There is no bound on Borel classes of the graphs in the Luzin-Novikov theorem

Abstract

We show that for every ordinal $\alpha \in [1, \omega_1)$ there is a closed set $F \subset 2^\omega \times \omega^\omega$ such that for every $x \in 2^\omega$ the section $\{y\in \omega^\omega; (x,y) \in F\}$ is a two-point set and $F$ cannot be covered by countably many graphs $B(n) \subset 2^\omega \times \omega^\omega$ of functions of the variable $x \in 2^\omega$ such that each $B(n)$ is in the additive Borel class $\boldsymbol \Sigma^0_\alpha$. This rules out the possibility to have a quantitative version of the Luzin-Novikov theorem. The construction is a modification of the method of Harrington who invented it to show that there exists a countable $\Pi^0_1$ set in $\omega^\omega$ containing a non-arithmetic singleton. By another application of the same method we get closed sets excluding a quantitative version of the Saint Raymond theorem on Borel sets with $\sigma$-compact sections.