Partial Orders realized by Prime Ideals on commutative rings The closed ordered subspaces of $\{0,1\}^{V}$ are precisely the Priestley spaces. Priestley duality states that the Priestley spaces are precisely the spaces of prime ideals on bounded distributive lattices. Furthermore, De Groot duality gives a Stone-type duality between the category of all Priestley spaces and the category of all Zariski topologies on commutative rings.

Are there always large discrete families of normal measures? The Stone space topology on $m(\kappa)$ is generated by a complete uniformity, namely the uniformity generated by the equivalence relation entourages $E_{f}$ where $(M,N)\in E_{f}$ iff $[f]_{N}=[f]_{M}.$ I do not know how this uniformity helps though.