Splitting Todorčević’s set mapping from 𝔟 = ω_{1}

J. Tatch Moore

August 16, 2004

In this note I will present a combinatorial object which exists if 𝔟 = ω_{1}. Currently this object is without an interesting application. It was originally used in a failed attempt to construct an example to Katětov’s problem assuming 𝔟 = ω_{1}. It is reproduced here in hopes that someone will find some use for it or the techniques involved. The results and techniques may be used with appropriate citation. The prose surrounding the construction is rough and unpolished but the lemma of interest has survived the scrutiny of two referees and the proof should be fairly polished.

We will need the following objects, which will be chosen and then fixed for the duration of the construction.

1. A coherent sequence e_{ξ} (ξ < ω_{1}) of finite-to-one maps.^{1}

2. A cofinal ω-sequence C_{δ} in all limit ordinals δ < ω_{1}. Also, define C_{α+1} = {α}

3. A base B for the topology on ω^{ω} which is countable, consists of clopen sets, and is closed under finite unions and complements.

4. A map s from ω_{1} to B which takes all values stationarily often.

As is conventional, if f and g are in ω^{ω}, Δ(f, g) will be used to denote the first coordinate on which f and g differ. Similarly, Δ(e_{α}, e_{β}) is the least γ such that e_{α}(γ) ≠ e_{β}(γ) (Δ(e_{α}, e_{β}) = α if e_{α} is an initial part of e_{β}). If for some k we have that f(n) < g(n) for all n ≥ k then we will write f <^{∗} g.

1Here coherence means that if α < β < ω_{1} then {ξ < α : e_{α}(ξ) ≠ e_{β}(ξ)} is finite. See [3] or [1] for more information on coherent sequences.

We will also write f <^{k} g if we wish to specify the k where the eventual domination must begin.

We will also need the notion of a walk from β to α as introduced in [1] and given a modern exposition in [3]. If α < β then define β^{0}(α) = β and β^{i+1}(α) = min(C_{β^{i}(α)} \ α) if β^{i}(α) ≠ α. This gives a decreasing sequence of ordinals

α = β^{k}(α) < β^{k−1} < … < β^{1}(α) < β^{0}(α) = β

which is called the trace of the walk from β to α and which is denoted tr(α, β) [1]. Our main interest will be in the square bracket operation of [1]

[αβ] = min(tr(Δ(e_{α}, e_{β}), β) \ α).

For the majority of the paper we will be working under the assumption that 𝔟 = ω_{1}. To this end fix a sequence f_{ξ} (ξ < ω_{1}) which is unbounded and strictly increasing in ω^{ω}. Define

H_{β} = {f_{α} : (α < β) and (e_{β}(α) < f_{β}(Δ(f_{α}, f_{β})))}
H_{β}^{+} = {f_{α} ∈ H_{β} : f_{α} ∈ s[αβ]}
H_{β}^{−} = {f_{α} ∈ H_{β} : f_{α} ∉ s[αβ]}.

Note that for each β < ω_{1} and s ∈ {+, −}, if H_{β}^{s} is infinite, it converges to f_{β}. The reader should note that the definition of H_{β} originates in Chapter 2 of [2]. The new ingredient here is the partitioning of these sets into two pieces in such a way that each piece is, in a sense, large. This is made precise by the following key combinatorial lemma (compare this to Lemma 2.0 of [2] and Lemma 2.5^{2} of [3]).

Before we proceed further, it will be helpful to fix some notation. If ᾱ is in [ω_{1}]^{n}, I will use α^{i} to denote the i-least element of ᾱ. Similarly if S is a set, s̄ is in S^{n}, and i < n then s^{i} will be used to denote the i^{th} coordinate of s̄. I will use fᾱ to denote the tuple (f_{α^{0}}, …, f_{α^{n−1}}). Statements like α < β̄ and ᾱ < β abbreviate α < min β̄ and max ᾱ < β respectively.

If β̄ is in [ω_{1}]^{n} then eβ̄ will denote the tuple (e_{β^{0}}, …, e_{β^{n−1}}). If γ < min β̄ then

eβ̄ ↾ γ = (e_{β^{0}} ↾ γ, …, e_{β^{n−1}} ↾ γ).

