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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 (${ℰ}^{n},D\right)$ of normal, fuzzy convex sets on the base space ${ℝ}^{n}$, 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 ${ℰ}^{n}$, 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 (${ℰ}^{n},D\right)$. Our main result is that a closed subset of ${ℰ}^{n}$ is compact if and only if the support sets are uniformly bounded in ${ℝ}^{n}$ 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 ${ℝ}^{n}$. To this end we note that the support functions provides a means of embedding all of the space ${ℰ}^{n}$ in a Banach space, which we exhibit explicitly, not just the subspace ${ℰ}_{Lip}^{n}$ of ‘Lipschitzian’ fuzzy sets considered by Puri and Ralescu.

##### MSC:
 54A40 Fuzzy topology