Hans-Peter A. Künzi

Nonsymmetric Distances and Their Associated Topologies: About the Origins of Basic Ideas in the Area of Asymmetric Topology

We begin with some remarks explaining the structure of this article. After some introductory statements in the following paragraphs, we summarize the historic development of what is now often called “Nonsymmetric or Asymmetric Topology” in Section 2. In the following, more specific sections we discuss the historic development of some of the main ideas of the area in greater detail. The list of sections and keywords given above should help the specialist to find his way through the various sections.

Partial quasi-metrics

In this article we introduce and investigate the concept of a partial quasi-metric and some of its applications. We show that many important constructions studied in Matthewss theory of partial metrics can still be used successfully in this more general setting. In particular, we consider the bicompletion of the quasi-metric space that is associated with a partial quasi-metric space and study its applications in groups and BCK-algebras.

Weighted Quasi‐Metrics

ABSTRACT: We study the class of topologies which are induced by weighted quasi‐metrics (equivalently, partial metrics). Partial metrics were introduced by S. Matthews in his study of topological models appropriate for the denotational semantics of programming languages.

On simultaneous extension of continuous partial functions

For a metric space X let Cvc(X) (that is, the set of all graphs of real-valued continuous functions with a compact domain in X) be equipped with the Hausdorff metric induced by the hyperspace of nonempty closed subsets of X×R. It is shown that there exists a continuous mapping Φ : Cvc(X) → Cb(X) satisfying the following conditions: (i) Φ(f)| dom f = f for all partial functions f. (ii) For every nonempty compact subset K of X, Φ|Cb(K) : Cb(K) → Cb(X) is a linear positive operator such that Φ(1K) = 1X .

Completely regular ordered spaces

We present an example of a completely regular ordered space that is not strictly completely regular ordered. Furthermore, we note that a completely regular ordered I-space is strictly completely regular ordered provided that it satisfies at least one of the following three conditions: It is locally compact, it is a C-space, it is a topological lattice.

Functorial admissible quasi-uniformities on topological spaces

Abstract Let T denote the forgetful functor from the category Quu of quasi-uniform spaces and quasi-uniformly continuous maps to the category Top of topological spaces and continuous maps. In the first part of this paper we develop a general method to construct nontransitive admissible quasi-uniformities on topological spaces. We use this technique to answer a question of G.C.L. Brummer in the negative by exhibiting a nontransitive T -section that is coarser than the (transitive) locally finite covering quasi-uniformity. In the second part of this paper we show that the Pervin functor restricted to the category Haus of Hausdorff spaces and continuous maps is the coarsest functor that puts compatible quasi-uniformities on the Hausdorff spaces.

A note on sequentially compact quasi-pseudo-metric spaces

It is asked in [3] whether sequential compactness is equivalent to compactness in a quasi-pseudo-metric space. In this note we give a counterexample and a proposition relating to this question.

The Ultra-Quasi-Metrically Injective Hull of a T0-Ultra-Quasi-Metric Space

Jointly with E. Kemajou, in previous work we constructed the so-called Isbell-hull of a T0-quasi-metric space. In this article we continue these investigations by presenting a similar construction in the category of T0-ultra-quasi-metric spaces and contracting maps. Comparable studies in the area of ultra-metric spaces have been conducted before by Bayod and Martínez-Maurica.

Stability in quasi-uniform spaces and the inverse problem☆

Kiinzi, H.-P.A., and H.J.K. Junnila, Stability in quasi-uniform spaces and the inverse problem, Topology and its Applications 49 (1993) 175-189. We study the concept of a stable quasi-uniform space due to D. Doitchinov. In particular quasi-pseudo-metric spaces (X, d) whose associated quasi-uniformities %(d) or *U(d-‘) are stable are investigated.

A double completion for an arbitrary T 0 -quasi-metric space

Abstract We present a conjugate invariant method for completing any T 0 -quasi-metric space. The completion is built as an extension of the bicompletion of the original space. For balanced T 0 -quasi-metric spaces our completion yields up to isometry the completion due to Doitchinov. The question which uniformly continuous maps between T 0 -quasi-metric spaces can be extended to the constructed completions leads us to introduce and investigate a new class of maps, which we call balanced maps.