September 2011 Forcing properties of ideals of closed sets
Marcin Sabok, Jindřich Zapletal
J. Symbolic Logic 76(3): 1075-1095 (September 2011). DOI: 10.2178/jsl/1309952535


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


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Marcin Sabok. Jindřich Zapletal. "Forcing properties of ideals of closed sets." J. Symbolic Logic 76 (3) 1075 - 1095, September 2011.


Published: September 2011
First available in Project Euclid: 6 July 2011

zbMATH: 1245.03076
MathSciNet: MR2849260
Digital Object Identifier: 10.2178/jsl/1309952535

Primary: 03E15 , 03E40 , 26A21 , 54H05

Keywords: Forcing , Ideals , Katětov order

Rights: Copyright © 2011 Association for Symbolic Logic


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Vol.76 • No. 3 • September 2011
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