Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-05-05T13:22:47.976Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bartoszyński, Tomek and Judah, Haim, Set theory. On the structure of the real line, A K Peters. Wellesley, MA, 1995.Google Scholar
[2]Bartoszyński, Tomek and Shelah, Saharon, Closed measure zero sets. Annals of Pure and Applied Logic, vol. 58 (1992), pp. 93–110.CrossRefGoogle Scholar
[3]Hrušák, Michael, Combinatorics of filters and ideals, (2009), preprint.Google Scholar
[4]Jayne, John E. and Rogers, C. Ambrose, First level Borel functions and isomorphisms. Journal de Mathématiques Pures et Appliquées. Neuvième Série, vol. 61 (1982), no. 2, pp. 177–205.Google Scholar
[5]Jech, Thomas, Set theory. Academic Press, San Diego, 1978.Google Scholar
[6]Katětov, Miroslav, Products of filters, Commentationes Mathematicae Universitatis Carolinae. vol. 9 (1968), pp. 173–189.Google Scholar
[7]Kechris, Alexander, Louveau, Alain, and Woodin, Hugh, The structure of a-ideals of compact sets. Transactions of American Mathematical Society, vol. 301 (1987), pp. 263–288.Google Scholar
[8]Kechris, Alexander S., Classical descriptive set theory. Springer Verlag, New York, 1994.Google Scholar
[9]Meza-Alcántara, David, Ideals andfilters on countable sets, Ph.D. thesis, Universidad Nacional Autónoma Méxieo, México, 2009.Google Scholar
[10]Sabok, Marcin, Forcing, games and families of closed sets, Transactions of the American Mathematical Society, to appear.Google Scholar
[11]Solecki, Sławomir, Covering analytic sets by families of closed sets, this Journal, vol. 59 (1994), pp. 1022–1031.Google Scholar
[12]Solecki, Slawomir, Decomposing Borel sets and functions and the structure of Baire class 1 functions. Journal of American Mathematical Society, vol. 11 (1998), pp. 521–550.CrossRefGoogle Scholar
[13]Solecki, Sławomir, Filters and sequences, Fundamenta Mathematicae, vol. 163 (2000), pp. 215–228.CrossRefGoogle Scholar
[14]Zapletal, Jindřich, Forcing idealized, Cambridge Tracts in Mathematics 174, Cambridge University Press, Cambridge, 2008.CrossRefGoogle Scholar
[15]Zapletal, Jindřich, Preserving P-points in definable forcing, Fundamenta Mathematicae, vol. 204 (2009), no. 2, pp. 145–154.CrossRefGoogle Scholar