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

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 the ‘almost and Baire 1 definability’ of types assuming NIP. We show that a formula φ(x, y) has the strict order property if and only if there is a convergent sequence of continuous functions on the space of φ-types such that its limit is not sequentially continuous. We deduce from this a theorem of Shelah and point out the correspondence between this theorem and the Eberlein-Šmulian theorem.