Topological properties on the space of fuzzy sets
The authors introduce the Skorokhod metric on the space \({\mathcal F}(\mathbb{R}^p)\) of fuzzy sets, where \({\mathcal F}(\mathbb{R}^p)\) consists of those fuzzy sets \(\widetilde\mu: \mathbb{R}^p\to [0,1]\) which are normal, i.e., \(\widetilde \mu(x) =1\) for some \(x\in \mathbb{R}^p\); upper semicontinuous and \(\text{supp} \widetilde \mu=\overline {\{x\in \mathbb{R}^p: \mu(x)> 0\}}\) is compact. It is proved that the space \(\mathbb{R}^p\) with Skorokhod metric is separable and complete [\textit{S. Y. Joo} and \textit{Y. K. Kim}, Fuzzy Sets Syst. 111, No. 3, 497-501 (2000; Zbl 0961.54024)]. A characterization of compact subsets of \({\mathcal F}(\mathbb{R}^p)\) equipped with the Skorokhod topology is provided [\textit{P. Diamond} and \textit{P. Kloeden}, ibid. 29, No. 3, 341-348 (1989; Zbl 0661.54011)] and several nice example are given.
scientific article; zbMATH DE number 1496311