Interpretable sets in dense o-minimal structures

TL;DR

This paper constructs a dense o-minimal structure with an interpretable set that cannot be parametrically bijected with any definable set, and shows that interpretable sets have well-behaved, tame topological properties.

Contribution

It provides a counterexample to a question about interpretable sets in dense o-minimal structures and establishes their tame topological characteristics.

Findings

Existence of a definable quotient that cannot be eliminated

Interpretable sets admit Hausdorff, definably homeomorphic neighborhoods

Definable functions are piecewise continuous and sets have finitely many connected components

Abstract

We give an example of a dense o-minimal structure in which there is a definable quotient that cannot be eliminated, even after naming parameters. Equivalently, there is an interpretable set which cannot be put in parametrically definable bijection with any definable set. This gives a negative answer to a question of Eleftheriou, Peterzil, and Ramakrishnan. Additionally, we show that interpretable sets in dense o-minimal structures admit definable topologies which are "tame" in several ways: (a) they are Hausdorff, (b) every point has a neighborhood which is definably homeomorphic to a definable set, (c) definable functions are piecewise continuous, (d) definable subsets have finitely many definably connected components, and (e) the frontier of a definable subset has lower dimension than the subset itself.