On stable Baire classes

Abstract

We introduce and study adhesive spaces. Using this concept we obtain a characterization of stable Baire maps ${f : X\to Y}$ of the class ${\alpha}$ for wide classes of topological spaces. In particular, we prove that for a topological space X and a contractible space Y a map ${f : X \to Y}$ belongs to the nth stable Baire class if and only if there exist a sequence ${(f_k)_{k=1}^\infty}$ of continuous maps ${f_k : {X \to Y}}$ and a sequence ${(F_k)_{k=1}^\infty}$ of functionally ambiguous sets of the nth class in X such that ${f|_{F_k}=f_k|_{F_k}}$ for every k. Moreover, we show that every monotone function ${f : \mathbb{R} \to \mathbb{R}}$ is of the ${\alpha}$ th stable Baire class if and only if it belongs to the first stable Baire class.