Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T15:48:36.328Z Has data issue: false hasContentIssue false
  • English
  • Français

More on Convergence of Continuous Functions and Topological Convergence of Sets

Published online by Cambridge University Press:  20 November 2018

Gerald Beer
Gerald Beer*
Affiliation:
Department of Mathematics, California State UniversityLos Angeles, CA (Visiting U.C. Davis)
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 C(X, Y) denote the set of continuous functions from a metric space X to a metric space Y. Viewing elements of C(X, Y) as closed subsets of X × Y, we say {fn} converges topologically to f if Li fn = Lsfn = f. If X is connected, then topological convergence in C(X,R) does not imply pointwise convergence, but if X is locally connected and Y is locally compact, then topological convergence in C(X, Y) is equivalent to uniform convergence on compact subsets of X. Pathological aspects of topological convergence for seemingly nice spaces are also presented, along with a positive Baire category result.

Keywords

40A3054C3554B20Topological convergence of setspointwise convergenceuniform convergence on compact subsetsequicontinuityKuratowski convergence of sets
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 28 , Issue 1 , 01 March 1985 , pp. 52 - 59
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Beer, G., Upper semicontinuous functions and the Stone approximation theorem, J. Approximation Theory 34(1982), pp. 1–11.Google Scholar
2. Beer, G., On uniform convergence of continuous functions and topological convergence of sets, Can. Math. Bull. 26(1983), pp. 418–424.Google Scholar
3. Kuratowski, K., Topology, Academic Press, New York, 1966.Google Scholar
4. Naimpally, S., Graph topology for function spaces, Trans. Amer. Math. Soc. 123 (1966), pp. 267–272.Google Scholar
5. Popov, V., Approximation of convex functions by algebraic polynomials in Hausdorff metric, Serdica 1 (1975), pp. 386–398.Google Scholar
6. Royden, H., Real Analysis, Macmillan, New York, 1968.Google Scholar
7. Sendov, B., Convergence of Vallée-Poussin sums in H ausdorff distance, C. R. Acad. Bulgare Sci. 26 (1973), pp. 1431–1432.Google Scholar
8. Sendov, B., and Penkov, B., Hausdorffsche metrik und approximationen, Numer. Math. 9 (1966), pp. 214–226.Google Scholar