  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.
