On the determination of fuzzy topological spaces and fuzzy neighbourhood spaces by their level-topologies
type: Item
instance of: scholarly article
title: On the determination of fuzzy topological spaces and fuzzy neighbourhood spaces by their level-topologies
zbMATH Open document ID: 0574.54004
DOI: 10.1016/0165-0114(84)90051-4
author: P. Wuyts
published in: Fuzzy Sets and Systems
publication date: 1984-01-01T00:00:00Z
review text: In J. Math. Anal. Appl. 64, 446-454 (1978; Zbl 0381.54004) \textit{R. Lowen} introduced for an arbitrary fuzzy topological space (X,\(\Delta)\) the family \(\{\iota_{\alpha}(\Delta)\); \(\alpha\in [0,1[\}\) of its level- topologies, and showed how properties of these topologies can sometimes be used to characterize properties of (X,\(\Delta)\). In Fuzzy Sets Syst. 7, 165-189 (1982; Zbl 0487.54008) he showed that in the case of a fuzzy neighbourhood space these level-topologies always form a descending chain. The first of these results raises the question to know in how far a fuzzy topological space is determined by the family of its level- topologies, while the second one already shows that, at least in the special case of fuzzy neighbourhood spaces, the family of the level- topologies cannot be given arbitrarily a priori. We give a fairly complete answer to the questions raised above. Given a family \({\mathcal F}=\{{\mathcal T}_{\alpha}\); \(\alpha\in [0,1[\}\) of topologies on a set X we give a necessary and sufficient condition under which there exists at least one fuzzy topology \(\Delta\) on X, having \({\mathcal F}\) as its family of level-topologies (i.e. such that \(\iota_{\alpha}(\Delta)={\mathcal T}_{\alpha}\) for each \(\alpha =[0,1[)\), and prove that the set of all these fuzzy topologies always has a maximum but in general no minimum. Further we give necessary and sufficient conditions under which there exists at least one fuzzy neighbourhood space having \({\mathcal F}\) as its family of level-topologies and we prove that in this case the solution is always unique and coincides with the above mentioned maximum solution.
Mathematics Subject Classification ID: 54A40
zbMATH DE Number: 3917204
zbMATH Keywords: fuzzy topological space
zbMATH Keywords: fuzzy neighbourhood space
zbMATH Keywords: family of level- topologies
MaRDI profile type: Publication
full work available at URL: 0165-0114(84)90051-4
OpenAlex ID: W4240947037
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description: scientific article; zbMATH DE number 3917204