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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.

##### MSC:
 54A40 Fuzzy topology
Full Text:
##### References:
 [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), BirkhĂ¤user 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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