Does anyone know of a useful general topological application of the algebraic properties of the semiring of open subsets of some topological space? Or examples of any such nontrivial properties at all?

Perhaps less vaguely, I'm looking for an application of the methods and results of what is generally known as "commutative algebra" to the structure of this commutative associative unital semiring.

An extension of the Galois theory of Grothendieck. – Zhen Lin Aug 6 '12 at 10:32