JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0894-0347(XX)0000-0

A SOLUTION TO THE L SPACE PROBLEM

JUSTIN TATCH MOORE

1. Introduction

In [23], Todorcevic gives a survey of basis problems in combinatorial set theory, listing nine theorems and six working conjectures^{1} — all in the presence of PFA — including the following three of interest to us here:

Conjecture 1. (Todorcevic; [23]) If R is a binary relation, then either R ≤ ℵ_{0} · ω_{1} or [ω_{1}]^{<ℵ_{0}} ≤ R.

Conjecture 2. (Hajnal, Juhasz; [5]) If X is a regular Hausdorff space, then the following are equivalent:

(i) X is hereditarily separable. (ii) X is hereditarily Lindelöf. (iii) X does not contain an uncountable discrete subspace.

Conjecture 3. (Gruenhage; [3]) The uncountable regular Hausdorff spaces have a three element basis consisting of a set of reals of cardinality ℵ_{1} with the metric, the Sorgenfrey, and the discrete topology.

According to the survey article [7], Kunen formulated the partition relation

ω_{1} → (ω_{1}, (ω_{1}; ω_{1}))^{2},

while studying Conjecture 2 in the 1970s. He showed that it would imply positive resolution of Conjecture 2 and asked whether it is consistent. Afterward Laver showed that under MA_{ℵ_{1}} this partition relation is equivalent to the formally weaker partition relation

ω_{1} → (ω_{1}; ω_{1})_{2}^{2}.

Around the same time, Galvin showed that the positive square bracket relation ω_{1} → [ω_{1}]_{5}^{2} implies ω_{1} → (ω_{1}; ω_{1})_{2}^{2}.

In [19] Todorcevic showed that ω_{1} ↛ [ω_{1}]_{5}^{2} by employing a new technique — the method of minimal walks — which has also proved useful in many other applications (see [24]). This technique will be employed in this paper where I will prove the following results which refute ω_{1} → (ω_{1}; ω_{1})_{2}^{2} and Conjectures 1–3.

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

2000 Mathematics Subject Classification. Primary: 54D20, 54D65, 03E02, 03E75; Secondary:

54F15.

Key words and phrases. L space, negative partition relation, Tukey order, hereditarily Lindelöf, non-separable, basis.

The research presented in this paper was funded by NSF grant DMS–0401893.

1Many of these were not phrased as conjectures by their authors but rather as problems.

Moreover, they were often phrased as “Is it consistent that ...?” rather than “Does PFA imply that ...?”.

Theorem 1.1. There is a function f : [ω_{1}]^{2} → ω_{1} such that whenever A, B ⊆ ω_{1} are uncountable and ξ < ω_{1} there are α in A and β in B such that α < β and f(α, β) = ξ. In particular ω_{1} ↛ (ω_{1}; ω_{1})_{2}^{2}.

Theorem 1.2. There is a family ℱ of binary relations which contains an antichain of cardinality 2^{ℵ_{1}}, is downwards directed, and such that for all R in ℱ, R ≰ ℵ_{0} · ω_{1} and [ω_{1}]^{<ℵ_{0}} ≰ R.

Theorem 1.3. There is a non-separable, hereditarily Lindelöf subspace ℒ of 𝕋^{ω_{1}}. Moreover the closure in 𝕋^{ω_{1}} of any uncountable subset of ℒ contains a canonical homeomorphic copy of 𝕋^{ω_{1}}.

Theorem 1.4. Any basis for the uncountable topological spaces has cardinality strictly greater than ℵ_{1}.

It is worth noting that it was known early on that the existence of an L space alone gives a refutation of these conjectures. The above results, however, take the failures of these conjectures a step further and require slightly different combinatorial refinements.

The theorems in this paper are consequences of an analysis of coherent sequences of finite-to-one functions ⟨e_{β} : β < ω_{1}⟩ and lower trace functions L : [ω_{1}]^{2} → [ω_{1}]^{<ω_{1}}. These are combinatorial objects which can be routinely constructed using the method of minimal walks. If we let osc(α, β) denote the number of changes in the relation between corresponding entries of e_{α} ↾ L(α, β) and e_{α} ↾ L(α, β) from ≤ to >, then the main result of the paper can be stated as follows.

Theorem 1.5. For every 𝒜 ⊆ [ω_{1}]^{k} and ℬ ⊆ [ω_{1}]^{l} which are uncountable families of pairwise disjoint sets and every natural number n, there are a in 𝒜 and b_{m} (m < n) in ℬ such that for all i < k, j < l, and m < n:

a < b_{m}
osc (a(i), b_{m}(j)) = osc (a(i), b_{0}(j)) + m.

This can be likened to the following two results of Todorcevic. The first is used to draw a number of conclusions about Conjecture 2 in [21]. The second yields one of many proofs in Section 4 of [19] that ω_{1} ↛ [ω_{1}]_{ω}^{2}. See [19] and Section 2 below for undefined notation.^{2}

Theorem 1.6. [20] If X and Y are unbounded and countably directed in (ω^{ω}, <^{∗}) and consist of monotonic functions, then there is a natural number l such that for all n there are x and y in X and Y respectively such that

osc(x, y) = n + l.

Theorem 1.7. [19] If X is an uncountable subset of ω_{1}, then the set of all values

osc ( Tr(ζ, α), Tr(ζ, β))

for α ≠ β in X with ζ = Δ(α, β) contains arbitrarily long intervals of integers.

This paper is organized as follows. Section 2 provides some background on the method of minimal walks and introduces the lower trace function which is used in the statement of the main theorem on oscillations. Section 3 provides a review of elementary submodels. The main combinatorial results of the paper are proved in Section 4. Theorem 1.1 is deduced in Section 5 and a two place function o : [ω_{1}]^{2} → 𝕋 is introduced which harnesses most of the strength of the more technical main theorem. Theorem 1.2 is then deduced in Section 6 after some motivation is provided. The paper closes with Section 7 where Theorems 1.3 and 1.4 are proved. This final section also contains a basic analysis of the space in Theorem 1.3.

