Rosenthal compacta and NIP formulas

Abstract

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 Minvariant 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 |M | coheirs (the bound a priori being 2 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 κ . 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 l 1 . He showed that a separable Banach space B does not contain a closed subspace isomorphic to l 1 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 closureĀ contains non-measurable functions or all functions in the * Partially supported by ValCoMo (ANR-13-BS01-0006) and by MSRI, Berkeley.

Remarks on the NIP in a model

“…Again we stick with the set up in § 1.2. We make use of a theorem from in addition to [, Theorem 2F] to obtain: Proposition Suppose that has the NIP in A where A has cardinality at most κ. Then the cardinality of the set of such that p is finitely satisfiable in A is at most 2 κ .…”

“…We are working again in the model theory context of , A , , , where we identify A with the collection of continuous functions from X to given by formulas for a in A . Following [, Definition 2.1], is the set of ‐valued functions f on X such that has empty interior, and is the set of ‐valued functions f on X such that for every nonempty closed subset Y of X . Now [, Theorem 2F(vi)] says that has the NIP in A .…”

“…And part (iii) of Theorem 2F says that for every “Čech‐complete” subset Y of X , is relatively compact in . (We are here using the fact that A consists of ‐valued functions as well as the compatibility of [, Definition 1A] with [, Definition 2.1] that we have above.) As every closed subset Y of X is Čech‐complete (cf.…”

“…But then . Hence by definition, We now want to apply [, Theorem 2.3]. The assumption that X is a compact Hausdorff 0‐dimensional space with a basis of size at most κ is satisfied.…”

“…If has the NIP in A (in the sense of Definition below) then part (v) of Corollary says that when A is countable then φ‐types over A have at most continuum many global coheirs. It is natural to ask what happens when A is not necessarily countable, and Proposition gives the result, using additional results from .…”