A representation theorem for fuzzy pseudometrics
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), which is separated (resp. Hausdorff) provided that the pseudometric \(m\) in question is a metric (Proposition~4.1, Corollary~4.3 on page~96). It also appears that a family of pseudometrics and its induced KM-fuzzy pseudometric space generate the same uniform and topological structures (Proposition~4.5 on page~97). The manuscript ends with a wish to generalize the obtained results to partial (pseudo)metrics in the sense of~[\textit{S.~G. Matthews}, in: Papers on general topology and applications. Papers from the 8th summer conference at Queens College, New York, NY, USA, June 18--20, 1992. New York, NY: The New York Academy of Sciences. Ann. N. Y. Acad. Sci. 728, 183--197 (1994; Zbl 0911.54025)]. The paper is well written, nicely self-contained and is easy to follow, which will certainly benefit the topologically-minded readers of the fuzzy community.
lower semicontinuous family of real-valued maps
scientific article; zbMATH DE number 6109297