2What I am referring to as Tr(α, β) is denoted as F_{α}(β) in [19]. Also, the usage of osc(·, ·) in Theorem 1.7 is different as it counts the oscillations between finite subsets of ω_{1} — see [19].

I would like to thank Boban Veličković for his careful reading of early drafts of this paper. The clarity of the present arguments owe much to Todorcevic’s and Veličković’s suggestions. A number of other people — too many to name — have also made the generous contribution of reading the paper and offering suggestions. Finally, special thanks is due to the diligent referees for their careful reading of the paper.

This paper is intended to be accessible to any interested reader who is fluent in set theory. Basic background can be found in, e.g., [10]. Elementary submodels will be employed at a number of points in the argument. Ironically, this represents the only non-elementary technique used in the proofs. The essentials are reviewed in Section 3. The reader is referred to III.1 of [6] for more information on elementary submodels and stationary sets. The proofs will also employ the method of minimal walks introduced in [19]. The necessary background is presented in Section 2. The reader is referred to [24] for further reading on minimal walks.^{3}

The notation is fairly standard. All ordinals are von Neumann ordinals — they are the set consisting of their predecessors. In particular, n = {0, 1, …, n − 1} and the first infinite ordinal ω is the set of all finite ordinals {0, 1, 2, …} and is identified with the natural numbers. All counting starts at 0. We will also need the unit circle viewed as the set

𝕋 = {z ∈ ℂ : |z| = 1}

equipped with the usual multiplication operation. If k is a natural number and X is a set, then [X]^{k} is the set of all k-element subsets of X. If X has a canonical linear ordering associated with it and a is in [X]^{k}, then a will be identified with the increasing sequence which enumerates it. If a and b are finite subsets of ω_{1}, then a < b will be used to abbreviate “α < β whenever α is in a and β is in b.” Similarly one defines statements such as α < b and a < β if α and β are ordinals.

This paper is dedicated to Stevo Todorcevic for teaching me how to traverse ω_{1} and for his inspirational [23].

2. The trace functions

In this section I will provide the necessary background on minimal walks; see [24] for further reading. With the exception of the definition and axiomatization of L, everything discussed in this section is contained either explicitly or implicitly in [19]. Minimal walks are facilitated by a C-sequence which one uses to “walk” from an ordinal β down to a smaller ordinal α.

Definition 2.1. A C-sequence is a sequence ⟨C_{α} : α < ω_{1}⟩ such that C_{α} is a cofinal subset of α and if γ < α then C_{α} ∩ γ is finite. It will be useful at certain points to assume that 0 is an element of every C_{α}.

3[24] is an updated, expanded, and clearer account of the methods and results of [19]. My references to [19] are for historical accuracy.

The following two functions will be of interest to us. The upper trace will not be necessary, but is useful in making the other definitions more transparent.

Definition 2.2. [24] (upper trace) If α ≤ β, then Tr(α, β) is defined recursively by

Tr(α, α) = ∅,
Tr(α, β) = Tr (α, min(C_{β} \ α)) ∪ {β}.

Definition 2.3. (lower trace) If α ≤ β, then L(α, β) is defined recursively by

L(α, α) = ∅,
L(α, β) = L(α, min(C_{β} \ α)) ∪ {max(C_{β} ∩ α)} \ max(C_{β} ∩ α).

Hence the upper trace Tr(α, β) is enumerated by the sequence β = β_{0} > β_{1} > ⋯ > β_{l−1} where β_{i+1} = min(C_{β_{i}} \ α) and α = β_{l} is in C_{β_{l−1}}. It is easily verified that the lower trace L(α, β) is listed as ξ_{0} ≤ ξ_{1} ≤ ⋯ ≤ ξ_{l−1} where

ξ_{k} = max ⋃_{j=0}^{k}(C_{β_{j}} ∩ α).

While the upper trace Tr(α, β) and the full lower trace F (α, β) of a walk are well studied (see [24]), this is, to my knowledge, the first explicit analysis of the lower trace L(α, β) ⊆ F (α, β).

For our purposes, the lower trace can be axiomatized by the following facts.

Fact 1. If α ≤ β ≤ γ and L(β, γ) < L(α, β), then

L(α, γ) = L(α, β) ∪ L(β, γ).

Proof. Let α, β, and γ satisfy the hypothesis of the fact. Observe that L(β, γ) < α and hence C_{ζ} ∩ α = C_{ζ} ∩ β whenever ζ is in Tr(β, γ). Hence β is in Tr(α, γ) and Tr(α, γ) = Tr(α, β) ∪ Tr(β, γ). Let

γ = γ_{0} > γ_{1} > … > γ_{l−1}

enumerate Tr(α, γ) and let l_{0} be such that β = γ_{l_{0}}. Then L(α, γ) is listed as

ξ_{0} ≤ ξ_{1} ≤ ⋯ ≤ ξ_{l−1}

where

ξ_{k} = max ⋃_{j=0}^{k} C_{γ_{j}} ∩ α.

If k < l_{0}, then C_{γ_{k}} ∩ α = C_{γ_{k}} ∩ β and so L(β, γ) = {ξ_{k}}_{k=0}^{l−1}. On the other hand,

max (C_{γ_{l_{0}}} ∩ α) > ξ_{l_{0}−1}

and so if k ≥ l_{0}

max ⋃_{j=0}^{k} (C_{γ_{j}} ∩ α) = max ⋃_{j=l_{0}}^{k} (C_{γ_{j}} ∩ α)

and hence L(α, β) = {ξ_{k}}_{k=l_{0}}^{l−1}. □

Fact 2. If α < β, then L(α, β) is a non-empty finite set and if 0 < β is a limit ordinal, then

lim_{α→β} min L(α, β) = β.

