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Some remarks on quasi-uniform spaces

Published online by Cambridge University Press:  18 May 2009

Hans-Peter A. Künzi
Hans-Peter A. Künzi
Affiliation:
Department of Mathematics, University of Berne, Siddlerstrasse 5, 3012 Berne, Switzerland
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A topological space is called a uqu space [10] if it admits a unique quasi-uniformity. Answering a question [2, Problem B, p. 45] of P. Fletcher and W. F. Lindgren in the affirmative we show in [8] that a topological space X is a uqu space if and only if every interior-preserving open collection of X is finite. (Recall that a collection ℒ of open sets of a topological space is called interior-preserving if the intersection of an arbitrary subcollection of ℒ is open (see e.g. [2, p. 29]).) The main step in the proof of this result in [8] shows that a topological space in which each interior-preserving open collection is finite is a transitive space. (A topological space is called transitive (see e.g. [2, p. 130]) if its fine quasi-uniformity has a base consisting of transitive entourages.) In the first section of this note we prove that each hereditarily compact space is transitive. The result of [8] mentioned above is an immediate consequence of this fact, because, obviously, a topological space in which each interior-preserving open collection is finite is hereditarily compact; see e.g. [2, Theorem 2.36]. Our method of proof also shows that a space is transitive if its fine quasi-uniformity is quasi-pseudo-metrizable. We use this result to prove that the fine quasi-uniformity of a T1 space X is quasi-metrizable if and only if X is a quasi-metrizable space containing only finitely many nonisolated points. This result should be compared with Proposition 2.34 of [2], which says that the fine quasi-uniformity of a regular T1 space has a countable base if and only if it is a metrizable space with only finitely many nonisolated points (see e.g. [11] for related results on uniformities). Another by-product of our investigations is the result that each topological space with a countable network is transitive.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 31 , Issue 3 , September 1989 , pp. 309 - 320
Copyright
Copyright © Glasgow Mathematical Journal Trust 1989

References

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