Subspaces of smooth fuzzy topologies and initial smooth fuzzy structures (Q1304253)

The concept of a smooth fuzzy topology [\textit{A. A. Ramadan}, ibid. 48, No. 3, 371-375 (1992; Zbl 0783.54007)] has been introduced as a ``second order'' fuzzification of an ordinary topology and a ``first order'' generalization of a fuzzy topology in the original sense of \textit{C. L. Chang} [J. Math. Anal. Appl. 24, 182-190 (1968; Zbl 0167.51001)]. Later on \textit{R. Lowen} [ibid. 56, 621-633 (1976; Zbl 0342.54003)] has modified Chang's structure by adding all the constants to the fuzzy topology. In this paper the author describes a modification of Ramadan's concept of a smooth fuzzy topology according to Lowen's modification of Chang's concept. After repeating the construction process of a smooth fuzzy topology generated by a given gradation mapping, the author gives a (corrected) proof of a construction process for a subspace of a (modified) smooth fuzzy topology. Further, the relation(s) between a (modified) smooth fuzzy topology and the weak and strong \(\alpha\)-cuts are studied. Next, the concept of smooth fuzzy continuity of a mapping between two smooth fuzzy topological spaces is introduced and studied with respect to the decomposition of this structure using \(\alpha\)-cuts. Finally, initial smooth fuzzy structures are constructed and it is shown that this initial structure commutes with the decomposition of a smooth fuzzy topology into fuzzy topologies in Lowen's sense.