Fuzzy topology on fuzzy sets: Fuzzy semicontinuity and fuzzy semiseparation axioms
\textit{C. L. Chang} [J. Math. Anal. Appl. 24, 182--190 (1968; Zbl 0167.51001)] introduced fuzzy topology as a pair \((X,\tau)\), where \(X\) is an ordinary nonempty set, and \(\tau\subseteq I^X\) \((I= [0,1])\) satisfies the usual conditions. In such a setting, the concepts of fuzzy semi-open and semi-closed sets and different separation axioms in terms of such sets, further semi-continuity of ordinary functions \(f: X\to Y\) where \(X\), \(Y\) are fts's, have been defined and studied by many researchers. Later, the concepts of fuzzy topology on a fuzzy set and a type of functions, called fuzzy proper functions, between two fuzzy sets were introduced. In the present paper, the authors give, in the usual way, the definitions of fuzzy semi-open and semi-closed sets in the latter setting. Some generalized forms of these definitions are also considered by using the generalized open and closed fuzzy sets of \textit{A. A. Allam} and \textit{A. M. Zahran} [Bull. Fac. Sci. Assiut Univ. A 14, No. 2, 81--91 (1986)]. Several separation axioms are then defined in terms of all these introduced concepts. The definition of semi-continuity of a fuzzy preopen function is also given in the paper. Finally, some routine verifications of some elementary properties concerning all these introduced concepts and a few relevant ones are done.
scientific article; zbMATH DE number 2081722