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## Forcing over the poset of nonempty open subsets of a nice topological space

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 at 7:57

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.