On the determination of fuzzy topological spaces and fuzzy neighbourhood spaces by their level-topologies
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.
scientific article; zbMATH DE number 3917204