Rosenthal compacta and NIP formulas

We apply the work of Bourgain, Fremlin and Talagrand on compact subsets of the first Baire class to show new results about φ-types for φ NIP. In particular, we show that if M is a countable model, then an M -invariant φ-type is Borel definable. Also the space of M invariant φ-types is a Rosenthal compactum, which implies a number of topological tameness properties. Shelah introduced the independence property (IP) for first order formulas in 1971 [13]. Some ten years later, Poizat [10] proved that a countable theory T does not have the independence property (is NIP) if and only if for any model M of T and type p ∈ S(M), p has at most 2 | coheirs (the bound a priori being 2 |M| ). Another way to state this result is to say that for any model M , the closure in S(M) of a subset of size at most κ has cardinality at most 2. Thus NIP is equivalent to a topological tameness condition on the space of types. At about the same time, Rosenthal [11] studied Banach spaces not embedding l1. He showed that a separable Banach space B does not contain a closed subspace isomorphic to l1 if and only if the unit ball of B is relatively sequentially compact in the bidual B, if and only if B has the same cardinality as B. Note that an element of B is by definition a function on B, the topology on B is that of pointwise convergence, and B, identified with a subset of B, is dense. Shortly after this work, Rosenthal [12] and then Bourgain, Fremlin and Talagrand [2] extended the ideas of this theorem and studied systematically the pointwise closure of subsets A of continuous functions on a Polish space. It turns out that there is a sharp dichotomy: either the closure Ā contains non-measurable functions or all functions in the ∗Partially supported by ValCoMo (ANR-13-BS01-0006) and by MSRI, Berkeley.