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Uniqueness of Compatible Quasi-Uniformities

Published online by Cambridge University Press:  20 November 2018

Charles I. Votaw
Charles I. Votaw*
Affiliation:
University of Kansas, Lawrence, Kansas
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Abstract

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It is shown that a topological space has a unique compatible quasi-uniformity if its topology is finite. Examples are given to show the converse is false for T1 and for normal second countable spaces. Two sufficient conditions are given for a topological space to have a compatible quasi-uniformity strictly finer than the associated Császár-Pervin quasi-uniformity. These conditions are used to show that a Hausdorff, semi-regular or first countable T1 space has a unique compatible quasi-uniformity if and only if its topology is finite. Császár and Pervin described, in quite different ways, quasi-uniformities which induce a given topology. It is shown that, for a given topological space, Császár and Pervin described the same quasi-uniformity.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 4 , December 1972 , pp. 575 - 583
Copyright
Copyright © Canadian Mathematical Society 1972

References

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