Ultra-Quasi-Metrically Tight Extensions of Ultra-Quasi-Metric Spaces

Abstract

The concept of the tight extension of a metric space was introduced and studied by Dress. It is known that Dress theory is equivalent to the theory of the injective hull of a metric space independently discussed by Isbell some years earlier. Dress showed in particular that for a metric space the tight extension is maximal among the tight extensions of . In a previous work with P. Haihambo and H.-P. Künzi, we constructed the tight extension of a 0 -quasi-metric space. In this paper, we continue these investigations by presenting a similar construction in the category of -metric spaces and nonexpansive maps.

On the ultra-quasi-tight extensions

“…In this article, we introduce the concept of tightness and essentiality of nonexpansive maps in the category of ultra-quasi-metric spaces and nonexpansive maps that we call ultra-quasi-tight and ultra-quasi-essential, respectively. We point out that the approach used in this article is different to the ultra-tree construction approach used in [1], but our findings extend the results from [3] and [6] on metric and quasi-metric settings, respectively. We establish, among other results, that ultra-quasi-tightness and ultra-quasi-essentiality of an extension of an ultra-quasimetric space are equivalent.…”

“…Let (Y, v) be an ultra-tight extension of X. From Proposition 3.5, one observes that if the map v : Y → (ν q (X, v), N ) is defined by v(y) = f y for all y ∈ Y , then v is a unique isometric embedding (see [1]). Therefore, the ultra-quasi extension (Y, v) of X is seen as a subspace of the extension (ν q (X, v), N ) of X.…”

“…Their results were used to study endpoints in T 0 -quasi-metric spaces. Furthermore, Agyingi [1] introduced tight extensions in the category of ultra-quasi-metric spaces and nonexpansive maps by extending the results from [2] on the tight extensions from quasi-metric point of view to the framework of ultraquasi-metric spaces.…”

Structure of u- Injective Hull in Ultra-Quasi-Pseudo Metric Space

“…Since one of the most interesting concepts in the theory of hyperconvex metric spaces is the notion of the injective hull of a metric space [5], it is very natural to wonder about the existence of an injective hull in the field of generalized metric spaces. From that point forward a considerable measure from claiming worth of effort have been carried in regards to those hyperconvexity for nonexpansive mappings (see [ 2,4,7,9]). …”