Proof. This follows immediately from the observation that

min L(α, β) = max(C_{β} ∩ α).
□

Remark 2.4. Notice that if 0 < β < γ are fixed and β is a limit ordinal, then Fact 2 implies that for all but a bounded set of α < β, the inequality in Fact 1 holds.

At this point I recall the following definition.

Definition 2.5. If ⟨e_{β} : β < ω_{1}⟩ is a sequence of functions such that e_{β} : β → ω for all β < ω_{1}, then the sequence is coherent if whenever β ≤ β′ < ω_{1}, e_{β′} ↾ β differs from e_{β} only on a finite set.

For brevity, if I say that ⟨e_{β} : β < ω_{1}⟩ is a coherent sequence of finite-to-one functions, then it is implicit that each e_{β} is a function from β to ω. The main combinatorial result of the paper will be concerned with counting oscillations between pairs of elements of such sequences. The following construction provides a standard example.

Definition 2.6. [19](maximal weight) If α ≤ β, then ϱ_{1}(α, β) is defined recursively by

ϱ_{1}(α, α) = 0
ϱ_{1}(α, β) = max (|C_{β} ∩ α|, ϱ_{1}(α, min(C_{β} \ α))).

Alternately, ϱ_{1}(α, β) is the maximum value of the form |C_{ζ} ∩ α| where ζ ranges over Tr(α, β). For each β < ω_{1}, define e_{β} : β → ω by e_{β}(α) = ϱ_{1}(α, β).

Fact 3. [19] ⟨e_{β} : β < ω_{1}⟩ is a coherent sequence of finite-to-one functions.

Proof. Let β ≤ β′ < ω_{1} and n < ω be given and set D equal to the set of all α < β such that either e_{β}(α) ≤ n or e_{β}(α) ≠ e_{β′}(α). It suffices to show that D has no limit points.

To this end, suppose δ ≤ β. It is easy to check that there is a δ_{0} < δ such that

Tr(α, β) = Tr(α, δ) ∪ Tr(δ, β),
Tr(α, β′) = Tr(α, δ) ∪ Tr(δ, β′),
|C_{δ} ∩ α| > n

whenever δ_{0} < α < δ. Notice that if δ is a successor, then this can be made vacuously true by letting δ_{0} = δ − 1. If δ_{0} < α < δ, then

e_{β}(α) ≥ |C_{δ} ∩ α| > n
e_{β}(α) = e_{δ}(α) = e_{β′}(δ)

and hence α is not in D. Consequently δ is not a limit point of D. □

Remark 2.7. It is interesting to note that while coherence and the finite-to-one property are at tension with each other, the verifications of these properties in the previous fact are virtually identical.

Remark 2.8. It follows from Fact 3 that T (ϱ_{1}) = {e_{β} ↾ α : α ≤ β < ω_{1}} is an Aronszajn tree — its levels and chains are countable.

Throughout this paper we will assume that L and ⟨e_{β} : β < ω_{1}⟩ satisfy Facts 1–3. We will also need the following two place function.

Definition 2.9. If α < β < ω_{1}, then let Δ(α, β) be the least ordinal ξ such that either e_{α}(ξ) ≠ e_{β}(ξ) or ξ = α.

3. Basic facts about elementary submodels

In the proof of the main theorem, we will need the following facts about elementary submodels of H(ℵ_{1}^{+}). Recall that H(ℵ_{1}^{+}) is the collection of all sets of hereditary cardinality at most ℵ_{1}. This structure is of interest to us since it contains all of the objects relevant to us (e.g. ω_{1}, L, and ⟨e_{β} : β < ω_{1}⟩), is closed under taking subsets, and satisfies all of the axioms of ZFC except the power set axiom. An elementary submodel M of H(ℵ_{1}^{+}) is a subset of H(ℵ_{1}^{+}) such that whenever ϕ is a logical formula with parameters in M, M satisfies ϕ if and only if H(ℵ_{1}^{+}) satisfies ϕ. The following fact is well known.

Fact 4. If M is a countable elementary submodel of H(ℵ_{1}^{+}), then M ∩ ω_{1} is an ordinal. Moreover, if F is a countable subset of H(ℵ_{1}^{+}), then the set of all ordinals of the form M ∩ ω_{1} such that M is a countable elementary submodel of H(ℵ_{1}^{+}) with F ⊆ M contains a club.^{4}

The following standard facts are very useful when working with countable elementary submodels.

Fact 5. If M is a countable elementary submodel of H(ℵ_{1}^{+}) which contains some element X, then X is countable iff X ⊆ M.

Fact 6. If M is a countable elementary submodel of H(ℵ_{1}^{+}) which contains as an element some subset A of ω_{1}, then A is uncountable iff A ∩ M ∩ ω_{1} is unbounded in M ∩ ω_{1}.

Fact 7. If M is a countable elementary submodel of H(ℵ_{1}^{+}), X is in H(ℵ_{1}^{+}) and X is definable from a logical formula with parameters in M, then X is in M.

Remark 3.1. The above facts remain true if ℵ_{1}^{+} is replaced with any uncountable regular cardinal.

4. Oscillations on the lower trace

In this section I will prove the main combinatorial theorem of the paper. We will take the following as our definition of the oscillation function.

Definition 4.1. Suppose that s and t are two functions defined on a common finite set of ordinals F. Let Osc(s, t; F) be the set of all ξ in F \ {min F } such that s(ξ^{−}) ≤ t(ξ^{−}) and s(ξ) > t(ξ) where ξ^{−} is the greatest element of F less than ξ.^{5}

The following notation will be convenient.

Definition 4.2. If α < β < ω_{1}, let Osc(α, β) denote

Osc (e_{α}, e_{β}; L(α, β))

and osc(α, β) denote the cardinality of Osc(α, β).

4A subset of ω_{1} is club if it is closed and unbounded.

