Baire numbers, uncountable Cohen sets and perfect-set forcing

Abstract

The Baire number of the real line (see 1.1(c)) has uncountable cofinality [Mi]. This number is equal to the Baire number of the space 2ω and also to the Baire number (1.1(d)) of every countable partial order. Our research is motivated by the following open question (1.13) (see also [BPS] and [Mi]): can the cofinality of nκ, the Baire number of the space (2κ)κ (1.1(b)), be less than or equal to κ? This question is nontrivial when κ is regular and 2κ = κ (1.3). A. Miller proved that the answer is “no” if κ is strongly inaccessible (see [Mi] and 1.11). Assuming CH, is equal to the Baire number of the Stone space of ℘(ω)/Fin.In §1 we discuss , the Baire number of the space (2λ)κ. If κ is regular, then , and using 1.3 and 1.5, we may rule out the easy cases and restrict ourselves to cases where κ is regular with 2<κ = κ, and κ ≤ λ 2κ. If λ > κ, then it is consistent to have cof()=κ ([Mi] and 1.7). We do not know whether is consistent (1.9). Next, the cardinal number eκ is defined (1.10). It is easy to see that nκ ≤ eκ and that cof(eκ) > κ. If κ is strongly inaccessible (s.i.), then nκ = eκ ([Mil], [Bar], [MF] and 1.11). For κ accessible, the question of whether nκ = eκ is open.Another open question (Fremlin) is stated as follows: assume W ⊇ V and f ∈ ωω ∩ W is such that, for every g ∈ ω ω ∩ V, ∣(f = g)∣ = ℵ0, does this imply that W contains sets which are ω-Cohen (1.1(a)) over V? If κ is accessible regular with 2<κ = κ, then the answer to the analog of Fremlin's question for κ is “no”. In fact, any κ-closed forcing that adds new subsets of κ, also adds a new function f ∈ κκ such that, for every g ∈ κκ ∩V, (f = g) is stationary (4.1(a)). On the other hand, perfect-set forcing on κ (see [Ka] and §5) adds a new subset of κ but produces a minimal extension (5.4(b)), and hence adds no sets which are κ-Cohen over V.

Questions on generalised Baire spaces

“…Recall the cardinal invariant from Definition . In the classical setting, this is always equal to , and the same holds for strongly inaccessible κ by . It is also known that if κ is successor and , then .…”

“…Cummings and Shelah [18] proved some analogies between these two different versions, but the following remained open: Recall the cardinal invariant in(κ) from Definition 2.15. In the classical setting, this is always equal to cov(M), and the same holds for strongly inaccessible κ by [45]. It is also known that if κ is successor and κ <κ = κ, then in(κ) = d(κ).…”

“…Usually the above definition is only applied to cardinals satisfying . In the absence of this assumption, by [, Lemma 1.3 (d)], and by [, Proposition 4.15]. The exact values of and may still be non‐trivial.…”

“…We also introduce a new characteristic, which is equal to in the classical case but may be more complicated in the generalised case. It was first isolated and studied by Landver in , who, among other things, proved that it is equal to for strongly inaccessible κ; cf. Question .…”

Higher Miller Forcing May Collapse Cardinals

“…Interesting non-equivalent variants of κ-Sacks forcing were introduced by Landver in [14]. He replaces closure of splitting nodes by the requirement that the splitting nodes along each branch must be in a given normal filter over κ.…”