Models of an Abstract Elementary Class as a Generalized Polish Space

TL;DR

The paper generalizes the topological perspective on model spaces from first-order theories to Abstract Elementary Classes by employing the Baldwin–Boney Relational Presentation Theorem. It defines a $G$-Polish topology on models of size $\kappa=LS(\mathbf{K})$ (and extends to any $\lambda\ge LS(\mathbf{K})$ under a set-theoretic assumption) via an expanded relational language that uniquely expands each model. The main result is that, under these conditions, the class of models forms a Generalized Polish Space in the sense of Generalized Descriptive Set Theory. This work connects AECs with generalized descriptive set theory and provides a framework for analyzing model-theoretic properties through topological and topometric methods, opening avenues for further study of atomicity and related notions in the AEC context.

Abstract

In first order logic, it is known that you can define a topology so that the countable models of some theory $T$ form a Polish Space (i.e. completely metrizable second countable space). In this paper we use the Baldwin- Boney Relational Presentation Theorem (from [3]; cf. 2.3) to generalize this result to the models of an Abstract Elementary Class (AEC). More specifically, we define a topology on the models of an AEC of size $λ\geq κ$, where $κ$ is the L$\ddot{o}$wenheim-Skolem number and $λ$ has to satisfy a set-theoretic assumption (see Section 4) and prove that these models form a Generalized Polish Space (i.e. a generalization of Polish Spaces i.e. completely $G$-metrizable space with weight $\leq κ$).

Theorems & Definitions (26)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Theorem 1.4
• Definition 1.5
• Theorem 1.6
• Theorem 1.7
• Definition 1.8
• Lemma 1.9
• Corollary 1.10