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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 (${\cal E}\sp n,D)$ of normal, fuzzy convex sets on the base space ${\bbfR}\sp n$, with D the supremum over the Hausdorff distances between corresponding level sets. We mention in particular the fuzzy random variables of {\it M. L. Puri} and {\it D. A. Ralescu} [Ann. Probab. 13, 1373-1379 (1985; Zbl 0583.60011)], the fuzzy differential equations of {\it 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 ${\cal E}\sp n$, 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 (${\cal E}\sp n,D)$. Our main result is that a closed subset of ${\cal E}\sp n$ is compact if and only if the support sets are uniformly bounded in ${\bbfR}\sp n$ and the support functions of Puri and Ralescu are equileftcontinuous in the membership grade variable $\alpha$ uniformly on the unit sphere $S\sp{n-1}$ of ${\bbfR}\sp n$. To this end we note that the support functions provides a means of embedding all of the space ${\cal E}\sp n$ in a Banach space, which we exhibit explicitly, not just the subspace ${\cal E}\sp n\sb{Lip}$ of `Lipschitzian’ fuzzy sets considered by Puri and Ralescu.

##### MSC:
 54A40 Fuzzy topology
Full Text:
##### 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) · 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. Second international congress of the international fuzzy system association (July 1987) [7] Kolmogorov, A. N.; Fomin, S. V.: Introductory real analysis. (1975) · 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