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Supremum metric and relatively compact sets of fuzzy sets

✍ Scribed by Gabriele H. Greco; Maria Pia Moschen

Publisher
Elsevier Science
Year
2006
Tongue
English
Weight
147 KB
Volume
64
Category
Article
ISSN
0362-546X

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✦ Synopsis

The supremum metric D between fuzzy subsets of a metric space is the supremum of the Hausdorff distances of the corresponding level sets. In this paper some new criteria of compactness with respect to the distance D are given; they concern arbitrary fuzzy sets (see Theorem 7), fuzzy sets having no proper local maximum points (see Theorem 12) and, finally, fuzzy sets with convex sendograph (see Theorem 13). In order to compare results with a previous characterization of compactness of Diamond-Kloeden, the criteria will be expressed by equi-(left/right)-continuity. In the proofs a first author's purely topological criterion of D-compactness and a variational convergence (calledconvergence) which was introduced by De Giorgi and Franzoni, are fundamental.

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