2This is the Lemma concerning the behavior of [··] in finite powers of ω_{1} — this numbering comes from a preliminary version of [3].

Lemma 0.1. Let n be a positive integer. If F ⊆ [ω_{1}]^{n} is an uncountable pairwise disjoint family then for each t̄ ∈ {+, −}^{n} there are ᾱ, β̄ in F such that for all i < n, f_{α^{i}} is in H_{β^{i}}^{t^{i}}.

Proof. Now let n, F ⊆ [ω_{1}]^{n}, and t̄ ∈ {+, −}^{n} be given as in the statement of the Lemma. By refining F if necessary, we may assume that for some B in B and every ᾱ in F, if i < n then f_{α^{i}} ∈ B iff t^{i} = +. We will also assume that for some k if β̄ is in F and 0 < i then

f_{β^{0}} <^{k} f_{β^{i}}.

Let T be the collection of all τ for which there are uncountably many β̄ in F such that τ = eβ̄ ↾ α for some α < β̄.

Claim 0.2. There is a closed and unbounded set of limit ordinals δ < ω_{1} such that for all β̄ > δ in F, m < ω, and γ < δ there is a τ in T such that:

1. The height of τ is less than δ and greater than γ.

2. τ^{i} is incompatible with ei ; β̄ for all i < n.

3. τ ↾ γ = eβ̄ ↾ γ.

4. For all ν < δ there is an ᾱ in F such that

(a) ν < ᾱ < δ,

(b) τ is a restriction of eᾱ, and

(c) fᾱ ↾ m + 1 = fβ̄ ↾ m + 1.

Proof. Let M be an elementary submodel of H_{ω_{2}} containing F, ⟨e_{α} : α < ω_{1}⟩, and T. It is sufficient to show that a τ satisfying the above conditions can be found for δ = M ∩ ω_{1}. To this end, let β̄ ∈ F, δ < β̄, m < ω, and γ < δ be given. Since every level of T is countable and γ < M ∩ ω_{1}, the restriction eβ̄ ↾ γ is in M. Also, the restriction fβ̄ ↾ m + 1 is in M since it is hereditarily finite.

Now consider the collection S of all elements σ in T such that for uncountably many ᾱ in F with γ < ᾱ we have that σ is a restriction of eᾱ and fᾱ ↾ m+1 = fβ̄ ↾ m+1. Since S is in M and eβ̄ ↾ ν is in S, for cofinally many ν in M ∩ ω_{1}, S is uncountable. Since none of T ’s coordinate projections have any branches, there is an element τ of S such that τ ↾ γ = eβ̄ ↾ γ but τ^{i} is not a restriction of ei ; β̄ for any i < n. τ now has all of the desired properties by elementarity of M.

Now select a δ satisfying the claim with the property that s(δ) = B.

Claim 0.3. There is a sequence β̄_{j} (j < ω) from F and an m < ω such that for all j < j′

1. δ < β̄_{j},

2. eβ̄_{j} ↾ δ = eβ̄_{j′} ↾ δ,

3. tr(δ, β_{j}^{i}) and tr(δ, β_{j′}^{i}) have the same size and furthermore if ξ and ξ′ are respective elements of these traces occupying the same places in their increasing enumeration, C_{ξ} ∩ δ = C_{ξ′} ∩ δ,

4. fβ̄_{j} ↾ m = fβ̄_{j′} ↾ m, and

5. if i < n then f_{β_{j}^{i}}(m) < f_{β_{j′}^{i}}(m).

Proof. First observe that there are only countably many sequences of the form

⟨C_{ξ} ∩ δ : ξ ∈ tr(δ, β)⟩

for β a countable ordinal above δ. Hence it is possible to refine F to an uncountable subset F′ whose elements satisfy items 1–3.

Now the set {f_{β^{0}} : β̄ ∈ F′} is unbounded and therefore it is possible to find a ū in (ω^{n})^{<ω} of length at least k such that for infinitely many l there is a β̄ in F′ such that the concatenation (u^{0})ˆ l is an initial part of f_{β^{0}} and ū is an initial part of fβ̄. Setting m to be equal to the length of ū and combining the above property of ū with our arrangement that f_{β^{0}} <^{k} f_{β^{i}} for all i < n and β̄ in F, it is possible to recursively select the desired sequence of β̄_{j}’s.

