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2 - Basics

Published online by Cambridge University Press:  20 August 2009

Jindrich Zapletal
Affiliation:
University of Florida
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Summary

Forcing with ideals

The key definition

Definition 2.1.1.Suppose that X is a Polish space and I is a σ-ideal on the space X. The symbol PIdenotes the partial order of I-positive Borel sets ordered by inclusion.

I will always tacitly assume that the Polish space X is uncountable and the ideal I contains all singletons. There are several cases in which this will not hold, and they will be pointed out explicitly. Note that the poset PI depends only on the membership of Borel sets in the ideal I, but it will frequently be of interest to look at the membership of non-Borel sets in I.

It is clear that the partial order PI is not separative, and its separative quotient is the σ-algebra B(X) mod I. There is exactly one property all partial orders of this kind share.

Proposition 2.1.2.The poset PIadds an elementgenof the Polish space X such that for every Borel set BX coded in the ground model, BG iffgenB.

Proof. It is easy to see that the closed sets contained in the generic filter form a collection closed under intersection which contains sets of arbitrarily small diameter.

Type
Chapter
Information
Forcing Idealized , pp. 15 - 32
Publisher: Cambridge University Press
Print publication year: 2008

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  • Basics
  • Jindrich Zapletal, University of Florida
  • Book: Forcing Idealized
  • Online publication: 20 August 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511542732.002
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  • Basics
  • Jindrich Zapletal, University of Florida
  • Book: Forcing Idealized
  • Online publication: 20 August 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511542732.002
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Basics
  • Jindrich Zapletal, University of Florida
  • Book: Forcing Idealized
  • Online publication: 20 August 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511542732.002
Available formats
×