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On the determination of fuzzy topological spaces and fuzzy neighbourhood spaces by their level-topologies. (English) Zbl 0574.54004
In J. Math. Anal. Appl. 64, 446-454 (1978; Zbl 0381.54004) 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.

54A40 Fuzzy topology
Full Text: DOI
[1] Dubois, D; Prade, H, Fuzzy sets and systems: theory and applications, (1980), Academic Press New York · Zbl 0444.94049
[2] Lowen, R, Fuzzy topological spaces and fuzzy compactness, J. math. anal. appl., 56, 621-633, (1976) · Zbl 0342.54003
[3] Lowen, R, A comparison of different compactness notions in fuzzy topological spaces, J. math. anal. appl., 64, 446-454, (1978) · Zbl 0381.54004
[4] Lowen, R, Convergence in fuzzy topological spaces, Topology and appl., 10, 147-160, (1979) · Zbl 0409.54008
[5] Lowen, R, Fuzzy neighborhood spaces, Fuzzy sets and systems, 7, 165-189, (1982) · Zbl 0487.54008
[6] Negoita, C.V; Ralescu, D.A, Applications of fuzzy sets to systems analysis, (1975), Birkhäuser Verlag Basel · Zbl 0326.94002
[7] Wuyts, P; Lowen, R, On separation axioms in fuzzy topological spaces, fuzzy neighborhood spaces and fuzzy uniform spaces, J. math. anal. appl., 93, 27-41, (1983) · Zbl 0515.54004
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