Tight extensions of T₀-quasi-metric spaces

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

“…In [1] a concept of tight extension that is appropriate in the category of 0 -quasi-metric spaces and nonexpansive maps was studied. In particular such an extension was constructed and it was shown that this extension is maximal among the tight extensions.…”

“…In this paper we will show how the studies of [1] can be modified in order to obtain a theory that is appropriate for -metric spaces. By UQP-metric space in the following, we mean 0 -ultra-quasi-metric spaces.…”

“…By UQP-metric space in the following, we mean 0 -ultra-quasi-metric spaces. Even though our studies follow essentially [1,2], we found it imperative to work out every detail of this theory in this paper.…”

Versatile asymmetrical tight extensions

“…Further, we provide another proof of the result on quasi-tight extensions (see Proposition 10) which also appears in [1,2]. …”

Endpoints inT0-Quasimetric Spaces: Part II

“…We will call a function pair minimal on ( , ) (among the ample function pairs on ( , )) if it is ample and whenever is ample on ( , ), and for each ∈ we have 1 ( ) ≤ 1 ( ) and 2 ( ) ≤ 2 ( ) (for any function pairs and satisfying this relation we will write ≤ ); then = . It is well known that Zorn's Lemma implies that below each ample function pair there is a minimal ample pair (cf., e.g., [2,20]). By we will denote the set of all minimal ample pairs on ( , ) equipped with the restriction of to × , which we will also denote by .…”