Quantales and continuity spaces

Applied General Topology

We identify two categories of quantale-valued convergence tower spaces that are isomorphic to the categories of quantale-valued metric spaces and quantale-valued partial metric spaces, respectively. As an application we state a quantale-valued metrization theorem for quantale-valued convergence tower groups.

2020

In the early 80's, Alain Quilliot presented an approach of ordered sets and graphs in terms of metric spaces, where instead of positive real numbers, the values of the distance are elements of an ordered monoid equipped with an involution. This point of view was further developed in a series of papers by Jawhari, Misane, Pouzet, Rosenberg and Kabil. Some results are currently published by Rosenberg, Kabil and Pouzet, Bandelt, Pouzet and Saidane, Khamsi and Pouzet. Special aspects were developed by the authors of the present paper. A survey on generalized metric spaces is in print. In this paper, we review briefly the salient aspects of the theory of generalized metric spaces, then we illustrate the properties of the preservation, by operations, of sets of relations, notably binary relations, and particularly equivalence relations.

In this paper we define the fuzzy metric space by using the usual definition of the metric space and vise versa, so we can obtain each one from the other. We prove some fixed point theorems on the fuzzy metric spaces.

Transactions of the American Mathematical Society, 1986

Families of finite metric spaces are investigated. A notion of metric type is introduced and it is shown that for Banach spaces it is consistent with the standard notion of type. A theorem parallel to the Maurey-Pisier Theorem in Local Theory is proved. Embeddings of Zp-cubes into the ¡i-cube (Hamming cube) are discussed.

Modern Real Analysis, 2015

Acknowledgements vii Dedication ix Table of Contents xi List of Tables xv List of Figures xvii 6.4 Automata accepting strings of the form {0 n , 0 n 1} 2n and strings not of that form.

The problem of constructing a satisfactory theory of fuzzy metric spaces has been investigated by several authors from different points of view. In particular, and by modifying a definition of fuzzy metric space given by Kramosil and Michalek, George and Veeramani have introduced and studied the following interesting notion of a fuzzy metric space: A fuzzy metric space is an ordered triple (X , M, * ) such that X is a set, * is a continuous t-norm and M is a function defined on X × X ×]0, +∞[ with values in ]0, 1] satisfying certain axioms and M is called a fuzzy metric on X . It is proved that every fuzzy metric M on X generates a topology τ M on X which has as a base the family of open sets of the form B(x, ε, t) = {x ∈ X , 0 < ε < 1, t > 0} where B(x, ε, t) = {y ∈ X : M(x, y , t) > 1 − ε} for all ε ∈]0, 1[ and t > 0. The topological space (X , τ ) is said to be fuzzy metrizable if there is a fuzzy metric M on X such that τ = τ M . Then, it was proved that a topological space is fuzzy metrizable if and only if it is metrizable. From then, several fuzzy notions which are analogous to the corresponding ones in metric spaces have been given. Nevertheless, the theory of fuzzy metric completion is, in this context, very different to the classical theory of metric completion: indeed, there exist fuzzy metric spaces which are not completable. This class of fuzzy metrics can be easily included within fuzzy systems since the value given by them can be directly interpreted as a fuzzy certainty degree of nearness, and in particular, recently, they have been applied to colour image filtering, improving some filters when replacing classical metrics In this lecture we survey some results and questions obtained in recent years about this class of fuzzy metric spaces.

arXiv: General Topology, 2016

For a quantale V we introduce V-approach spaces via V-valued point-set-distance functions and, when V is completely distributive, characterize them in terms of both, so-called closure towers and ultrafilter convergence relations. When V is the two-element chain 2, the extended real half-line [0,∞], or the quantale ∆ of distance distribution functions, the general setting produces known and new results on topological spaces, approach spaces, and the only recently considered probabilistic approach spaces, as well as on their functorial interactions with each other.

We introduce the notion of -metric as a generalization of a metric by replacing the triangle inequality with a more generalized inequality. We investigate the topology of the spaces induced by a -metric and present some essential properties of it. Further, we give characterization of well-known fixed point theorems, such as the Banach and Caristi types in the context of such spaces.

2019

The aim of this paper is to introduce some characterization of various fuzzy bounded sets in fuzzy metric spaces & some fixed-point theorems for multi valued mappings are also defined. The original concept of a KM-fuzzy metric—we call this modification by a GV-fuzzy metric. This modification allows many natural examples of fuzzy metrics, in particular, fuzzy metrics constructed from metrics. GV-fuzzy metrics appear to be more appropriate also for the study of induced topological structures. Along with the principal interest of many researchers in the theoretical aspects of the theory of fuzzy metrics—in particular, the topological and sequential properties of fuzzy metric spaces, their completeness, fixed points of mappings, etc.—fuzzy metrics have also aroused interest among specialists working in various applied areas of mathematics. Among others, fuzzy metrics have been used in decision making problems with uncertain and imprecise information and other engineering problems.