Processing math: 0%

Wykorzystujemy pliki cookies aby ułatwić Ci korzystanie ze strony oraz w celach analityczno-statystycznych.

A+ CATEGORY SCIENTIFIC UNIT

PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

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

Volume 576 / 2022

Petr Holický, Miroslav Zelený Dissertationes Mathematicae 576 (2022), 1-77 MSC: Primary 03E15; Secondary 28A05, 54H05. DOI: 10.4064/dm831-11-2021 Published online: 24 March 2022

Abstract

We show that for every ordinal 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.

Authors

  • Petr HolickýCharles University
    Faculty of Mathematics and Physics
    Department of Mathematical Analysis
    Sokolovská 83
    Praha 8, 186 75
    Czech Republic
    e-mail
  • Miroslav ZelenýCharles University
    Faculty of Mathematics and Physics
    Department of Mathematical Analysis
    Sokolovská 83
    Praha 8, 186 75
    Czech Republic
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image