Abstract
During the Vth Seminar on Fuzzy Set Theory (Linz, Austria, September 1983) U. Cerruti and U. Hohle 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 e X. During the seminar we solved this question in the negative by the example following Corollary 2.3. based on [3] and conjectured that it could be generalized to arbitrary pseudometric spaces. In this paper we prove this conjecture. We show that for a large class of fuzzy equalities the relationship between X and P(X) is as between a pseudometric space and its completion and between E and [Etilde] is as between a uniformly continuous map and its continuous extension. A first proof i...
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