On stable Baire classes

TL;DR

This paper characterizes stable Baire maps using adhesive spaces and provides conditions under which functions belong to specific stable Baire classes, including a characterization for monotone real functions.

Contribution

It introduces the concept of adhesive spaces to characterize stable Baire maps and establishes a criterion for functions to belong to stable Baire classes, especially for monotone functions.

Findings

Characterization of stable Baire maps via adhesive spaces.

Equivalence of stable Baire class membership for monotone functions.

Conditions involving sequences of continuous maps and functionally ambiguous sets.

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.