Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-22T04:25:45.716Z Has data issue: false hasContentIssue false
  • English
  • Français

Topological Completeness of Function Spaces Arising in the Hausdorff Approximation of Functions

Published online by Cambridge University Press:  20 November 2018

Gerald Beer
Gerald Beer*
Affiliation:
Department of Mathematics California State University, Los Angeles Los Angeles, California 90032
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.

Let X be a complete metric space. Viewing continuous real functions on X as closed subsets of X × R, equipped with Hausdorff distance, we show that C(X, R) is completely metrizable provided X is complete and sigma compact. Following the Bulgarian school of constructive approximation theory, a bounded discontinuous function may be identified with its completed graph, the set of points between the upper and lower envelopes of the function. We show that the space of completed graphs, too, is completely metrizable, provided X is locally connected as well as sigma compact and complete. In the process, when X is a Polish space, we provide a simple answer to the following foundational question: which subsets of X × R arise as completed graphs?

Keywords

54C3554B2054E50function spaceHausdorff metriccompleted graphtopological completenesscomplete metric space
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 35 , Issue 4 , 01 December 1992 , pp. 439 - 448
Copyright
Copyright © Canadian Mathematical Society 1992

References

[At] Atsuji, M., Uniform continuity of continuous functions of metric spaces, Pacific J. Math. 8(1958), 11–16.Google Scholar
[Au] Aubin, J. P., Applied abstract analysis, Wiley, New York, 1977.Google Scholar
[Be1] Beer, G. A., Approximate selections for upper semicontinuous convex valued multifunctions, J. Approximation Theory 39(1983), 172–184.Google Scholar
[Be2] Beer, G. A., The approximation of real functions in the Hausdorjf metric, Houston J. Math. 10(1984), 325– 338.Google Scholar
[Be3] Beer, G. A., Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance, Proc. Amer. Math. Soc. 95(1985), 653–658.Google Scholar
[Be4] Beer, G. A., More on convergence of continuous functions and topological convergence ofsets, Canad. Math. Bull. 28(1985), 52–59.Google Scholar
[Be5] Beer, G. A., More about metric spaces on which continuous functions are uniformly continuous, Bull. Australian Math. Soc. 38(1986), 397–406.Google Scholar
[Be6] Beer, G. A., Complete subsets ofC(X, Y) with respect to Hausdorff distance, Math. Balkanica, 2 (new series), (1988), 78-84.Google Scholar
[CV] Castaing, C. and Valadier, M., Convex analysis and measurable multifunctions, Lecture notes in mathematics 580, Springer-Verlag, Berlin, (1977).Google Scholar
[En] Engelking, R., General topology, PWN, Warsaw, 1977.Google Scholar
[Ha] Hausdorff, F, Set Theory, Chelsea, New York, 1962.Google Scholar
[Ho] Hola, L., Hausdorff metric convergence of continuous functions, in Proc. Sixth Prague Topological Symposium 1986, Heldermann Verlag, Berlin, 1988,263–271.Google Scholar
[HN] Hola, L. and Neubrunn, T., On almost uniform convergence and convergence in Hausdorff metric, Radovi Mat. 4(1988), 193–205.Google Scholar
[KT] Klein, E. and Thompson, A., Theory of correspondences, Wiley, New York, 1984.Google Scholar
[MH] McCoy, R. and Hola, L., The Fell topology on C(X), Proc. 1990 Sum. Conf. on Gen. Top. Appl., New York Acad. Sci., to appear.Google Scholar
[Na] Naimpally, S., Graph topology for function spaces, Trans. Amer. Math. Soc. 123(1966), 267–272.Google Scholar
[Po] Poppe, H., Einige bemurkungen iiber den raum der abgeschlossenen mengen, Fund. Math. 59(1966), 159–169.Google Scholar
[Ra] Rainwater, J., Spaces whose finest uniformity is metric, Pacific J. Math. 9(1959), 567–570.Google Scholar
[RZ] Revalski, J. and Zhivkov, N., Well-posed optimization problems in metric spaces, J. Opt. Theory Appl., to appear.Google Scholar
[Se] Sendov, BI., Hausdorffapproximations, Bulgarian Academy of Sciences, Sofia, 1979 (in Russian); English version published by Kluwer, Dordrecht, Holland, 1990.Google Scholar
[To] loader, Gh., On a problem of Nagata, Mathematica (Cluj) 20 (43) (1978), 78–79.Google Scholar
[Wa] Waterhouse, W., On UC spaces, Amer. Math. Monthly, 72(1965), 634–635.Google Scholar