5Actually, we will later see that we are really interested in counting oscillations between the relations = and ≠ (rather than the more conventional < and >). The above definition is a compromise between the two.

Theorem 4.3. For every 𝒜 ⊆ [ω_{1}]^{k} and ℬ ⊆ [ω_{1}]^{l} which are uncountable families of pairwise disjoint sets and every n < ω, there are a in 𝒜 and b_{m} (m < n) in ℬ such that for all i < k, j < l, and m < n:

a < b_{m}
osc (a(i), b_{m}(j)) = osc (a(i), b_{0}(j)) + m.

Theorem 4.3 will be derived from the following lemma.

Lemma 4.4. Let 𝒜 ⊆ [ω_{1}]^{k} and ℬ ⊆ [ω_{1}]^{l} be uncountable and pairwise disjoint. There is a closed and unbounded set of δ < ω_{1} such that if a is in 𝒜 \ δ, b is in ℬ \ δ, and R is in {=, >}, then there are a^{+} in 𝒜 \ δ, and b^{+} in ℬ \ δ such that for all i < k and j < l:^{6}

(i) max L(δ, b(j)) is less than both Δ(a(i), a^{+}(i)) and Δ(b(j), b^{+}(j)).

(ii) L(δ, b(j)) is a proper initial part of L(δ, b^{+}(j)).

(iii) If ξ is in L^{+} = L(δ, b^{+}(j)) \ L(δ, b(j)), then e_{a^{+}(i)}(ξ) R e_{b^{+}(j)}(ξ).

Proof. Let M be a countable elementary submodel of H(ℵ_{1}^{+}) which contains everything relevant and let δ = M ∩ ω_{1}. By Fact 4, it suffices to show that δ satisfies the conclusion of the theorem.

First suppose that R is =. Applying Fact 3, find a γ_{0} < δ satisfying the following conditions:

(1) If j < l, then L(δ, b(j)) < γ_{0}.

(2) If i < k, j < l, and γ_{0} < ξ < δ, then e_{a(i)}(ξ) = e_{b(j)}(ξ).

Applying Fact 2, pick a γ < δ such that if γ < ξ < δ, then γ_{0} < L(ξ, δ). Consider the set D ⊆ ω_{1} consisting of all δ^{+} such that for some a^{+} in 𝒜 \ δ^{+} and b^{+} in ℬ \ δ^{+} the following conditions are satisfied:

(3) e_{a^{+}(i)} ↾ γ_{0} = e_{a(i)} ↾ γ_{0} and e_{b^{+}(j)} ↾ γ_{0} = e_{b(j)} ↾ γ_{0} for all i < k and j < l.

(4) L(δ^{+}, b^{+}(j)) = L(δ, b(j)) for all j < l.
(5) If γ < ξ < δ^{+}, then γ_{0} < L(ξ, δ^{+}).
(6) If γ_{0} < ξ < δ^{+}, i < k, and j < l, then e_{a^{+}(i)}(ξ) = e_{b^{+}(j)}(ξ).

Observe that for all β ≥ γ_{0}, e_{β} ↾ γ_{0} is in M since by Fact 3 it differs from e_{γ_{0}} on a finite set. Hence D is definable from the parameters γ_{0}, γ, e_{a(i)} ↾ γ_{0}, etc. which are all elements of M. Therefore D is in M by Fact 7. Since D has δ as a member, it is uncountable by Fact 5. Hence there is a δ^{+} > δ in D. Let a^{+} ∈ 𝒜 \ δ^{+} and b^{+} ∈ ℬ \ δ^{+} witness that δ^{+} is in D.

Now let i < k and j < l be arbitrary. First observe that

γ_{0} ≤ Δ(a(i), a^{+}(i)),
γ_{0} ≤ Δ(b(j), b^{+}(j)).

Put L^{+} = L(δ, δ^{+}). Notice that L(δ, b(j)) = L(δ^{+}, b^{+}(j)) < L^{+} and hence

L(δ, b^{+}(j)) = L(δ^{+}, b^{+}(j)) ∪ L^{+} = L(δ, b(j)) ∪ L^{+}

holds by Fact 1. Since δ < δ^{+}, L^{+} is non-empty. If ξ is in L^{+}, then γ_{0} < ξ < δ^{+} and so

e_{a^{+}(i)}(ξ) R e_{b^{+}(j)}(ξ).

6It will be convenient to let 𝒜 \ δ denote the set of all a in 𝒜 such that δ ≤ a whenever 𝒜 is a collection of finite subsets of ω_{1} and δ < ω_{1}.

Now suppose that R is >. Put E equal to the set of all limit ν < ω_{1} such that for all a_{0} in 𝒜 \ ν, all ν_{0} < ν, ε < ω_{1}, n < ω, and finite L^{+} ⊆ ω_{1} \ ν there is an a_{1} in 𝒜 \ ε with

ν_{0} ≤ Δ(a_{0}(i), a_{1}(i)),
e_{a_{1}(i)}(ξ) > n

whenever i < k and ξ is in L^{+}. By Fact 7, E is an element of M.

Claim 4.5. δ is in E. In particular, E is uncountable by Fact 5.

Proof. Let a_{0}, ν_{0}, ε, n, and L^{+} be given as in the definition of E for ν = δ. By Fact 3 we may assume without loss of generality that ν_{0} is an upper bound for all ξ < δ such that e_{a_{0}(i)}(ξ) ≤ n for some i < k. Now, applying elementarity of M, there is a δ^{+} above ε, δ, and max L^{+} and an a_{1} in 𝒜 \ δ^{+} such that the following conditions are satisfied:

(7) For all i < k, e_{a_{0}(i)} ↾ ν_{0} = e_{a_{1}(i)} ↾ ν_{0}.

(8) If ν_{0} < ξ < δ^{+} and i < k, then e_{a_{1}(i)}(ξ) > n.

