A Topology for Galois Types in AECs

TL;DR

This paper introduces a topology for Galois types in abstract elementary classes, linking model-theoretic properties with topological features, and highlighting tameness as a separation principle.

Contribution

It develops a topological framework for Galois types in AECs, extending syntactic topologies and connecting model properties with topological concepts.

Findings

Topologies refine syntactic structures in first order logic.

Model-theoretic properties correspond to topological features.

Tameness is characterized as a topological separation property.

Abstract

We present a way of topologizing sets of Galois types over structures in abstract elementary classes with amalgamation. In the elementary case, the topologies thus produced refine the syntactic topologies familiar from first order logic. We exhibit a number of natural correspondences between the model-theoretic properties of classes and their constituent models and the topological properties of the associated spaces. Tameness of Galois types, in particular, emerges as a topological separation principle.