FORCING PROPERTIES OF IDEALS OF CLOSED SETS

With every sigma-ideal I on a Polish space we associate the sigma-ideal I* generated by the closed sets in I. We study the forcing notions of Borel sets modulo the respective sigma-ideals I and I* 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. We also study the I-I or constant property of sigma-ideals. i.e., the property that every Borel function defined on a Borel positive set can be restricted to a positive Borel set on which it either 1-1 or constant. We prove the following dichotomy: if I is a sigma-ideal generated by closed sets, then either the forcing P(I) adds a Cohen real, or else I has the 1-1 or constant property.