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Published online by Cambridge University Press: 20 August 2009
Continuous reading of names
The continuous reading of names (CRN for short) is one of the more common and more slippery properties of forcing notions of the form PI.
Definition 3.1.1.Suppose that I is a σ-ideal on a Polish space X such that the forcing PIis proper. The forcing PIis said to have continuous reading of names (CRN) if for every Borel function f : B → 2ωwith an I-positive Borel domain B ⊂ X there is an I-positive Borel set C ⊂ B such that the function f • C is continuous.
How common this property is will be obvious from the list of examples below, and anyone acquianted with the combinatorial approach to forcing can appreciate it immediately. The slippery part is that the continuous reading of names may (or then again, may not) depend on the presentation of the forcing PI. Note that extending the topology on the underlying Polish space without changing the Borel structure will make more functions continuous and so it can bring about the continuous reading of names in a forcing in which it was originally not present. This actually happens in the case of Steprāns forcing of Section 4.2.3. One simple observation is that every forcing with the CRN has a presentation on the Baire space with the CRN.
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