a1 Department of Mathematics, University of Berne, Siddlerstrasse 5, 3012 Berne, Switzerland
A topological space is called a uqu space  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  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, p. 29]).) The main step in the proof of this result in  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  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 , 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.  for related results on uniformities). Another by-product of our investigations is the result that each topological space with a countable network is transitive.
(Received July 07 1988)