Finite Combinations of Baire Numbers

Let κ be a regular cardinal. Consider the Baire numbers of the spaces (2)κ for various θ ≥ κ. Let l be the number of such different Baire numbers. Models of set theory with l = 1 or l = 2 are known and it is also known that l is finite. We show here that if κ > ω, then l could be any given finite number. The Baire number of a topological space with no isolated points is the minimal cardinality of a family of dense open sets whose intersection is empty. The Baire number (also called the Novák number [V]) of a partial order is the minimal cardinality of a family of dense sets that has no filter [BS] (i.e. no filter on the given partial order intersecting all these dense sets non-trivially). Fnκ(θ, 2) is the collection of all partial functions p : θ → 2 such that |p| < κ, and is partially ordered by reverse inclusion. For κ regular and θ ≥ κ we consider the spaces (2)κ whose points are functions from θ to 2 and a typical basic open set is {f : θ → 2 | p ⊂ f} where p ∈ Fnκ(θ, 2). We denote the Baire number of (2 )κ by n θ κ. It is not hard to see that nκ is also the Baire number of Fnκ(θ, 2). Let us now list some known facts (see [L] §1). Facts. Let κ be a regular cardinal and let θ ≥ κ. Then 1. κ ≤ nκ ≤ 2 . 2. If 2 > κ, then nκ = κ . 3. If θ1 ≤ θ2, then n θ2 κ ≤ n θ1 κ and therefore {n θ κ : θ ≥ κ is a cardinal} is finite. 4. If θ1 ≤ θ2 and n θ2 κ = θ1, then n θ1 κ = θ1. 5. If θ = n κ κ , then θ is the unique cardinal with n θ κ = θ and for every θ1 ≥ θ,