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.

Automated summary

This paper addresses several open questions regarding structures with o-minimal open core, presenting examples and constructions to illustrate concepts such as proper expansions, dense function graphs, and Skolem functions within the context of o-minimal structures.