Journal of Symbolic Logic

Forcing properties of ideals of closed sets

Marcin Sabok and Jindřich Zapletal

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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

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

First available in Project Euclid: 6 July 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E40: Other aspects of forcing and Boolean-valued models 03E15: Descriptive set theory [See also 28A05, 54H05] 54H05: Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets) [See also 03E15, 26A21, 28A05] 26A21: Classification of real functions; Baire classification of sets and functions [See also 03E15, 28A05, 54C50, 54H05]

forcing ideals Katětov order


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.

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