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

TL;DR

The paper demonstrates that for every countable ordinal, there exists a closed set in a product space with sections of size two that cannot be covered by countably many graphs of functions in any given Borel class, showing limits of a quantitative Luzin-Novikov theorem.

Contribution

It constructs specific closed sets that defy coverage by countably many Borel-class graphs, establishing a fundamental limit on the quantitative extension of the Luzin-Novikov theorem.

Findings

Existence of closed sets with two-point sections not coverable by countably many Borel-class graphs.

Demonstrates the impossibility of a quantitative Luzin-Novikov theorem.

Provides a method to construct sets excluding a quantitative Saint Raymond 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.