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Forcing Properties of Ideals of Closed Sets

Published online by Cambridge University Press:  12 March 2014

Marcin Sabok  and
Jindřich Zapletal
Marcin Sabok
Affiliation:
Instytut Matematyczny Uniwersytetu Wrocławskiego, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland Instytut Matematyczny Polskiej Akademii Nauk, Ul. Śniadeckich 8, 00-956 Warszawa, Poland, E-mail: sabok@math.uni.wroc.pl
Jindřich Zapletal
Affiliation:
Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, CZ - 115 67 Praha 1, Czech Republic Department of Mathematics University of Florida, 358 Little Hall Po Box 118105 Gainesville, FL 32611-8105, USA, E-mail: zapletal@math.cas.cz
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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 P1 adds a Cohen real, or else I has the 1–1 or constant property.

Keywords

03E4003E1554H0526A21forcingidealsKatětov order
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 3 , September 2011 , pp. 1075 - 1095
Copyright
Copyright © Association for Symbolic Logic 2011

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