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On topological spaces with a unique compatible quasi-uniformity

Published online by Cambridge University Press:  18 May 2009

Lawrence M. Brown
Lawrence M. Brown
Affiliation:
Hacettepe University, Department of Mathematics, Beytepe Campus, Ankara, Turkey
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It is shown in [2] that a uqu space satisfies the following conditions.

(DC) There is no infinite, strictly decreasing sequence of open sets with open intersection.

(IC) There is no infinite, strictly increasing sequence of open sets.

In this note we show that for a transitive space these conditions are sufficient for the space to be uqu. This will follow as a consequence of the following result.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 18 , Issue 1 , January 1977 , pp. 11 - 12
Copyright
Copyright © Glasgow Mathematical Journal Trust 1977

References

REFERENCES

1.Birkhoff, G., Lattice theory, 3rd ed., Amer. Math. Soc. Colloquium Publications 25 (1967).Google Scholar
2.Lindgren, W. F., Topological spaces with a unique compatible quasi-uniformity, Canad. Math. Bull. 14 (1971), 369–372.CrossRefGoogle Scholar
3.Szasz, G., Einführung in die Verbandtheorie (Teubner, Leipzig, 1962).Google Scholar