ON THE RELATION BETWEEN FUZZY EQUALITIES PSEUDOMETRICS AND UNIFORMITIES

Abstract

During the Vth Seminar on Fuzzy Set Theory (Linz, Austria, September 1983) U. Cerruti and U. Höhle stated the following open problem from their work on fuzzy equalities. “16 (X,E) is a set with [0,1]-valued fuzzy equality E, and (P(X),E) is the set of fuzzy points with the induced fuzzy equality E, what is the precise relationship between X and P(X) and between E and E?” In particular they were unable to show that P(X) in general is “larger” than X and wondered if each fuzzy point ρ was of the type ρ(z) = E(x, z) for some x ε X.