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

TL;DR

This paper links model-theoretic properties like stability, NIP, and NSOP to topological and measure-theoretic features of function spaces, providing new characterizations and insights into their structure.

Contribution

It introduces topological and measure-theoretic characterizations of stability, NIP, and NSOP, connecting model theory with functional analysis and topology.

Findings

Characterizes stability, NIP, and NSOP via topological properties of function spaces.

Proves almost definability and Baire~1 definability of coheirs assuming NIP.

Links the strict order property to convergence of functions and relates it to classical theorems.

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 the `almost 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.