LAUSR

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 is a complete lattice. The equivalence between neighbourhood systems, Kuratowski interior operations and pseudo-frame sets is proved. Furthermore the categories of internal neighbourhood spaces is shown to be topological. Regular epimorphisms of categories of internal neighbourhood spaces are described and conditions ensuring hereditary regular epimorphisms are probed. It is shown the category of internal neighbourhood spaces of topological spaces is the category of bitopological spaces, while in the category of locales every locale comes equipped with a natural internal topology.