Since L^{+} ⊆ δ^{+} \ δ, this completes the proof of the claim. □

Applying elementarity of M and uncountability of E, find an element γ_{0} of E such that L(δ, b(j)) < γ_{0} < δ for all j < l. Applying Fact 2, find a γ < δ such that if γ < ξ < δ, then γ_{0} < L(ξ, δ). Again using elementarity of M, select a limit δ^{+} > δ and a b^{+} in ℬ \ δ^{+} so that the following conditions are satisfied:

(9) e_{b^{+}(j)} ↾ γ_{0} = e_{b(j)} ↾ γ_{0} for all j < l.

(10) L(δ^{+}, b^{+}(j)) = L(δ, b(j)) for all j < l.

(11) If γ < ξ < δ^{+}, then γ_{0} < L(ξ, δ^{+}). Put L^{+} = L(δ, δ^{+}). Applying the definition of E, find an a^{+} in 𝒜 \ δ such that for all i < k, j < l, and ξ in L^{+}

L(δ, b(j)) < Δ(a(i), a^{+}(i))
e_{a^{+}(i)}(ξ) > e_{b^{+}(j)}(ξ).

The rest of the verification is as in the previous case. This completes the proof of Lemma 4.4. □

Now we are ready to prove Theorem 4.3

Proof. Let 𝒜 and ℬ be given and select a countable elementary submodel M of H(ℵ_{1}^{+}) containing everything relevant, setting δ = M ∩ ω_{1}. Since M contains 𝒜 and ℬ, the closed and unbounded set provided by Lemma 4.4 is in M and therefore δ is in this closed and unbounded set. Using Lemma 4.4, it is possible to select a_{m} (m < ω) in 𝒜 \ δ, b_{m} (m < ω) in ℬ \ δ, and ξ_{m} (m < ω) in δ so that for all m < ω the following conditions are satisfied whenever i < k and j < l:

(12) L(δ, b_{m}(j)) is a proper initial part of L(δ, b_{m+1}(j)).

(13) L(δ, b_{m+1}(j)) \ L(δ, b_{m}(j)) does not depend on j and contains ξ_{m} as an element.

(14) Osc (a_{m+1}(i), b_{m+1}(j); L(δ, b_{m+1}(j))) is formed by adding ξ_{m} to Osc (a_{m}(i), b_{m}(j); L(δ, b_{m}(j))).

(15) If m′ < m, then ξ_{m′} is strictly less than both Δ(a_{m}(i), a_{m+1}(i)) and Δ(b_{m}(j), b_{m+1}(j)).

(16) e_{a_{m}(i)}( max L(δ, b_{m}(j))) > e_{b_{m}(j)}( max L(δ, b_{m}(j))).

Notice that, once a_{m} and b_{m} have been selected, Lemma 4.4 is applied twice in order to obtain a_{m+1} and b_{m+1} — first with R being = and then with R being >. The ordinal ξ_{m} is the minimum of what might be described as

L(δ, b_{m+1}(j)) \ L(δ, b_{m+1/2}(j)).

Now let n be given. Pick a γ_{0} < δ which is an upper bound for each L(δ, b_{n}(j)) for j < l and all ξ < δ such that for some m, m′ ≤ n and j, j′ < l

e_{b_{m}(j)}(ξ) ≠ e_{b_{m′} (j′)}(ξ)

(the latter set is finite by Fact 3). Using elementarity of M and Fact 2, select an a in 𝒜 such that a < δ and for all i < k and j < l

L(δ, b_{n}(j)) < Δ(a(i), a_{n}(i)),
γ_{0} < L(a(i), δ).

Now let i < k, j < l, and m < n be fixed. It follows from Fact 1 that

L(a(i), b_{m}(j)) = L(a(i), δ) ∪ L(δ, b_{m}(j)).

Finally, e_{b_{m}(j)} ↾ L(a(i), δ) does not depend on m and therefore

Osc (a(i), b_{0}(j); L(a(i), δ)) = Osc (a(i), b_{m}(j); L(a(i), δ)).

By (16), Osc (a(i), b_{m}(j)) is the union

Osc (a(i), b_{m}(j); L(a(i), δ)) ∪ Osc (a(i), b_{m}(j); L(δ, b_{m}(j)))

(i.e. no new oscillation occurs at the “seam”). Hence, by (14),

Osc (a(i), b_{m}(j)) = Osc (a(i), b_{0}(j)) ∪ {ξ_{m′} : m′ < m}.

This completes the proof of Theorem 4.3. □

5. A negative partition relation and the coloring o

We will now consider the following function.

Definition 5.1. If α < β, define

o(α, β) = z_{α}^{osc(α,β)+1}.

This definition is motivated by the following theorem — commonly known as Kronecker’s Theorem — discovered independently by Kronecker and Tchebychef (see, e.g., [1]).

Kronecker’s Theorem. [8] [17] Suppose that z_{i} (i < k) are elements of 𝕋 which are rationally independent. For every ε > 0 there is a natural number n_{ε} such that if u, v are in 𝕋^{k}, there is an m < n_{ε} such that for all i < k

|u_{i}z_{i}^{m} − v_{i}| < ε.

Remark 5.2. The usual formulation of this theorem is that n_{ε} depends on u and v. The uniformity of n_{ε} as u and v vary follows from a standard compactness argument.

For the remainder of the paper, fix a sequence ⟨z_{α} : α < ω_{1}⟩ of elements of 𝕋 which are rationally independent. This is possible since, given any countable rationally independent subset I of 𝕋, there are only countably many z for which I ∪ {z} is rationally dependent.

The following function will also be useful.

Definition 5.3. The function ∗ : 𝕋 → ω is defined by letting ∗(z) = n iff there is a θ in (\frac{2π}{n+2}, \frac{2π}{n+1}] such that z = e^{iθ}. If f is a function taking values in 𝕋, then f^{∗} will denote the composition of f followed by ∗.

