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

On balancedness and D-completeness of the space of semi-Lipschitz functions

“…Following the modern terminology [10], Cauchy sequences in the sense of Doitchinov will be called, in the sequel,D-Cauchy sequences and complete extended quasi-metric spaces will be called D-(sequentially) complete extended quasi-metric spaces. We say that the normed cone…”

A Common Mathematical Framework for Asymptotic Complexity Analysis and Denotational Semantics for Recursive Programs Based on Complexity Spaces

“…Following (Künzi, 1993), a quasi-metric on a nonempty set X is a function d : X × X → R + such that for all x, y, z ∈ X :…”

“…On the other hand, in the literature it has been introduced the so-called domain of words as a possible mathematical foundation of Denotational Semantics (Davey & Priestley, 1990;Künzi, 1993;Matthews, 1994). Such a mathematical structure can be constructed as follows:…”

Fuzzy quasi-metric spaces

“…There exist many different notions of quasi-uniform and quasi-metric completeness in the literature (see [10]). Then, by Lemma 3.1 and Remark 3.3, one can define in a natural way the corresponding notions of completeness in a fuzzy setting and easily deduce several properties taking into account the well-known completeness properties of quasi-uniform and quasi-metric spaces (compare with [6], where these ideas are used to study completeness in the fuzzy metric case).…”

“…Following the modern terminology (see Section 11 of [10]), by a quasi-metric on X we mean a quasi-pseudo-metric d on X that satisfies the following condition: d(x, y) = d(y, x) = 0 if and only if x = y. If the quasi-pseudo-metric d satisfies: d(x, y) = 0 if and only if x = y, then we say that d is a T 1 quasi-metric on X.…”