Paper detail

Forcing properties of ideals of closed sets

With every $σ$-ideal $I$ on a Polish space we associate the $σ$-ideal $I^*$ generated by the closed sets in $I$. We study the forcing notions of Borel sets modulo the respective $σ$-ideals $I$ and $I^*$ and find connections between their forcing properties. To this end, we associate to a $σ$-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 $σ$-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 $σ$-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.