Journal of Symbolic Logic 76 (3):1075 - 1095 (2011)

Abstract
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 P I adds a Cohen real, or else I has the 1—1 or constant property
Keywords forcing   ideals   Katětov order
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DOI 10.2178/jsl/1309952535
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References found in this work BETA

Covering Analytic Sets by Families of Closed Sets.Sławomir Solecki - 1994 - Journal of Symbolic Logic 59 (3):1022-1031.
Closed Measure Zero Sets.Tomek Bartoszynski & Saharon Shelah - 1992 - Annals of Pure and Applied Logic 58 (2):93-110.
Set Theory.Thomas Jech - 1999 - Studia Logica 63 (2):300-300.

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Hypergraphs and Proper Forcing.Jindřich Zapletal - 2019 - Journal of Mathematical Logic 19 (2):1950007.

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