A model theoretic Baire category theorem for simple theories

TL;DR

This paper establishes a model theoretic Baire category theorem for certain sets in countable simple theories with the extension property, and applies it to classify minimal types in nfcp theories.

Contribution

It introduces a new Baire category theorem for $ ilde\tau_{low}^f$-sets and proves a trichotomy for minimal types in countable nfcp theories.

Findings

A Baire category theorem for $ ilde\tau_{low}^f$-sets in simple theories

A trichotomy classification for minimal types in nfcp theories

Identification of conditions for interpretability of groups or minimal formulas

Abstract

We prove a model theoretic Baire category theorem for $\tilde\tau_{low}^f$-sets in a countable simple theory in which the extension property is first-order and show some of its applications. We also prove a trichotomy for minimal types in countable nfcp theories: either every type that is internal in a minimal type is essentially-1-based by means of the forking topology or $T$ interprets an infinite definable 1-based group of finite $D$-rank or $T$ interprets a strongly-minimal formula.