Geodesic
On Projective Mappings
Matematičeskie zametki, Tome 72 (2002) no. 3, pp. 323-329.

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Let $X,Y$ be Polish spaces, and let $\mathscr B_k$ be the $\sigma $-algebra generated by the projective class $L_{2k+1}$. A mapping $f\colon X\mapsto Y$ is called $K$-projective if $f^{-1}(E)\in \mathscr B_k$ for any Borel subset $E\subset Y$. The following theorem is our main result: for any $k$-projective mapping $f\colon X\mapsto Y$ there exist a Polish space $\widetilde X_S$, a dense subset $X_S\in \mathscr B_k$, and two continuous mappings $f_0, i: \widetilde X_S\to Y$ such that i) $f_0|_{X_S}=f\circ i|_{X_S}$; ii) $i|_{X_S}$ is a bijection. 
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S. S. Gabrielyan. On Projective Mappings. Matematičeskie zametki, Tome 72 (2002) no. 3, pp. 323-329. http://geodesic.mathdoc.fr/item/MZM_2002_72_3_a0/

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