Local Compactness and the Baire Category Theorem in Abstract Stone Duality

Abstract

Abstract Stone Duality is a re-axiomatisation of general topology intended to make it recursive. By turning the idea of the Scott topology on its head, notions that involve directed (infinitary) joins are reformulated using functions of higher type instead. Here we consider compactness and the way below relation ≪ used for continuous lattices. A new characterisation of local compactness is formulated in terms of an effective basis, i.e. one that comes with a dual basis. Any object that is definable in the monadic λ-calculus for Abstract Stone Duality (including the lattice structure and Scott principle) has such a basis. This is used to prove a form of Baire's category theorem, that, for any countable family of open dense subsets, the intersection is also dense.