Abstract
We show that for every ordinal alpha E [1, omega 1) there is a closed set F* C 2" x omega" such that for every x E 2" the section {y E omega"; (x, y) E F*} is a two-point set and F* cannot be covered by countably many graphs B(n) C 2" x omega" of functions of the variable x E 2" such that each B(n) is in the additive Borel class sigma 0a. 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 01 set in 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.