Some questions about Zadeh's extension on metric spaces

Abstract

For a continuous function f on a Hausdorff space X, we prove that [fˆ(u)]α=f(uα) for each u∈F(X) and α∈[0,1], where fˆ is the Zadeh's extension of f. By means of this result, some results on (locally) compact spaces and the Zadeh's extension are generalized. Given a metric space (X,d), we introduce Skorokhod's metric d0 on the set F(X) of the family of all upper semicontinuous fuzzy sets u:X→[0,1] with compact support and such that u−1(1) is non-empty. We show that if f:(X,d)→(X,d) is a continuous function, then its Zadeh's extension fˆ to (F(X),d0) is also continuous and that (X,d) is separable if and only if (F(X),d0) is separable. We also present a fuzzy version of the so-called Hutchinson operator, a valuable tool in fractal theory.