Some remarks on quasi-uniform spaces
Hans-Peter A. Künzi
1989
Glasgow Mathematical Journal
Introduction. 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 <# of open sets of a topological space is called interior-preserving if the intersection of an arbitrary subcollection of % is open (see e.g. [2,
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... . 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 71 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 T x 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. Recently there has been some interest in the construction of uqu spaces (compare [4]). In this connection our observation that the product of finitely many uqu spaces is a uqu space may be useful. We prove this result in the second section of this note. Topological spaces admitting a unique quasi-proximity are called uqp spaces in [10]. Each hereditarily compact space is a uqp space [10, Theorem 2.4]. Answering a question [2, Problem B, p. 45] of P. Fletcher and W. F. Lindgren in the negative, we show in [6] that a (first-countable) uqp space need not be hereditarily compact. In fact, it is proved in [9, Proposition 4] that a uqp space X is hereditarily compact if and only if each ultrafilter on X has an irreducible convergence set. (Recall that a nonempty subset A of a topological space is called irreducible (see e.g. [9, p. 238]) if each pair of nonempty A-open subsets has a nonempty intersection.) In particular each uqp Hausdorff space is hereditarily compact (and, thus, finite) [2, Theorem 2.36]. It seems to be unknown whether the uqp T r spaces can be characterized in a similar way. The last result contained in this paper shows that, at least, each uqp T x space with countable pseudo-character is hereditarily compact (and, thus, countable). (Recall that a topological space ^h a s countablepseudo-character if each point of X is a G a -set in X.) Glasgow Math. J. 31 (1989) 309-320.
doi:10.1017/s0017089500007874
fatcat:x5wnjwrzy5ebni2s5nqgcauhoa