Definable V-topologies, Henselianity and NIP

Definable and Invariant Types in Enrichments of NIP Theories

On Logical Characterization of Henselianity

We give some sufficient conditions under which any valued field that admits quantifier elimination in the Macintyre language is henselian. Then, without extra assumptions, we prove that if a valued field of characteristic (0, 0) has a Z-group as its value group and admits quantifier elimination in the main sort of the Denef-Pas style language LRRP then it is henselian. In fact the proof of this...

Stable embeddedness and NIP

We give some sufficient conditions for a predicate P in a complete theory T to be “stably embedded”. Let P be P with its “induced ∅definable structure”. The conditions are that P (or rather its theory) is “rosy”, P has NIP in T and that P is stably 1-embedded in T . This generalizes a recent result of Hasson and Onshuus [6] which deals with the case where P is o-minimal in T . Our proofs make u...

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 indepen...

On NIP and invariant measures

We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if p = tp(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over bdd...