Pathological examples of structures with o-minimal open core

TL;DR

This paper explores complex examples of structures with o-minimal open core, demonstrating various properties and limitations, including expansions, dense functions, and definability issues.

Contribution

It constructs specific structures with o-minimal open core that exhibit novel behaviors, answering open questions in the field.

Findings

Constructed an expansion with a proper o-minimal open core.

Provided an example with dense graph of a definable function.

Showed a structure with Skolem functions that is not o-minimal.

Abstract

This paper answers several open questions around structures with o-minimal open core. We construct an expansion of an o-minimal structure $\mathcal{R}$ by a unary predicate such that its open core is a proper o-minimal expansion of $\mathcal{R}$. We give an example of a structure that has an o-minimal open core and the exchange property, yet defines a function whose graph is dense. Finally, we produce an example of a structure that has an o-minimal open core and definable Skolem functions, but is not o-minimal.