Completeness properties of hyperspaces of compact fuzzy sets

Abstract

0. Introduction. On an arbitrary uniform space there are two types of compactlike fuzzy sets which are widely used in applications: u.s.c. fuzzy sets with compact support ( we denote this collection $C(X)) and u.s.c. fuzzy sets with compact levelsets (we denote this collection *iv(X)) [2], [12]. Always $C(X) C $w(X) but the converse holds only if X itself is compact. In the first part of our paper we prove that for the global fuzzy hyperspace structure [8], [9] the completeness of X is equivalent to the completeness of $ c (X) and to either the completeness or the ultracompleteness of $w (X) [6], [7]. In the second part we then prove the rather surprising result that the completion of $c(-^0 [7] is isomorphic to $jy (X) where X denotes the completion of X. These results not only generalize K. Morita's results on hyperspace of compact subsets [11] to the setting of fuzzy hyperspaces of compactlike fuzzy subsets but moreover via the isomorphism of the uniform modification of $C(X) and $w (X) with hyperspaces of closed subsets of X x [0,1] [9], they also include an extension of K. Morita's classical results to classes of closed subsets of X x [0,1] which are in general not compact.