A representation theorem for fuzzy pseudometrics
type: Item
instance of: scholarly article
title: A representation theorem for fuzzy pseudometrics
zbMATH Open document ID: 1255.54014
published in: Fuzzy Sets and Systems
publication date: 2012-11-21T00:00:00Z
review text: Since the inception of the notion of fuzzy set in~[\textit{L.~A. Zadeh}, Inf. Control 8, 338--365 (1965; Zbl 0139.24606)], there were many attempts to come up with a proper fuzzification of metric spaces. In particular,~[\textit{I.~Kramosil} and \textit{J.~Michalek}, Kybernetika, Praha 11, 336--344 (1975; Zbl 0319.54002)] presented the notion of fuzzy metric, which was inspired by the already existing theory of probabilistic metric spaces~[\textit{B.~Schweizer} and \textit{A.~Sklar}, Pac. J. Math. 10, 313--334 (1960; Zbl 0091.29801)]. Motivated by the former concept, the paper under review introduces the notion of \textit{KM-fuzzy pseudometric space} (``KM'' is the abbreviation for ``I.~Kramosil and J.~Michalek''), which is a triple \((X,m,T)\), where \(X\) is a non-empty set, \(T\) is a t-norm, and \(m:X\times X\times[0,\infty)\rightarrow[0,1]\) is a map such that for every \(x,y,z\in X\) and every \(t,s\in[0,\infty)\), (1) \(m(x,y,0)=0\); (2) \(m(x,x,t)=1\) for every \(t>0\); (3) \(m(x,y,t)=m(y,x,t)\); (4) \(T(m(x,y,t),m(y,z,s))\leqslant m(x,z,t+s)\); (5) \(m(x,y,-):[0,\infty)\rightarrow[0,1]\) is left-continuous, and \(\text{lim}_{t\rightarrow\infty}m(x,y,t)=1\). \noindent If the map \(m\) also satisfies (6) \(m(x,y,t)=1\) and \(t>0\) imply \(x=y\); \noindent then \((X,m,T)\) is said to be a \textit{KM-fuzzy metric space} (Definition~2.1 on page~92). Additionally, given a subset \(I\) of the real line, a family of real-valued maps \(\{d_i\,|\,i\in I\}\) is called \textit{lower semicontinuous} (\textit{LSC}, for short) provided that \(d_i=\bigwedge_{j>i}d_j\) for every \(i\in I\) (Definition~3.1 on page~93). With the above two notions in hand, the authors establish a bijection between the set of KM-fuzzy pseudometrics on \(X\) (under the t-norm \(\wedge\))and \([0,1)\)-indexed LSC families of ordinary pseudometrics on \(X\) (Theorem~3.12 on page~95). Moreover, they show that every KM-fuzzy pseudometric space gives rise to a uniform space (resp. topological space),…
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 03E72
Mathematics Subject Classification ID: 06A15
Mathematics Subject Classification ID: 54A40
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 54E70
zbMATH DE Number: 6109297
zbMATH Keywords: pseudo metric
zbMATH Keywords: fuzzy pseudo metric
zbMATH Keywords: fuzzy metric
zbMATH Keywords: Galois connection
zbMATH Keywords: Hausdorff space
zbMATH Keywords: \(T_1\) space
zbMATH Keywords: lower semicontinuous family of real-valued maps
zbMATH Keywords: metrizable space
zbMATH Keywords: partial metric
zbMATH Keywords: probabilistic metric space
zbMATH Keywords: t-norm
zbMATH Keywords: topological space
zbMATH Keywords: uniform space
zbMATH Keywords: uniformity subbase
author: Iraide Mardones-Pérez
author: María Angeles de Prada Vicente
reviewed by: Sergejs Solovjovs
MaRDI profile type: Publication
full work available at URL: j.fss.2011.11.008
OpenAlex ID: W2074569934
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DOI: 10.1016/J.FSS.2011.11.008
description: scientific article; zbMATH DE number 6109297