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Is there anything sensible to be said concerning a notion of forcing given by the poset of nonempty open subsets of the sort of topological space that comes up in ($e.g.$ algebraic) topology? If so, are interesting topological properties somehow coded in the resulting forcing extennsion. For example, would ${\mathbb S}^1$ versus ${\mathbb S}^2$ (or the open interval versus the closed interval versus the Hawaiian earring) yield a detectable difference? I suppose what's really at issue is how much topological information is lost on passage to the complete Boolean algebra of regular open subsets: to what extent can a space be reconstructed from that structure?

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This question is in some sense equivalent to asking about the subtopos of $\lnot \lnot$-sheaves inside the topos of ordinary sheaves on a topological space. –  Zhen Lin Jan 9 '13 at 7:57
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up vote 5 down vote accepted

The property that you are describing is called coabsoluteness. In other words, two regular spaces are said to be coabsolute if their regular open algebras are isomorphic. In the paper, A Characterization of Coabsoluteness for a Class of Metric Spaces by Catherine Gates, theorem 2.3 says that two locally compact metric spaces $X$ and $Y$ are coabsolute if and only if $d(X)=d(Y)$. Here $d(X)$ denotes the density of $X$, i.e. the $d(X)$ is smallest cardinal such that there is a dense subset of $X$ of cardinality $d(X)$. In particular, the open interval, closed inverval, $S^{1}$, $S^{2}$ and the Hawaiian earring are all locally compact separable metric spaces, so they are all coabsolute. Therefore these spaces all have isomorphic regular open algebras.

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Cool! How about (not necessarily compact, not necessarily metrizable) CW complexes? –  Adam Epstein Jan 9 '13 at 0:46
    
@Adam Epstein, a more fruitful exercise would be to instead focus on infinite CW complexes and produce new ones generically, then ask just how much you can monkey around with the constructions of algebraic topology. –  Michael Blackmon Jan 9 '13 at 1:16
    
@Michael Blackmon "generically" as in "generic extension"? –  Adam Epstein Jan 9 '13 at 1:36
    
CW complexes are normal Hausdorff spaces. The coabsoluteness ruslts of Gates apply even more generally to all regular Hausdorff spaces. –  Adam Epstein Jan 9 '13 at 1:42
    
@Adam Epstein, yes and by infinite I mean infinite dimensional. –  Michael Blackmon Jan 9 '13 at 1:47
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