Forcing properties of ideals of closed sets

With every σ-ideal I on a Polish space we associate the σ-ideal generated by closed sets in I. We study the forcing notions of Borel sets modulo the respective σ-ideals 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. In a futher application, we show when does a hypersmooth equivalence relation admit a Borel I-positive independent set.