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 $n$'th 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 $n$'th 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.