functions are uniformaly continuous and Hausdorff distance”, Proc. Amer. Math. Soc. 95 (1985), 653–658.CrossRefGoogle Scholar
[5]Berge, C., Topological spaces, (Oliver and Boyd, Edinburgh, 1963.).Google Scholar
[6]Castaing, C. and Valadier, M., Convex analysis and measurable multifunctions, (Springer-Verlag, Berlin, 1977).CrossRefGoogle Scholar
[7]Deutsch, F. and Lambert, J., “On continuity of metric projections”, J. Approx. Theory 29 (1980), 116–131.CrossRefGoogle Scholar
[8]Effros, E., “Convergence of closed subsets in a topological space”, Proc. Amer. Math. Soc. 16 (1965), 929–931.CrossRefGoogle Scholar
[9]Francaviglia, S., Lechicki, A. and Levi, S., “Quasi-uniformization of hyperspaces and convergence of nets of semi-continuous multifunctions”, J. Math. Anal. Appl. 112 (1985), 347–370.CrossRefGoogle Scholar
[10]Hildenbrand, W., Core and equilibria of a large economy, (Princeton University Press, Princeton, N.J. 1974).Google Scholar
[11]Kelley, J., General topology, (Van Nostrand, Princeton, N.J. 1955).Google Scholar
[12]Klein, E. and Thompson, A., Theory of correspondences, (Wiley, New York, 1984).Google Scholar
[13]Kuratowski, K., Topology, (Academic Press, New York, 1966).Google Scholar
[14]Lechicki, A. and Levi, S., “Wijsman convergence in the hyperspace of a metric space”, (to appear, Boll, Un. Mat. Ital.).Google Scholar
[15]Michael, E., “Topologies on spaces of subsets”, Trans. Amer. Math. Soc. 71 (1951), 152–182.CrossRefGoogle Scholar
[16]Mrowka, S., ”On the convergence of nets of sets”, Fund. Math. 45 (1958) 237–246.CrossRefGoogle Scholar
[17]Ramsey, F., “On a problem of formal logic”, Proc. London Math. Soc. 30 (1930), 264–286.CrossRefGoogle Scholar
[18]Salinetti, G. and Wets, R., “Weak convergence of probability measures revisited”, (in preparation).Google Scholar
[19]Waterhouse, W., “On UC spaces”, Amer. Math. Monthly 72 (1965), 634–635.CrossRefGoogle Scholar