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Background: I'm doing some expository writing on intuitionistic logic and I have been toying with the idea of demonstrating its applicability via models where the denotations are taken from a Heyting algebra. A topology is nothing more than a complete Heyting algebra of sets, so we have a natural and rich source of Heyting algebras to build models of intuitionistic propositional logic with. The Sierpiński space, for example, is already sufficient to falsify double negation elimination and some other tautologies of classical logic.

Dually, Kuratowski's result for the complement–closure problem can be viewed as a reflection of the fact that $\lnot \lnot \lnot p \dashv \vdash \lnot p$ is valid intuitionistically. In particular, this proves that $A^{\circ c \circ c \circ c \circ} = A^{\circ c \circ}$, since $B^{c \circ}$ is precisely the Heyting pseudocomplement of $B$ in the Heyting algebra of open sets.

Question: Are there similar applications for the cofinite topology in particular? I was hoping to give an example of how intuitionistic logic could be used to reason about "almost-everywhere" truths, but I haven't been able to figure out anything non-trivial.

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I'm not sure what you mean by either "similar" or "application" here. –  Qiaochu Yuan Jun 10 '11 at 16:51
    
@Qiaochu: I could elaborate on the two examples I mentioned, but I get the feeling that's not the problem? –  Zhen Lin Jun 10 '11 at 17:15

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