Some questions about Zadeh's extension on metric spaces

Abstract

For a continuous function f on a Hausdorff space X , we prove that [ ̂ f(u) ] α = f(u α ) for each u ∈ F (X) and α ∈[ 0 , 1 ] , where ̂ f 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 d 0 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 ̂ f to ( F (X), d 0 ) is also continuous and that (X, d) is separable if and only if ( F (X), d 0 ) is separable. We also present a fuzzy version of the so-called Hutchinson operator, a valuable tool in fractal theory.