Characterization of compact subsets of fuzzy sets.

*(English)*Zbl 0661.54011Many 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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\textit{P. Diamond} and \textit{P. Kloeden}, Fuzzy Sets Syst. 29, No. 3, 341--348 (1989; Zbl 0661.54011)

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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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