Enriched closure spaces as a novel framework for domain theory

We propose a generalization of continuous lattices and domains through the concept of enriched closure space, defined as a closure space equipped with a preclosure operator satisfying some compatibility conditions. In this framework we are able to define a notion of waybelow relation; an appropriate definition of continuity then naturally follows. Characterizations of continuity of the enriched closure space and necessary and sufficient conditions for the interpolation property are proved. We also draw a link between continuity and the possibility for the subsets that are open with respect to the preclosure operator to form a topology.