Priestley space

Summary

A Priestley space is a compact topological space with a partial order in which any two points $x \not\le y$ are separated by a clopen upper set. Priestley spaces are the duals of bounded distributive lattices in Priestley duality. Forgetting the order, every Priestley space is a Stone space, and a Stone space with the equality order is a Priestley space. So Priestley spaces extend Stone duality from Boolean algebras to distributive lattices.