On projective mappings (Q1810193)

The main result of this paper is a straightforward application of the theory of topology refinements to Polish spaces. In modern notation (the author uses the notation of Kuratowski's 1966 book \textit{Topologie}), let \({\mathcal B}_n\) be the \(\sigma\)-algebra generated by the \(\pmb{\Sigma}^1_n\) sets (\textit{i.e.}, \({\mathcal B}_0\) is the standard Borel \(\sigma\)-algebra). The following is a classical result: Let \(\langle X,\tau_X\rangle\) and \(\langle Y,\tau_Y\rangle\) be Polish spaces, and \(f:X\to Y\) a \({\mathcal B}_0\)-measurable function. Then there is a Polish space \(\langle \tilde X,\tau\rangle\) such that (1) \(X\) is a \(\tau\)-closed subset of \(\tilde X\), (2) the subspace topology of \(\tau\) on \(X\) is a refinement of \(\tau_X\), and (3) \(f\) is continuous as a map from \(\langle X,\tau\rangle\) to \(\langle Y,\tau_Y\rangle\). We usually express this as ``a Borel function \(f:X\to Y\) can be made continuous by embedding \(X\) as a closed subset into a larger Polish space''. The author generalizes this to his main theorem (Theorem 1): a \({\mathcal B}_n\)-measurable function \(f:X\to Y\) can be made continuous by embedding \(X\) as a \({\mathcal B}_n\) subset into a larger Polish space. This should be seen in the context of standard results on Borel sets and projective sets (cf., e.g., \textit{A. S. Kechris}, Classical descriptive set theory, Graduate Texts in Mathematics. 156. Berlin: Springer-Verlag (1995; Zbl 0819.04002)]), and it should be compared with the following two unpublished theorems of H.~Becker (early 1990s): Call a subset \(A\) of a Polish space \textbf{\(n\)-refinable} if \(A\) is clopen in a topology refinement of the Polish topology that has weight \(\pmb{\delta}^1_n\) and is strong Choquet. Theorem A (Becker). If Projective Determinacy holds, then every \(\Sigma^1_n\) subset of a Polish space is \(n\)-refinable. Theorem B (Becker). If the Axiom of Real Determinacy holds, then every \(n\)-refinable subset of a Polish space is \(\Sigma^1_n\).