  The main aim of this paper is to provide a description of neighbourhood
operators in finitely complete categories with finite coproducts and a proper
factorisation system such that the semilattice of admissible subobjects make a
distributive complete lattice. The equivalence between neighbourhoods,
Kuratowski interior operators and pseudo-frame sets is proved. Furthermore the
categories of internal neighbourhoods is shown to be topological. Regular
epimorphisms of categories of neighbourhoods are described and conditions
ensuring hereditary regular epimorphisms are probed. It is shown the category
of internal neighbourhoods of topological spaces is the category of
bitopological spaces, while in the category of locales every locale comes
equipped with a natural internal topology.
