Stability, NIP, and NSOP; Model Theoretic Properties of Formulas via Topological Properties of Function Spaces

Abstract: We study and characterize stability, NIP and NSOP in terms of topological and measure theoretical properties of classes of functions. We study a measure theoretic property, Talagrand's stability', and explain the relationship between this property and NIP in continuous logic. Using a result of Bourgain, Fremlin and Talagrand, we prove thealmost definability' and `Baire~1 definability' of coheirs assuming NIP. We show that a formula $\phi(x,y)$ has the strict order property if and only if there is a convergent sequence of continuous functions on the space of $\phi$-types such that its limit is not continuous. We deduce from this a theorem of Shelah and point out the correspondence between this theorem and the Eberlein-\v{S}mulian theorem.