  We consider the question of when an expansion of a topological structure has
the property that every open set definable in the expansion is definable in the
original structure. This question is related to and inspired by recent work of
Dolich, Miller and Steinhorn on the property of having o-minimal open core. We
answer the question in a fairly general setting and provide conditions which in
practice are often easy to check. We give a further characterisation in the
special case of an expansion by a generic predicate.
