It is known that if K is a compact subset of the (separable complete) metric Urysohn space ( ${\mathbb U}$ , d) and f is a Katětov function on the subspace K of ( ${\mathbb U}$ , d), then there is z ∈ ${\mathbb U}$ such that d(z, x) = f(x) for all x ∈ K.

Answering a question of Normann, we show in this article that the supseparable bicomplete q-universal ultrahomogeneous T 0-quasi-metric space (q ${\mathbb U}$ , D) recently discussed by the authors satisfies a similar property for Katětov function pairs on subsets that are compact in the associated metric space (q ${\mathbb U}$ , Ds ).

We construct a T1-space that is not hereditarily compact, although each of its open sets is the intersection of two compact open sets. The search for such a space was motivated by a problem in the theory of quasi-proximities.

Let Cb be the admissible functorial quasi-uniformity on the completely regular bispaces which is spanned by the upper quasi-uniformity on the real line. Answering a question posed by B. Banaschewski and G. C. L. Brümmer in the affirmative we show that CbX is transitive for every strongly zero-dimensional bispace X.

A characterization of the topological spaces that possess a bicomplete fine quasi-uniformity is obtained. In particular we show that the fine quasi-uniformity of each sober space, of each first-countable T1-space and of each quasi-pseudo-metrizable space is bicomplete. Moreover we give examples of T1-spaces that do not admit a bicomplete quasi-uniformity.

We obtain several conditions under which the semi-continuous quasi-uniformity of a topological space is bicomplete and observe that the well-monotone covering quasiuniformity of a topological space is bicomplete if and only if the space is quasi-sober.

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.

We characterize the generalized ordered topological spaces X for which the uniformity (X) is convex. Moreover, we show that a uniform ordered space for which every compatible convex uniformity is totally bounded, need not be pseudocompact.

A topological space X is said to be somewhat normal provided that for each open cover is a normal cover of X. We show that a completely regular somewhat normal space need not be normal, thereby answering a question of W. M. Fleischman. We note that a collectionwise normal somewhat normal space need not be almost 2-fully normal, as had previously been asserted, and that neither the perfect image nor the perfect preimage of a somewhat normal space has to be somewhat normal.

We show that a topological space X admits a unique quasiuniformityif and only if every interior-preserving open collection of X isfinite.