Topology of interval-valued fuzzy sets
Following the original notion of fuzzy set as introduced by \textit{L. A. Zadeh} [Inf. Control 8, 338-353 (1965; Zbl 0139.24606)] the present authors introduce a new function \(A:X\to D[0,1]\) and term it as an interval-valued fuzzy set (IVF) where \(D[0,1]\) denotes the set of all closed subintervals of the unit interval \([0,1]\). Some elementary facts are established in order to define the notion of topology of IVF. Having defined the notion of topology the authors discuss various basic results from classical topology. It is observed that the class of all such topological spaces and continuous functions form a topological category. There is a set back in this new setting, that is, in case of product topology of IVF the projection maps are not necessarily open. This is however removed by adding all constant interval-valued fuzzy sets in the notion of topology which is in the line of \textit{R. Lowen}'s fuzzy topology [J. Math. Anal. Appl. 56, 621-633 (1976; Zbl 0342.54003)]. Finally, the authors introduce the notion of topology on IVF just following the notion of fuzzy topology on fuzzy sets as introduced by \textit{M. K. Chakraborty} and \textit{T. M. G. Ahsanullah} [Fuzzy Sets Syst. 45, No. 1, 103-108 (1992; Zbl 0754.54004)]. In this direction some results are achieved.
scientific article; zbMATH DE number 1275949