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Continuity of functions defined on the space of fuzzy sets. (English) Zbl 1051.54004
The centre of discussion in this paper is $$J(\mathbb{R}^p)$$ which denotes the family of all fuzzy sets $$\widetilde u:\mathbb{R}^p\to[0,1]$$ with the following properties: (1) $$\widetilde u$$ is normal, i.e, there exists $$x\in \mathbb{R}^p$$ such that $$\widetilde u(x)=1$$; (2) $$\widetilde u$$ is upper semicontinuous; (3) $$\text{supp}\, \widetilde u=\overline {\{x\in \mathbb{R}^p:\widetilde u(x)>0\}}$$ is compact. Addition and scalar multiplication are suitably defined and the space is endowed with a special type of metric named Skorokhod metric. The authors first deal with the question: at which points of $$I(\mathbb{R}^p)$$ is a given function continuous, $$J(\mathbb{R}^p)$$ being suitably topologized by the Skorokhod metric? Finally, the authors treat the continuity of the defined addition in $$J(\mathbb{R}^p)$$.

##### MSC:
 54A40 Fuzzy topology
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##### References:
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