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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

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