Submission date: 25 March 2017

Abstract:

In this short note, using results of Bourgain, Fremlin, and Talagrand [BFT], we show that for a countable structure M and a formula φ(x,y) the following are equivalent:
(i) φ(x,y) has NIP on M (see Definition below).
(ii) For every net (a_i)_{i\in I} of elements of M and a saturated elementary extension M^* of M, if tp_φ(a_i/M^*) → p', then the type p' is Baire 1 definable over M.

This implies, as it is well known, that if φ is NIP then every global M-invariant φ-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.

Mathematics Subject Classification:

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Full text arXiv 1703.08731: pdf, ps.