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...the space of fuzzy sets F n , with respect to two different metrics, the usual metric derived from the Hausdorff metric on the family of compact sets and the endograph metric defined by Kloeden =-=[4]-=-. Finally we determine a class of fuzzy sets where these metrics are equivalent. 1 Introduction The Zadeh extension is the way we produce a fuzzy transformation ˆf : F n sF n from a given f...

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...nificant early results about the topological structure of K(X ) were obtained by Wojdysławski [13,14]. Let I be the closed interval [0,1] and let Q be the Hilbert cube IN, the countable infinite Cartesian product of copies of I. Probably the most important theorem about K(X ) is by Curtis and Schori [4] and states that the hyperspace K(X ) is homeomorphic to Q if and only if X is a non-degenerate Peano continuum, that is, a connected and locally connected compactum. By the celebrated Hahn–Mazurkiewicz Theorem [6,9] the Peano continua are precisely the continuous images of I. Following Kloeden [7,8], who uses the concept to investigate fuzzy dynamical systems, we define a fuzzy compactum in X as an upper semi-continuous function f : A → I such that A ∈ K(X ) and {x ∈ A : f (x) > 0} is dense in A. That is, the compact subset A of X comes equipped with a density function. Let K(X ) denote the collection of all fuzzy compacta in X. Note that K(X ) is canonically imbedded in K(X ) by choosing the density function equal to one at each point. A function f : A → I is upper semi-continuous precisely if its hypograph hyp f = {(x, t) : 0 ≤ t ≤ f (x)} is closed ∗ Corresponding author. Tel.: +31 205...