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Characterization of compact subsets of fuzzy sets. (English) Zbl 0661.54011
Many applications of fuzzy sets restrict attention to the convenient metric space ($${\mathcal E}^ n,D)$$ of normal, fuzzy convex sets on the base space $${\mathbb{R}}^ n$$, with D the supremum over the Hausdorff distances between corresponding level sets. We mention in particular the fuzzy random variables of M. L. Puri and D. A. Ralescu [Ann. Probab. 13, 1373-1379 (1985; Zbl 0583.60011)], the fuzzy differential equations of O. Kaleva [Fuzzy Sets Syst. 24, 301-317 (1987; Zbl 0646.34019)], the fuzzy dynamical systems of the second author [Fuzzy Sets Syst. 7, 275-296 (1982; Zbl 0509.54040)] and the chaotic iterations of fuzzy sets of Diamond and Kloeden. In these papers specific results are often obtained for compact subsets of $${\mathcal E}^ n$$, which raises the question of how to characterize such compact subsets. The purpose of this is to present a convenient characterization of compact subsets of the metric space ($${\mathcal E}^ n,D)$$. Our main result is that a closed subset of $${\mathcal E}^ n$$ is compact if and only if the support sets are uniformly bounded in $${\mathbb{R}}^ n$$ and the support functions of Puri and Ralescu are equileftcontinuous in the membership grade variable $$\alpha$$ uniformly on the unit sphere $$S^{n-1}$$ of $${\mathbb{R}}^ n$$. To this end we note that the support functions provides a means of embedding all of the space $${\mathcal E}^ n$$ in a Banach space, which we exhibit explicitly, not just the subspace $${\mathcal E}^ n_{Lip}$$ of ‘Lipschitzian’ fuzzy sets considered by Puri and Ralescu.

MSC:
 54A40 Fuzzy topology
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References:
 [1] P. Diamond, Fuzzy chaos, J. Math. Anal. Appl. (submitted). · Zbl 0921.93026 [2] Graves, L.M, The theory of functions of real variables, (1946), McGraw-Hill New York · Zbl 0063.01720 [3] Kaleva, O, On the convergence of fuzzy sets, Fuzzy sets and systems, 17, 54-65, (1985) · Zbl 0584.54004 [4] O. Kaleva, The Cauchy problem for fuzzy differential equations, Preprint. · Zbl 0696.34005 [5] Kloeden, P.E, Fuzzy dynamical systems, Fuzzy sets and systems, 7, 275-296, (1982) · Zbl 0509.54040 [6] Kloeden, P.E, Chaotic mappings on fuzzy sets, (), Preprint · Zbl 0746.54010 [7] Kolmogorov, A.N; Fomin, S.V, Introductory real analysis, (1975), Dover New York · Zbl 0213.07305 [8] Puri, M.L; Ralescu, D.A, The concept of normality for fuzzy random variables, Ann. probub., 13, 1373-1379, (1985) · Zbl 0583.60011
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