2010 seminar talk: A dichotomy for σ-ideals generated by closed sets
Talk held by Marcin Sabok (Instytut Matematyczny Uniwersytetu Wrocławskiego, Wrocław, Poland, Instytut Matematyczny Polskiej Akademii Nauk, Warszawa, Poland, and KGRC) at the KGRC seminar on 2010-11-11.
We say that a σ-ideal I on a Polish space X has the "1-1 or constant" property if every Borel function defined on a Borel I-positive subset of X can be restricted to a Borel I-positive set, on which it is 1-1 or constant. In other words, this means that the forcing PI adds a minimal real degree. The classical well-known examples of σ-ideals with the 1-1 or constant property are the σ-ideal of countable sets (Sacks forcing) and the σ-ideal of σ-compact subsets of the Baire space (Miller forcing). On the other hand, an example for which this property drastically does not hold is the σ-ideal of meager sets (or any I for which PI adds a Cohen real). During the talk I will prove the following dichotomy: if I is a σ-ideal generated by closed sets, then
(i) either I has the 1-1 or constant property,
(ii) or else PI adds a Cohen real.
This is joint work with Jindra Zapletal.