A representation theorem for quasi-metric spaces

Abstract

We show that every quasi-metric space is isomorphic to a subspace of the hyperspace of a suitable metric space, endowed with the Hausdorff quasi-metric. Therefore a topological space is quasi-metrizable if and only if it can be embedded in a Hausdorff quasi-metric hyperspac

Semi-Lipschitz Functions and Best Approximation in Quasi-Metric Spaces

“…These advances have also permitted the development of generalizations, to the``nonsymmetric case,'' of classical mathematical theories: hyperspaces [15,22,33], function spaces [13,20,21,23,26], approximation and fixed point theory [6,7,25], linear lattices [3], etc. This paper is a contribution to the study of semi-Lipschitz functions and best approximation from a nonsymmetric point of view.…”

Representation theorems for fuzzy orders and quasi-metrics

“…Let d : S Â S ! ½0; 1 be a generalized extended metric space and C a class of subsets of S. Then we say that ðC; e d Þ is a Hausdorff excess space (see [9] and [10]). The proof of the following proposition is trivial.…”

“…Namely, the first theorem says that any extended quasi-metric space is isomorphic with a difference-based quasi-metric space. The second theorem says that any quasi-metric space is isomorphic with a Hausdorff excess space (see also [9] and [10]). …”

Logic of approximate entailment in quasimetric spaces

“…We propose to examine instead the question if quasimetric spaces can be suitably embedded in metric spaces, such that the quasimetric of the former corresponds to the Hausdorff quasimetric of the latter space. A result of this kind is contained in [21]; unfortunately, the proposed representation seems not to be applicable in the present context and an independent approach needs to be found.…”