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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.

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The semiring structure is not enough (you're ignoring the very fundamental property of closure under arbitrary unions). See en.wikipedia.org/wiki/Complete_Heyting_algebra . –  Qiaochu Yuan Aug 5 '12 at 23:09
Or lookup frames. –  Benjamin Steinberg Aug 5 '12 at 23:28
Good point! Does this mean that there are broad classes of spaces with pairwise isomorphic semirings? –  Igor Makhlin Aug 5 '12 at 23:32
Depends on how nasty you allow the spaces to be. The topological space is uniquely determined by its frame if the space is Hausdorff, or more generally, sober. –  Emil Jeřábek Aug 6 '12 at 10:17
The "commutative algebra" of frames and their "modules" is developed in Joyal and Tierney's memoir, An extension of the Galois theory of Grothendieck. –  Zhen Lin Aug 6 '12 at 10:32

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