Cofinality of the Nonstationary Ideal

We show that the reduced cofinality of the nonstationary ideal NSκ on a regular uncountable cardinal κ may be less than its cofinality, where the reduced cofinality of NSκ is the least cardinality of any family F of nonstationary subsets of κ such that every nonstationary subset of κ can be covered by less than κ many members of F . For this we investigate connections of the various cofinalities of NSκ with other cardinal characteristics of κκ and we also give a property of forcing notions (called manageability) which is preserved in <κ–support iterations and which implies that the forcing notion preserves non-meagerness of subsets of κκ (and does not collapse cardinals nor changes cofinalities). 0. Introduction Let κ be a regular uncountable cardinal. For C ⊆ κ and γ ≤ κ, we say that γ is a limit point of C if ⋃ (C ∩ γ) = γ > 0. C is closed unbounded if C is a cofinal subset of κ containing all its limit points less than κ. A set A ⊆ κ is nonstationary if A is disjoint from some closed unbounded subset C of κ. The nonstationary subsets of κ form an ideal on κ denoted by NSκ. The cofinality of this ideal, cof(NSκ), is the least cardinality of a family F of nonstationary subsets of κ such that every nonstationary subset of κ is contained in a member of F . The reduced cofinality of NSκ, cof(NSκ), is the least cardinality of a family F ⊆ NSκ such that every nonstationary subset of κ can be covered by less than κ many members of F . This paper addresses the question of whether cof(NSκ) = cof(NSκ). Note that κ ≤ cof(NSκ) ≤ cof(NSκ) ≤ 2, so under GCH we have cof(NSκ) = cof(NSκ). Let κ2 be endowed with the κ–box product topology, 2 itself considered discrete. We say that a set W ⊆ κ2 is κ–meager if there is a sequence 〈Uα : α < κ〉 of dense open subsets of κ2 such that W ∩ ⋂ α<κ Uα = ∅. The covering number for the category of the space κ2, denoted cov(Mκ,κ), is the least cardinality of any collection X of Received by the editors March 3, 2003. 2000 Mathematics Subject Classification. Primary 03E05, 03E35, 03E55.