A characterization of relatively compact sets of fuzzy sets

A purely topological characterization of relatively compact sets is given for the metric space (K(Y),D)(K(Y),D) of upper semicontinuous, compact-supported, normal fuzzy subsets of a metric space YY. The considered metric DD is that of the distance between fuzzy subsets, which is the supremum of the Hausdorff distances of the corresponding level sets. In the given proof the compactness of a variational convergence which was introduced by De Giorgi and Franzoni is fundamental.