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Published online by Cambridge University Press: 20 August 2009
Ideals σ-generated by closed sets
This is the historically first class of ideals discovered to generate proper forcings.
Definition 4.1.1.A σ-ideal I is σ-generated by closed sets if every set in I is contained in an Fσset in I.
Several results of this section are in fact special cases of the results of Section 4.2 concerning the abstract porosity ideals. Nevertheless I decided to treat them separately here because the abstract porosity ideals form a much more complex family.
One feature, which sets this class of ideals apart from the other classes considered in this chapter, is that it is not invariant under different presentations. Every forcing of the form PI where I is an ideal on a Polish space X σ-generated by closed sets has a presentation PJ on the Baire space ωω such that J is σ-generated by closed sets. To see this note that the set X is a one-to-one continuous image of a closed subset C of the Baire space, X = f″C. The ideal J on ωω generated by ωω\C and the preimages of I-small sets is σ-generated by closed sets since the open set ωω\C is Fσ and f-preimages of closed sets are closed. Clearly PI is naturally isomorphic to PJ.
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