Decomposing $\mathbf{\Pi}^1_1$ sets into closed sets - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T23:50:00Z http://mathoverflow.net/feeds/question/96639 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96639/decomposing-mathbf-pi1-1-sets-into-closed-sets Decomposing $\mathbf{\Pi}^1_1$ sets into closed sets Liang Yu 2012-05-11T03:26:26Z 2012-09-01T21:22:01Z <p>It is well known that every $\mathbf{\Pi}^1_1$-set is a union of $\aleph_1$-many Borel sets. I wonder whether it can be improved under certain reasonable set theory axioms assumption.</p> <p>For example, assuming $ZFC+CH$, then it is trivially true that every set is a union of $\aleph_1$-many closed sets. But this seems heavily depends on $CH$ since if $ZFC+\neg CH+MA$, then there is a lightface $\Pi^0_2$-set which cannot be a union of $\aleph_1$-many closed sets.</p> <p>So my question is: is it consistent with $ZFC+\neg CH$ that every $\mathbf{\Pi}^1_1$-set is a union of $\aleph_1$-many closed sets?</p> http://mathoverflow.net/questions/96639/decomposing-mathbf-pi1-1-sets-into-closed-sets/96775#96775 Answer by alephomega for Decomposing $\mathbf{\Pi}^1_1$ sets into closed sets alephomega 2012-05-12T14:49:07Z 2012-05-12T14:49:07Z <p>There is a theorem of my teacher Steve Jackson which says that assuming $ZFC + AD^{L(\mathbb{R})}$ every projective set is $\aleph_{\omega}$-Borel. So in particular this holds for $\Pi^1_1$ sets. The proof uses the theory of descriptions and every other technical tool from descriptive set theory (homogeneous trees, scales,...). Also, with respect to $MA$ and $CH$, $AD$ can't decide them, so maybe that result might be what you're looking for, I'm not sure. You can find the result in this survey of Jackson <a href="http://math.berkeley.edu/~steel/martin/jackson.martin.ps" rel="nofollow">"A survey of Determinacy"</a> somewhere in the end of the paper. </p>