A note on uniform ordered spaces

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 pseudocompac

Strongly zero-dimensional bispaces

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

Some remarks on quasi-uniform spaces

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, 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 transitiv

Bicompleteness of the fine quasi-uniformity

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-sobe

Ti-ordered reflections

[EN] We present a construction which shows that the Ti-ordered reflection (i ϵ {0, 1, 2}) of a partially ordered topological space (X, , τ, ≤) exists and is an ordered quotient of (X, τ, ≤). We give an explicit construction of the T0-ordered reflection of an ordered topological space (X, τ, ≤), and characterize ordered topological spaces whose T0-ordered reflection is T1-ordered.The first author would like to thank the South African Research Foundation for partial financial support under Grant Number 2068799.Künzi, HA.; Richmond, TA. (2005). Ti-ordered reflections. Applied General Topology. 6(2):207-216. https://doi.org/10.4995/agt.2005.1955SWORD2072166

The bicompletion of the Hausdorff quasi-uniformity

We study conditions under which the Hausdorff quasi-uniformity ${\mathcal U}_H$ of a quasi-uniform space $(X,{\mathcal U})$ on the set ${\mathcal P}_0(X)$ of the nonempty subsets of $X$ is bicomplete. Indeed we present an explicit method to construct the bicompletion of the $T_0$-quotient of the Hausdorff quasi-uniformity of a quasi-uniform space. It is used to find a characterization of those quasi-uniform $T_0$-spaces $(X,{\mathcal U})$ for which the Hausdorff quasi-uniformity $\widetilde{{\mathcal U}}_H$ of their bicompletion $(\widetilde{X},{\widetilde{\mathcal U}})$ on ${\mathcal P}_0(\widetilde{X})$ is bicomplete

Quasi-metric spaces, quasi-metric hyperspaces and uniform local compactness

We show that every locally compact quasi-metrizable Moore space admits a uniformly locally compact quasi-metric. We also observe that every equinormal quasi-metric is cofinally complete. Finally we prove that for any small-set symmetric quasi-uniform space, uniform local compactness is preserved by the Hausdorff-Bourbaki quasi-uniformity on compact sets. Several illustrative examples are given

The Katětov construction modified for a T0-quasi-metric space

AbstractWe discuss the existence and uniqueness of a T0-quasi-metric space qU defined by the following three conditions: (i) qU is bicomplete and supseparable, (ii) every isometry between two finite subspaces of qU extends to an isometry of qU onto itself, and (iii) qU contains an isometric copy of every supseparable T0-quasi-metric space