On Projective Mappings

Abstract

Let X,Y be Polish spaces, and let $\mathcal{B}_k$ be the σ-algebra generated by the projective class $L_{2k + 1}$. A mapping $f:X \mapsto Y$ is called $k$-projective if $f^{ - 1} (E) \in \mathcal{B}_k$ for any Borel subset E ⊂ Y. The following theorem is our main result: for any $k$-projective mapping $f:X \mapsto Y$ there exist a Polish space $\tilde X_s$, a dense subset $X_s \in \mathcal{B}_k$, and two continuous mappings $f_0 ,i:\tilde X_s \to Y$ such that i)

$${\text{i) }}f_0 {\text{|}}X_s = f \circ i|X_s ;$$

ii)

$$i|X_s {\text{ is a bijection}}{\text{.}}$$

.