Forcing properties of ideals of closed sets

Abstract

With every $\sigma$-ideal $I$ on a Polish space we associate the $\sigma$-ideal $I^{\ast }$ generated by the closed sets in $I$. We study the forcing notions of Borel sets modulo the respective $\sigma$-ideals $I$ and $I^{\ast }$ and find connections between their forcing properties. To this end, we associate to a $\sigma$-ideal on a Polish space an ideal on a countable set and show how forcing properties of the forcing depend on combinatorial properties of the ideal. For $\sigma$-ideals generated by closed sets we also study the degrees of reals added in the forcing extensions. Among corollaries of our results, we get necessary and sufficient conditions for a $\sigma$-ideal $I$ generated by closed sets, under which every Borel function can be restricted to an $I$-positive Borel set on which it is either 1-1 or constant.