Forcing properties of ideals of closed sets

Abstract

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. We also study the 1—1 or constant property of σ-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 σ-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.