## Journal of Symbolic Logic

### Forcing properties of ideals of closed sets

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

#### Article information

Source
J. Symbolic Logic, Volume 76, Issue 3 (2011), 1075-1095.

Dates
First available in Project Euclid: 6 July 2011

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1309952535

Digital Object Identifier
doi:10.2178/jsl/1309952535

Mathematical Reviews number (MathSciNet)
MR2849260

Zentralblatt MATH identifier
1245.03076

Keywords
forcing ideals Katětov order

#### Citation

Sabok, Marcin; Zapletal, Jindřich. Forcing properties of ideals of closed sets. J. Symbolic Logic 76 (2011), no. 3, 1075--1095. doi:10.2178/jsl/1309952535. https://projecteuclid.org/euclid.jsl/1309952535