Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-24T16:45:35.196Z Has data issue: false hasContentIssue false
  • English
  • Français

Nearness Convergence

Published online by Cambridge University Press:  20 November 2018

S. A. Naimpally  and
Mohan L. Tikoo
S. A. Naimpally
Affiliation:
Department of Mathematics Kuwait University P.O. Box 5969 SAFAT 130GO KUWAIT
Mohan L. Tikoo
Affiliation:
Department of Mathematic Southeast Missouri State University Cape Girardeau, Missouri 63701 U.S.A.
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.

In this paper, a concept of nearness convergence is introduced which contains the proximal convergence of Leader as a special case. It is proved that uniform convergence and this nearness convergence are equivalent on totally bounded uniform nearness spaces. One of the features of this convergence is that it lies between uniform convergence and pointwise convergence, and implies uniform convergence on compacta. Some other weaker notions of nearness convergence which are sufficient to preserve nearness maps are also discussed.

Keywords

54D2554E17Convergencenearnessuniformtotally boundedUniform convergenceproximal convergenceDini convergenceArzelà convergenceLeader convergenceUniform convergence on compacta
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 33 , Issue 3 , 01 September 1990 , pp. 268 - 272
Copyright
Copyright © Canadian Mathematical Society 1990

References

1. Bartle, R. G., On compactness in Functional Analysis, Trans. Amer. Math. Soc. 79 (1955), 35–77.Google Scholar
2. Di, A. Concilio and S. A. Naimpally, Proximal Convergence, Monatshefte Für Math. 103 (1987), 93–102.Google Scholar
3. Dunford, N. and Schwartz, J. T., Linear operators part I, Interscience, New York, 1958.Google Scholar
4. Herrlich, H., A Concept of nearness, General Topology and its applications 5 (1974), 191–212.Google Scholar
5. Hobson, E. W., The theory of functions of a real variable (Volume II), Dover Publications Inc., 1927.Google Scholar
6. Leader, S., On completion of proximity spaces by local clusters, Fund. Math. 48.Google Scholar
7. Nachman, L. J., Weak and strong constructions in proximity spaces, Ph.D. dissertation, The Ohio State University, 1968.Google Scholar
8. Naimpally, S. A., Reflective functors via nearness, Fund. Math. 85 (1974), 245–255.Google Scholar
9. Naimpally, S. A. and Warrack, B. D., Proximity spaces, Cambridge, University Press, UK (1970).Google Scholar
10. O. Njåstad, Some properties of proximity and generalized uniformity, Math. Scand. 12 (1963), 47–56.Google Scholar
11. Weil, A., Sur les espaces a structure uniforme et sur la topologie générale, Herman, Paris 1938.Google Scholar