FORCING PROPERTIES OF IDEALS OF CLOSED SETS

被引：12

作者：

Sabok, Marcin ^{[1
,2
]}

Zapletal, Jindrich ^{[3
,4
]}

机构：

[1] Uniwersytetu Wrociawskiego, Inst Matemat, PL-50384 Wroclaw, Poland

[2] Polskiej Akad Nauk, Inst Matemat, PL-00956 Warsaw, Poland

[3] Acad Sci Czech Republic, Inst Math, CZ-11567 Prague 11567, Czech Republic

[4] Univ Florida, Dept Math, Gainesville, FL 32611 USA

来源：

JOURNAL OF SYMBOLIC LOGIC | 2011年 / 76卷 / 03期

基金：

美国国家科学基金会;

关键词：

forcing; ideals; Katetov order;

D O I：

10.2178/jsl/1309952535

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

With every sigma-ideal I on a Polish space we associate the sigma-ideal I* generated by the closed sets in I. We study the forcing notions of Borel sets modulo the respective sigma-ideals I and I* and find connections between their forcing properties. To this end, we associate to a sigma-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 I-I or constant property of sigma-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 sigma-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.

引用

页码：1075 / 1095

页数：21