Forcing Properties of Ideals of Closed Sets

Abstract

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 1 adds a Cohen real, or else I has the 1–1 or constant property.