On stable Baire classes

We introduce and study adhesive spaces. Using this concept we obtain a characterization of stable Baire maps f : X → Y of the class α 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 → Y belongs to the n th stable Baire class if and only if there exist a sequence ( f k ) k = 1 ∞ of continuous maps f k : X → Y and a sequence ( F k ) k = 1 ∞ 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 : R → R is of the α th stable Baire class if and only if it belongs to the first stable Baire class.