The point is that ∗ takes all values on every dense subset of 𝕋. The notation for ∗ is chosen to mimic its usage in [24]. We will shortly see that o^{∗} defines a partition of [ω_{1}]^{2} with strong properties.

In [18], Todorcevic considered two variations of ω_{1} → (ω_{1}; ω_{1})_{2}^{2} which are closely related to (S) and (L) respectively:

ω_{1} → (ω_{1}, (ω_{1}; fin ω_{1}))^{2},
ω_{1} → (ω_{1}, (fin ω_{1}; ω_{1}))^{2}.

The statement ω_{1} → (ω_{1}, (ω_{1}; fin ω_{1}))^{2} asserts that if c : [ω_{1}]^{2} → 2, then either (A) there is an uncountable X ⊆ ω_{1} such that c is constantly 0 on [X]^{2} or (B) there are uncountable A ⊆ ω_{1} and uncountable pairwise disjoint ℬ ⊆

[ω_{1}]^{<ℵ_{0}} such that for all α in A and b in ℬ with α < b, there is a β in b with c(α, β) = 1.

The statement ω_{1} → (ω_{1}, (fin ω_{1}; ω_{1}))^{2} is similarly defined by replacing A ⊆ ω_{1} by a pairwise disjoint 𝒜 ⊆ [ω_{1}]^{<ℵ_{0}} and ℬ by B ⊆ ω_{1}. These imply (S) and (L) respectively (see [18]) and are each immediate consequences of ω_{1} → (ω_{1}; (ω_{1}; ω_{1}))_{2}^{2}.

In [18], Todorcevic showed that ω_{1} → (ω_{1}, (ω_{1}; fin ω_{1}))^{2} was relatively consistent with ZFC and in fact follows from PFA. We will now see that o^{∗} witnesses a strong failure of ω_{1} → (ω_{1}, (fin ω_{1}; ω_{1}))^{2}.

Theorem 5.4. Let 𝒜 ⊆ [ω_{1}]^{k} and ℬ ⊆ [ω_{1}]^{l} be uncountable families of pairwise disjoint sets. For every χ : k → ω and ϕ : k → l there are a in 𝒜 and b in ℬ such that a < b and for all i < k,

o^{∗}(a(i), b(ϕ(i))) = χ(i).

By a “stretching” argument which can be found at the beginning of Section 4 of [19], Theorem 5.4 yields the following corollary.

Corollary 5.5. ω_{1} ↛ [ω_{1}; ω_{1}]_{ω_{1}}^{2}.

Theorem 5.4 is an immediate consequence of the following theorem.

Theorem 5.6. Let 𝒜 ⊆ [ω_{1}]^{k} and ℬ ⊆ [ω_{1}]^{l} be uncountable families of pairwise disjoint sets. For every sequence U_{i} (i < k) of open neighborhoods in 𝕋 and every ϕ : k → l there are a in 𝒜 and b in ℬ such that a < b and for all i < k,

o(a(i), b(ϕ(i))) ∈ U_{i}.

Proof. Without loss of generality, we may assume that each U_{i} is an ε-ball about a point v_{i} for some fixed ε > 0. By refining 𝒜 if necessary, we may assume that the n_{ε} be given by Kronecker’s theorem for the sequence z_{a(i)} (i < k) is uniform for a in 𝒜. Let a in 𝒜 and b_{m} (m < n_{ε}) in ℬ be such that for all i < k, j < l, and m < n_{ε}

a < b_{m}
osc (a(i), b_{m}(j)) = osc (a(i), b_{0}(j)) + m.

For each i < k, put u_{i} = o(a(i), b_{0}(ϕ(i))). By choice of n_{ε}, there is an m < n_{ε} such that for all i < k

|u_{i}z_{a(i)}^{m} − v_{i}| < ε

or, equivalently, o(a(i), b_{m}(ϕ(i))) is in U_{i}. This finishes the proof. □

Question 5.7. (PFA^{7}) If c : [ω_{1}]^{2} → 2, are there A ⊆ ω_{1} and ℬ ⊆ [ω_{1}]^{2} which are uncountable with ℬ being pairwise disjoint and χ : 2 → 2 such that for all α ∈ A and b ∈ ℬ with α < b there is an i < 2 such that c(α, b(i)) ≠ χ(i)?

6. A family of binary relations on ω_{1}

In this section we will consider the family of binary relations on ω_{1} with the Tukey order.

Definition 6.1. [25] (Tukey order) If R and S are binary relations, then we write R ≤ S iff there are functions f : dom(R) → dom(S) and g : ran(S) → ran(R) such that

f(x) S y implies x R g(y).

This order was first considered by Tukey in the class of transitive relations [25]. It makes sense, however, to consider this order in the more general setting of the class of binary relations (see [26]).

Our focus will be on binary relations on ω_{1}. Two fundamental examples are the well order ω_{1} and the family [ω_{1}]^{<ℵ_{0}} of finite subsets of ω_{1} ordered by inclusion. We will also need a few standard operations on relations. If R and S are relations, then R ⊕ S is the relation which is the disjoint union of R and S. If m is a cardinal and R is a relation, then we will let m · R will denote the direct sum of m copies of R. If R and S are two binary relations, then R ∧ S and R ∨ S are the relations with domains dom(R) × dom(S) and ranges ran(R) × ran(S) such that

(a, b) R ∧ S (c, d) iff a R c or b S d,
(a, b) R ∨ S (c, d) iff a R c and b S d.

The “join” R ∨ S is often denoted R × S. It is easily verified that ∧ and ∨ give lower and upper bounds respectively.^{8}

It was observed that the binary relations on ω_{1} which one knew how to construct in ZFC were either below ℵ_{0} · ω_{1} or above [ω_{1}]^{<ℵ_{0}} in the Tukey order.

