Which Distributive Lattices are Lattices of Open Sets of P-Spaces?

摘要

Representing lattices by topologies has been studied to a great extent. In this paper, we prove that a complete lattice L is isomorphic to the lattice of open subsets of a P-space iff L is order generated by its countably prime elements. We establish the dual equivalence between the category of complete lattices order generated by their countably prime elements with morphisms preserving arbitrary sups and countable infs, and the category of countably sober P-spaces with morphisms of continuous maps. Finally, we show that the category of countably sober P-spaces is reflective in the category of P-spaces.