# Pro-definability of spaces of definable types

## TL;DR

This paper establishes the pro-definability of spaces of definable types across various classical first-order theories, revealing a unifying property with geometric and model-theoretic implications.

## Contribution

It proves the pro-definability of definable types in multiple theories by demonstrating their uniform definability through stable embeddedness, extending previous results.

## Key findings

- Pro-definability of definable types in o-minimal theories
- Pro-definability in Presburger arithmetic and p-adically closed fields
- Identification of geometrically interpretable subspaces

## Abstract

We show pro-definability of spaces of definable types in various classical complete first order theories, including complete o-minimal theories, Presburger arithmetic, $p$-adically closed fields, real closed and algebraically closed valued fields and closed ordered differential fields. Furthermore, we prove pro-definability of other distinguished subspaces, some of which have an interesting geometric interpretation. Our general strategy consists in showing that definable types are uniformly definable, a property which implies pro-definability using an argument due to E. Hrushovski and F. Loeser. Uniform definability of definable types is finally achieved by studying classes of stably embedded pairs.