Abstract

We study and characterize stability, the negation of the independence property (NIP) and the negation of the strict order property (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 the NIP in continuous logic. Using a result of Bourgain, Fremlin, and Talagrand, we prove almost definability and Baire 1 definability of coheirs assuming the NIP. We show that a formula 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 continuous. We deduce from this a theorem of Shelah and point out the correspondence between this theorem and the Eberlein‐Šmulian theorem.