Exercise 2.I of J. Kelley's book General Topology introduces a theorem by M.H. Stone concerning maximal ideals in distributive lattices. The statement is the following :
Let $A$ and $B$ be disjoint subsets of a distributive lattice $X$ such that $A$ is an ideal and $B$ is a dual ideal.
Then there are disjoint sets $A'$ and $B'$ such that $A'$ is an ideal containing $A$, $B'$ is a dual ideal containing $B$, and $X=A' \cup B'$.
Kelley writes that Stone's theorem is "the best form of one of the basic facts about ordered sets."
Stone's original proof is actually hard to follow, but I am interested in applications of his result. For instance, Exercise 2.J of Kelley's book uses Stone's theorem to prove that any net has a subnet which is universal. Even the proof of Tychonoff's theorem about compact spaces can be based on Stone's theorem. Unfortunately, I cannot find any book or expository paper in which this approach based on ordered sets is used for proving results in general topology. Does anybody know a reference?