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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Cofinality of the nonstationary ideal


Authors: Pierre Matet, Andrzej Roslanowski and Saharon Shelah
Journal: Trans. Amer. Math. Soc. 357 (2005), 4813-4837
MSC (2000): Primary 03E05, 03E35, 03E55
Published electronically: June 29, 2005
MathSciNet review: 2165389
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Abstract: We show that the reduced cofinality of the nonstationary ideal ${\mathcal N S}_\kappa$ on a regular uncountable cardinal $\kappa$ may be less than its cofinality, where the reduced cofinality of ${\mathcal N S}_\kappa$ is the least cardinality of any family ${\mathcal F}$ of nonstationary subsets of $\kappa$ such that every nonstationary subset of $\kappa$ can be covered by less than $\kappa$ many members of ${\mathcal F}$. For this we investigate connections of the various cofinalities of ${\mathcal N S}_\kappa$ with other cardinal characteristics of ${}^{\textstyle\kappa}\kappa$ and we also give a property of forcing notions (called manageability) which is preserved in ${<}\kappa$-support iterations and which implies that the forcing notion preserves non-meagerness of subsets of ${}^{\textstyle\kappa}\kappa$ (and does not collapse cardinals nor changes cofinalities).


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Additional Information

Pierre Matet
Affiliation: Departement de Mathématiques, Université de Caen – CNRS, BP 5186, 14032 Caen Cedex, France
Email: matet@math.unicaen.fr

Andrzej Roslanowski
Affiliation: Department of Mathematics, University of Nebraska at Omaha, Omaha, Nebraska 68182-0243
Email: roslanow@member.ams.org

Saharon Shelah
Affiliation: Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel – and – Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08854
Email: shelah@math.huji.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9947-05-04007-9
PII: S 0002-9947(05)04007-9
Keywords: Nonstationary ideal, cofinality
Received by editor(s): March 3, 2003
Published electronically: June 29, 2005
Additional Notes: The second author thanks the University Committee on Research of the University of Nebraska at Omaha for partial support. He also thanks his wife, Małgorzata Jankowiak–Rosłanowska, for supporting him when he was preparing the final version of this paper.
The research of the third author was partially supported by the Israel Science Foundation. Publication 799
Article copyright: © Copyright 2005 American Mathematical Society