O-minimal open core is not an elementary property

TL;DR

This paper proves that having an o-minimal open core is not an elementary property by constructing a structure with an o-minimal open core whose elementary superstructure does not share this property.

Contribution

It demonstrates that o-minimal open core is not preserved under elementary equivalence through explicit construction.

Findings

Constructed a structure with o-minimal open core

Showed some elementary superstructures lack o-minimal open core

Answer to a question by Dolich, Miller, and Steinhorn

Abstract

Given a structure $\mathcal{M}$ with a definable topology, its open core is a structure defined on the same universe whose language consists of all open sets of all arities definable in $\mathcal{M}$. In response to questions raised by Dolich, Miller, and Steinhorn in their early work on open core, we prove that having an o-minimal open core is not an elementary property. In particular, we construct an expansion of the structure $(\mathbb{Q},<)$ that has an o-minimal open core, but some of its elementary superstructures do not.