Forcing over the poset of nonempty open subsets of a nice topological space - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T01:54:07Z http://mathoverflow.net/feeds/question/118407 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118407/forcing-over-the-poset-of-nonempty-open-subsets-of-a-nice-topological-space Forcing over the poset of nonempty open subsets of a nice topological space Adam Epstein 2013-01-08T23:58:13Z 2013-01-09T00:43:43Z <p>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?</p> http://mathoverflow.net/questions/118407/forcing-over-the-poset-of-nonempty-open-subsets-of-a-nice-topological-space/118409#118409 Answer by Joseph Van Name for Forcing over the poset of nonempty open subsets of a nice topological space Joseph Van Name 2013-01-09T00:38:46Z 2013-01-09T00:38:46Z <p>The property that you are describing is called <em>coabsoluteness</em>. 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.</p>