2 - Basics
Published online by Cambridge University Press: 20 August 2009
Summary
Forcing with ideals
The key definition
Definition 2.1.1.Suppose that X is a Polish space and I is a σ-ideal on the space X. The symbol PIdenotes the partial order of I-positive Borel sets ordered by inclusion.
I will always tacitly assume that the Polish space X is uncountable and the ideal I contains all singletons. There are several cases in which this will not hold, and they will be pointed out explicitly. Note that the poset PI depends only on the membership of Borel sets in the ideal I, but it will frequently be of interest to look at the membership of non-Borel sets in I.
It is clear that the partial order PI is not separative, and its separative quotient is the σ-algebra B(X) mod I. There is exactly one property all partial orders of this kind share.
Proposition 2.1.2.The poset PIadds an element ẋgenof the Polish space X such that for every Borel set B ⊂ X coded in the ground model, B ∈ G iff ẋgen ∈ B.
Proof. It is easy to see that the closed sets contained in the generic filter form a collection closed under intersection which contains sets of arbitrarily small diameter.
- Type
- Chapter
- Information
- Forcing Idealized , pp. 15 - 32Publisher: Cambridge University PressPrint publication year: 2008