NIP formulas and Baire 1 definability

Karim Khanaki

Published 2017-03-25Version 1

In this short note, using results of Bourgain, Fremlin, and Talagrand \cite{BFT}, we show that for a countable structure $M$ and a formula $\phi(x,y)$ the following are equivalent: (i) $\phi(x,y)$ has NIP on $M$ (see Definition~\ref{NIP-formula} below). (ii) For every net $(a_i)_{i\in I}$ of elements of $M$ and a saturated elementary extension $M^*$ of $M$, if $tp_\phi(a_i/M^*)\to p'$, then the type $p'$ is Baire 1 definable over $M$. This implies, as it is well known, that if $\phi$ is NIP then every global $M$-invariant $\phi$-type is Baire 1 definable over $M$. Then, we point out that Poizat's result about the numbers of coheirs of types in NIP theories holds in the framework of continuous logic.