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 ${\mathbf{\Sigma }}_{\alpha }^0$. 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 ${\Pi }_1^0$ 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.

References in corpus (1)

• Harrington's Solution to McLaughlin's Conjecture and Non-uniform Self-moduli