Example 6.2. Suppose that r_{α} (α < ω_{1}) is a sequence of distinct reals. Define αRβ iff α < β and r_{α} < r_{β}.

The relation R is essentially Sierpiński’s partition which witnesses ω_{1} ↛ (ω_{1})_{2}^{2}. In this case R ≤ ℵ_{0} · ω_{1} by the following Tukey maps:

f(α) = (min{n : r_{α} < q_{n}}, α)
g(α, n) = min{β : α < β ∧ q_{n} < r_{β}}

where {q_{n}}_{n<ω} is an enumeration the rationals. On the other hand, the following example shows that with an additional hypothesis such as ♢ one can construct more complex relations.

7Or any appropriate hypothesis; the same comment applies to Conjecture 4 below.

8In general these need not be optimal — they are in the class of directed relations.

Example 6.3. Let R be a tree order on a set T. If R ≤ ℵ_{0} ·ω_{1}, then T is the union of countably many R-chains. If [ω_{1}]^{<ℵ_{0}} ≤ R, then T contains an uncountable Rantichain. In particular, if (T, R) is a Suslin tree, then R ≰ ℵ_{0} ·ω_{1} and [ω_{1}]^{<ℵ_{0}} ≰ R.

Observations such as these led Todorcevic to make the following conjecture.

Conjecture. (PFA) If R is a binary relation, then either R ≤ ℵ_{0} · ω_{1} or else [ω_{1}]^{<ℵ_{0}} ≤ R.

This can be considered a basis conjecture in the following sense. Let ℛ denote the class of all binary relations R on ω_{1} such that R is not reducible to ℵ_{0} · ω_{1}. Then the above conjecture is just the assertion that ℛ has a single element basis consisting of [ω_{1}]^{<ℵ_{0}}. This conjecture was given further plausibility by the following theorem which implies that it is true for transitive relations.

Theorem 6.4. [22] (PFA) Every transitive relation on ω_{1} is Tukey equivalent to one of the following for some nonnegative integers n_{i} (i < 5):

(i) n_{0} · 1 ⊕ n_{1} · ω ⊕ n_{2} · ω_{1} ⊕ n_{3} · ω × ω_{1} ⊕ n_{4} · [ω_{1}]^{<ℵ_{0}}.

(ii) ℵ_{0} · 1 ⊕ n_{2} · ω_{1} ⊕ n_{3} · ω × ω_{1} ⊕ n_{4} · [ω_{1}]^{<ℵ_{0}}.

(iii) ℵ_{0} · ω_{1} ⊕ n_{4} · [ω_{1}]^{<ℵ_{0}}.

(iv) ℵ_{0} · [ω_{1}]^{<ℵ_{0}}.

(v) =.

We will now see, however, that this conjecture is provably false.

Definition 6.5. Let R denote the relation with domain and range ω_{1} defined by letting αRβ iff α = β or α < β and o^{∗}(α, β) > 0. If X is an uncountable subset of ω_{1}, let R_{X} denote the relation with domain X and range ω_{1} which is the restriction of R to X × ω_{1}.

It is easily verified that the enumeration map ε : ω_{1} → X and the identity map on ω_{1} witness ω_{1} ≤ R_{X} for every uncountable X. If X ⊆ Y, then the inclusion and identity maps witness R_{X} ≤ R_{Y}. The following shows that none of these relations are above [ω_{1}]^{<ℵ_{0}} in the Tukey order.

Theorem 6.6. [ω_{1}]^{<ℵ_{0}} ≰ R.

Proof. Suppose that f : [ω_{1}]^{<ℵ_{0}} → ω_{1} and g : ω_{1} → [ω_{1}]^{<ℵ_{0}} are functions. It suffices to show that they are not Tukey reductions.

Applying the pressing down lemma, find a stationary B ⊆ ω_{1} and a finite x_{0} ⊆ ω_{1} such that g(β) ∩ β = x_{0} for all β in B. Let 𝒳 = {x ∈ [ω_{1}]^{<ℵ_{0}} : x ⊈ x_{0}} and A = {f(x) : x ∈ 𝒳 }. By refining B if necessary, we may assume that if α is in A and β is in B with α < β, then there is an x in 𝒳 such that f(x) = α and x ⊆ β.

If A is countable, then we can find a α in A such that

{x ∈ 𝒳 : f(x) = α}

is uncountable. It is then easy to find an x in 𝒳 such that x is not contained in g(α) but f(x) = α and hence f(x) R β, witnessing that f, g are not Tukey maps.

If A is uncountable, apply Theorem 5.4 to obtain an α in A and a β in B such that α < β and o^{∗}(α, β) > 0. Now pick an x in 𝒳 such that f(x) = α and x ⊆ β and observe that x is not contained in g(β) and yet f(x) R β. □

Theorem 6.7. For all finite sets F of uncountable subsets of ω_{1},

⋀_{X∈F} R_{X} ≰ ℵ_{0} · ω_{1}.

Proof. By replacing F with a disjoint refinement if necessary, we may assume without loss of generality that F consists of pairwise disjoint sets. It suffices to show that if

f : ∏_{X∈F} X → ω × ω_{1}  and  g : ω × ω_{1} → ∏_{X∈F} ω_{1},

then f and g are not Tukey maps. For convenience, we will identify elements of ∏_{X∈F} X with their range. Let S denote the relation on ω × ω_{1} given by ℵ_{0} · ω_{1}. Find uncountable pairwise disjoint 𝒜 ⊆ [ω_{1}]^{|F |} and an n such that:

(17) For each element a of 𝒜, a ∩ X is a singleton for all X in F.

(18) f ↾ 𝒜 is an injection into {n} × ω_{1}.

(19) The ordering on F which is induced by an element a of 𝒜 — X is the i^{th}-least element of F if a(X) is the i^{th}-least coordinate of a — does not depend on a.

