On the convergence of fuzzy sets
Fuzzy Sets and Systems
Some notes on the characterization of compact sets of fuzzy sets with Lp metric
Fuzzy Sets and Systems
Information Sciences: an International Journal
A note on the sendograph metric of fuzzy numbers
Information Sciences: an International Journal
The topological structure of fuzzy sets with endograph metric
Fuzzy Sets and Systems
Some notes on Zadeh's extensions
Information Sciences: an International Journal
A t-norm embedding theorem for fuzzy sets
Fuzzy Sets and Systems
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Let K(Y) be the family of all fuzzy subsets of an arbitrary metric space Y, which are upper-semicontinuous, normal with nonempty compact support. The sendograph distance H between fuzzy sets is the Hausdorff distance of their ''reduced hypographs'' (called sendographs). Fan [Fuzzy Sets and Systems 143 (2004) 471-477] characterized compact sets of fuzzy numbers (=convex fuzzy subsets of the real line R) with respect to sendograph metric and declared open the problem of finding a criterion for convex fuzzy subsets of R^n. In this paper relatively compact subsets of (K(Y),H) are characterized (see Theorem 8) and, consequently, it is shown that Fan's theorem holds for arbitrary metric spaces without any convexity assumption. In the proofs a variational convergence (called @C-convergence), introduced by De Giorgi-Franzoni [Atti. Accad. Naz. Lincei Rend. Cl. Sc. Mat. Fis. Natur. 58(8) (1975) 842-850; Rend. Sem. Mat. Brescia 3 (1979) 63-101], and author's previous results are fundamental; no isometrical embedding theorem is used. co et al., Ann. Univ. Ferrara (Sez. VII, Sc. Mat.) 44 (1998) 27-39; Greco and Moschen, Nonlinear Anal. TMA, to appear]; no isometrical embedding theorem is used.