Fix a sequence β̄_{j} (j < ω) as in the second claim. Let γ < δ be a bound for all ordinals appearing in sets of the form C_{ξ} ∩ δ where ξ is chosen from some tr(δ, β_{j}^{i}). Such a γ can be found since by item 3 of the second claim the index j is irrelevant and hence there are only finitely many ordinals to bound. It is easily verified that γ has the following property: for all i < n, j < ω, and α with γ < α < δ,

tr(δ, β_{j}^{i}) = tr(α, β_{j}^{i}) \ δ.

Observe that by item 3 of the second claim we have also that if γ < δ and i < n then tr(γ, β_{j}^{i}) ∩ δ = tr(γ, β_{j′}^{i}) ∩ δ. Applying the first claim to β̄_{0}, m and γ, find a τ in T which satisfies the conclusion of the claim. Let ν < δ be an upper bound for all ordinals in sets of the form

tr(Δ(τ^{i}, e_{β_{j}^{i}}), β_{j}^{i}) ∩ δ

(there are only finitely many by our observation). Now by the 3rd clause of the first claim applied to ν there is a ᾱ in F such that ν < ᾱ < δ, τ is a restriction of eᾱ and fᾱ ↾ m + 1 = fβ̄_{0} ↾ m + 1.

Notice that for all i < n and 0 < j < ω

[α^{i}β_{j}^{i}] = δ,
s([α^{i}, β_{j}^{i}]) = B,
Δ(f_{α^{i}}, f_{β_{j}^{i}}) = m.

Since

lim_{j→∞} f_{β_{j}^{i}}(m) = ∞

for all i we can find a j < ω such that for all i < n

e_{β_{j}^{i}}(α^{i}) < f_{β_{j}^{i}}(Δ(f_{α^{i}}, f_{β_{j}^{i}})) = f_{β_{j}^{i}}(m).

It is now easily verified that for all i < n α^{i} is in H_{β_{j}^{i}}^{t^{i}} (i.e. ᾱ and β̄_{j} satisfy the conclusion of the lemma).

The following strengthening of Lemma 0.1 also holds.

Lemma 0.4. Let n be a positive integer. If F ⊆ [ω_{1}]^{n} is an uncountable pairwise disjoint family then for each t̄ ∈ {+, −}^{n} there is a sequence ᾱ_{k} (k < ω) of distinct elements of F and a β̄ in F such that for all i < n, f_{α_{k}^{i}} is in H_{β^{i}}^{t^{i}}.

Proof. Suppose that this is not the case and fix an uncountable pairwise disjoint F ⊆ [ω_{1}]^{n} and t̄ ∈ {+, −}^{n} for which the lemma fails. Enumerate F = {β̄_{ξ} : ξ < ω_{1}}. By refining F is necessary we may assume that if ξ < η then β̄_{ξ} < β̄_{η}.

If ξ < η, define c(ξ, η) to be equal to 1 if for every i < n

β_{ξ}^{i} ∈ H_{β_{η}^{i}}^{t^{i}}

and 0 otherwise. Applying the partition relation ω_{1} → (ω_{1}, ω + 1)^{2} it is possible to find either a set A ⊆ ω_{1} of order type ω + 1 such that c(ξ, η) = 1 for all ξ < η in A or else there is an uncountable B ⊆ ω_{1} such that c(ξ, η) = 0 for all ξ < η in B. It is now easily checked that the first possibility gives us the conclusion of the theorem (with β̄ being the last element of A and {ᾱ_{k} : k < ω} forming the rest of A) and that the second possibility is forbidden by Lemma 0.1.

References

[1] Stevo Todorčević. Partitioning pairs of countable ordinals. Acta Math., 159(3–4):261–294, 1987.

[2] Stevo Todorčević. Partition Problems In Topology. Amer. Math. Soc., 1989.

[3] Stevo Todorčević. Coherent sequences. In Handbook of Set Theory. North-Holland, (in preparation).