The Hausdorff fuzzy metric on compact sets
type: Item
instance of: scholarly article
title: The Hausdorff fuzzy metric on compact sets
zbMATH Open document ID: 1069.54009
author: Jesús Rodríguez-López
author: Salvador Romaguera
published in: Fuzzy Sets and Systems
publication date: 2004-11-23T00:00:00Z
review text: According to [\textit{A. George} and \textit{P. Veeramani}, Fuzzy Sets Syst. 64, 395--399 (1994; Zbl 0843.54014)], a fuzzy metric space is a triple \((X,M,*)\) where \(X\) is a set, \(*\) is a continuous \(t\)-norm and the fuzzy metric \(M\) is a mapping \(M: X\times X\times (0,\infty) \to [0,1]\) satisfying certain conditions. In the same paper the authors showed how this metric induces a topology \(\tau_M\) on the set \(X\). The authors of this paper define a fuzzy metric \(H_M\) on the set \(K(X)\) of all nonempty compact subsets of \((X,\tau_M)\), that is actually a fuzzy set \(H_M: K(X)\times K(X)\times (0,\infty) \to [0,1]\) which is a fuzzy analogue of the Hausdorff-Bourbaki metric, and study some properties of the corresponding space: such as completeness, completion and precompactness. A typical result states that the space is \((X,M,*)\) is precompact iff the space \((K(X),H_M,*)\) is precompact. Some examples are given. Possible applications of the obtained results are discussed.
Mathematics Subject Classification ID: 54A40
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 54B20
zbMATH DE Number: 2118379
zbMATH Keywords: Hausdorff fuzzy metric
zbMATH Keywords: fuzzy metric
zbMATH Keywords: continuous \(t\)-norm
zbMATH Keywords: complete
zbMATH Keywords: precompact
zbMATH Keywords: completable
reviewed by: Alexander Šostak
MaRDI profile type: Publication
full work available at URL: j.fss.2003.09.007
OpenAlex ID: W2005433363
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DOI: 10.1016/J.FSS.2003.09.007
description: scientific article; zbMATH DE number 2118379