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Topological properties on the space of fuzzy sets. (English) Zbl 0986.54012
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 [S. Y. Joo and 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 [P. Diamond and P. Kloeden, ibid. 29, No. 3, 341-348 (1989; Zbl 0661.54011)] and several nice example are given.

##### MSC:
 54A40 Fuzzy topology 03E72 Theory of fuzzy sets, etc.
##### Keywords:
Skorokhod metric; Skorokhod topology
##### Citations:
Zbl 0961.54024; Zbl 0661.54011
Full Text:
##### References:
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