There is no bound on Borel classes of graphs in the Luzin–Novikov theorem
Volume 576 / 2022
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 nonarithmetic 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.
