Pathological examples of structures with o‐minimal open core

This paper answers several open questions around structures with o‐minimal open core. We construct an expansion of an o‐minimal structure R by a unary predicate such that its open core is a proper o‐minimal expansion of 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.