A fuzzy version of Tietze's extension theorem

Abstract

The following theorem generalizing Tietze's Extension Theorem for real-valued functions is proved. Theorem. Let (X,(tau)) be an L-fuzzy topological space. If X is normal and fuzzy subsets K and K (WEDGE) K' are closed in X, and f : X (--->) {0.1}(L) is an L-fuzzy continuous function on K (LESSTHEQ) X , then there exists F : X (--->) {0,1}(L), an L-fuzzy continuous extension of f to X , such that (F(VBAR)K)('-1)(0+) (LESSTHEQ) F(0+) (LESSTHEQ) F(1-) (LESSTHEQ) (F(VBAR)K)('-1)(1-) (V) K'. Throughout the paper the conceptual difficulties of generalizing standard topological terms to L-fuzzy topological terms are discussed. In particular, a theory of relative topologies and relative functions for L-fuzzy topological spaces is developed. The extension of a relative L-fuzzy continuous function into the fuzzy unit interval is defined. The equivalence of L-fuzzy continuous functions and monotone families of open sets is proved. This equivalence is exploited to establish a fuzzy version of Tietze's Extension Theorem. A partial converse to the theorem is proved.