Cofinality of the nonstationary ideal

Matet, Pierre; Rosłanowski, Andrzej; Shelah, Saharon

doi:10.1090/S0002-9947-05-04007-9

Cofinality of the nonstationary ideal
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by Pierre Matet, Andrzej Rosłanowski and Saharon Shelah PDF

Trans. Amer. Math. Soc. 357 (2005), 4813-4837 Request permission

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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