Topological properties on the space of fuzzy sets
type: Item
instance of: scholarly article
title: Topological properties on the space of fuzzy sets
zbMATH Open document ID: 0986.54012
author: Sang Yeol Joo
author: Yun Kyong Kim
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-06-11T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 54A40
Mathematics Subject Classification ID: 03E72
zbMATH DE Number: 1496311
zbMATH Keywords: Skorokhod metric
zbMATH Keywords: Skorokhod topology
reviewed by: T. M. G. Ahsanullah
MaRDI profile type: Publication
full work available at URL: jmaa.2000.6820
OpenAlex ID: W1973516716
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DOI: 10.1006/JMAA.2000.6820
description: scientific article; zbMATH DE number 1496311