Quasi-metric tree inT0-quasi-metric spaces

Abstract

MSC: 54E15 54E35 54C15 54E55 54E50 Keywords:Metric interval Metric tree T 0 -quasi-metric Quasi-metric interval Quasi-metric treeIn his well-known paper dealing with metric trees and tight extension of metric spaces, Dress studied the relationship between metric trees and hyperconvex metric spaces. In the present work, we introduce similarly the notion of quasi-metric tree of a T 0 -quasi-metric space. In particular, we show that large parts of the theory of metric trees do not use the symmetry of the metric and, under appropriate modifications, it still holds essentially unchanged for T 0 -quasi-metrics.

Quasi-metric trees and $q$-hyperconvex hulls

“…The following proposition improves [9,Lemma 2]. We now prove that if w ∈ ⟨x, y⟩ d , then w ∈ ⟨x, z⟩ d ∪ ⟨z, y⟩ d .…”

“…In [9], we started investigating the concept of a metric tree in T 0 -quasi-metric spaces, which we called quasi-metric tree. In this article, we continue our study of this concept by generalizing some well-known results about metric trees from metric setting to the quasi-metric point of view.…”