Subordinations, closed relations, and compact Hausdorff spaces

(compact Hausdorff zero-dimensional spaces) and continuous maps. De Vries [12] generalized Stone duality to the category of compact Hausdorff spaces and continuous maps. Objects of the dual category are complete Boolean algebras B with a binary relation ≺ (called by de Vries a compingent relation) satisfying certain conditions that resemble the definition of a proximity on a set [10]. Another generalization of Stone duality is central to modal logic. We recall that modal algebras are Boolean algebras B with a unary function 2: B → B preserving finite meets, and modal spaces (descriptive frames) are Stone spaces X with a binary relation R satisfying certain conditions. Stone duality then generalizes to a duality between the categories of modal algebras and modal spaces (see, e.g., [5, 9, 3]). The dual of a modal algebra (B,2) is the modal space (X,R), where X is the Stone dual of B (the space of ultrafilters of B), while the binary relation R ⊆ X×X is the Jónsson-Tarski dual of 2 [8]. Unlike the modal case, in de Vries duality we do not split the dual space of (B,≺) in two components, the Ston