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Non-Metrizable Uniformities and Proximities on Metrizable Spaces

Published online by Cambridge University Press:  20 November 2018

P. L. Sharma
P. L. Sharma*
Affiliation:
Indian Institute of Technology, Kanpur, India; Southern Illinois University, Carbondale, Illinois
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In the literature there exist examples of metrizable spaces admitting nonmetrizable uniformities (e.g., see [3, Problem C, p. 204]). In this paper, this phenomenon is presented more coherently by showing that every non-compact metrizable space admits at least one non-metrizable proximity and uncountably many non-metrizable uniformities. It is also proved that the finest compatible uniformity (proximity) on a non-compact non-semidiscrete space is always non-metrizable.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 25 , Issue 5 , October 1973 , pp. 979 - 981
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Alfsen, E. M. and O. Njåstad, Proximity and generalized uniformity, Fund. Math. 52 (1963), 253–252.Google Scholar
2. Gillman, L. and Jerison, M., Rings of continuous functions (Van Nostrand, Princeton, 1960).Google Scholar
3. Kelley, J. L., General topology (Van Nostrand, Princeton, 1961).Google Scholar
4. Naimpally, S. A. and Warrack, B. D., Proximity Spaces, Cambridge Tract in Maths., No. 59 (Cambridge University Press, Cambridge, 1970).Google Scholar
5. Sharma, P. L., Compactification numbers (preprint).Google Scholar
6. Sharma, P. L., On p-classes of uniformities (preprint).Google Scholar
7. Reed, E. E. and Thron, W. J., m-bounded uniformities between two given uniformities, Trans. Amer. Math. Soc. 141 (1969), 71–77.Google Scholar
8. Thron, W. J., Topological structures, (Holt, Rinehart and Winston, New York, 1966).Google Scholar