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