This is possible since if ∏_{X∈F} X is partitioned into countably many pieces, then one piece must contain an uncountable pairwise disjoint family. If the g-image of {n} × ω_{1} is countable, then it is not possible for f and g to be Tukey maps; this is a consequence of the following argument in which ℬ̃ is replaced by the trivial Δ-system with a single element.

Hence we may select an uncountable ℬ̃ contained in the image of {n}×ω_{1} under g such that the collection of sets of coordinates of elements of ℬ̃ forms a Δ-system with root r. Let ℬ be the elements of this Δ-system with r removed and let k denote the uniform cardinality of the members of ℬ. By refining ℬ̃ if necessary, we may further assume that there is a function ϕ : |F | → k ∪ {−1} such that if X is the i^{th} element in the induced enumeration of F and b̃ is in ℬ̃, then b̃(X) is the ϕ(i)^{th}-least element of ran(b̃) \ r if b̃(X) is not in r and ϕ(i) = −1 otherwise. Let χ be constantly 0 on the subset of |F | on which ϕ is non-negative.

Applying Theorem 5.4, it is possible to find an a in 𝒜, a b̃ in ℬ̃, and a ζ with g(n, ζ) = b̃ such that r < a < min(b \ r), f(a) is S-related to (n, ζ), and for all X in F for which b̃(X) is not in r

o^{∗}(a(X), b̃(X)) = 0

where b denotes the set of coordinates of b̃.

If X is in F, then either b̃(X) is in r in which case b̃(X) < a(X) or else b̃(X) is not in r and c(a(X), b̃(X)) = 0. In either case a(X) is not R_{X}-related to b̃(X). Since this is true for arbitrary X, a is not ⋀_{X∈F} R_{X}-related to g(x) and yet f(a)Sx. Consequently f and g are not Tukey maps. □

The next theorem shows that the relations R_{X} (X ⊆ ω_{1}) are typically incomparable.

Theorem 6.8. If X \ Y is uncountable, then R_{X} ≰ R_{Y}.

Proof. Suppose for contradiction that f : X → Y and g : ω_{1} → ω_{1} are Tukey maps. To obtain a contradiction, it suffices to find an α in X and a β in ω_{1} such that f(α) < β, α < g(β), o^{∗}(f(α), β) = 1, and o^{∗}(α, g(β)) = 0. Since f and g are Tukey reductions it is possible to find a uncountable X_{0} ⊆ X \ Y and uncountable Z_{0} ⊆ ω_{1} such that f ↾ X_{0} and g ↾ Z_{0} are injections, the f-image of X_{0} is disjoint from X_{0}, the g-image of Z_{0} is disjoint from Z_{0} unless g is eventually equal to the identity, and the inequalities α < f(α), β = g(β), β < g(β) are uniformly true or false as α ranges over X_{0} and β ranges over Z_{0}. Put

𝒜 = {{α, f(α)} : α ∈ X_{0}}, ℬ = {{β, g(β)} : β ∈ Z_{0}}

and notice that these families are uncountable and consist of pairwise disjoint sets of uniform cardinality which, in the case of 𝒜, is two. By our uniformity assumption on the relations < and =, it is possible to find a functions ϕ : 2 → 2 and χ : 2 → 2 such that:

(20) {a(i), b(ϕ(i))} (i < 2) lists the pairs {α, g(β)} and {f(α), β}.

(21) χ(i) = 1 if {a(i), b(ϕ(i))} = {f(α), β} and χ(i) = 0 otherwise. Applying Theorem 5.4 to 𝒜, ℬ, ϕ, and χ, there are a in 𝒜 and b in ℬ such that a < b and

o^{∗}(a(i), b(ϕ(i))) = χ(i).

Translating the outcome, it is easily checked that α ∈ a ∩ X_{0} and β ∈ b ∩ Z_{0} are as desired. □

I will close this section with two questions.

Question 6.9. For which families ℱ ⊆ [ω_{1}]^{ℵ_{1}} is there an S such that S ≰ ℵ_{0} · ω_{1} and S ≤ R_{X} for all X in ℱ?

Question 6.10. Is the collection of all S with S ≰ ℵ_{0} · ω_{1} downwards-directed in the Tukey order?

7. An L space and the non-existence of a small basis for the regular Hausdorff spaces.

In this section I will give an example of an L space — a regular Hausdorff space which is hereditarily Lindelöf but not hereditarily separable. The question of the existence of such spaces was first asked explicitly in [5], though arguably this question can be traced to Sierpinski’s [14] where he constructed Hausdorff examples distinguishing HS and HL and to Suslin’s [15] where he posed his famous hypothesis. For instance, an immediate consequence of Kurepa’s [11] is that a Suslin line is an example of an L space. From the 1960s until the 1980s, there was a concerted effort to understand both L spaces and their “dual,” the S space. I refer the reader to Juhasz’s [7] and M. E. Rudin’s [13] as well as Roitman’s more recent [12] for more discussion on these developments. I have selected a few to mention here.

First, Zenor showed that there was a relationship between the existence of certain S and L spaces.

Theorem 7.1. [27] There is a strong S space iff there is a strong L space.^{9}

Hence the difference in the existence of S and L spaces lies in the properties of their finite powers. This gives some explanation as to why the existence of S and L spaces seem to be such similar hypotheses at first.

There are a number of results under MA_{ℵ_{1}} which limit the existence of S and L spaces.

9Here a space is a strong S (L) space iff all of its finite powers are S (L) spaces.

Theorem 7.2. [9] (MA_{ℵ_{1}}) There are no strong S or L spaces.

Theorem 7.3. [16] (MA_{ℵ_{1}}) There are no first countable L spaces.

Theorem 7.4. [2] (MA_{ℵ_{1}}) If K is a compact space which contains an L space, then K maps continuously onto [0, 1]^{ℵ_{1}}.

Finally, Todorcevic proved that S spaces do not exist assuming PFA.

Theorem 7.5. [18] (PFA) If a regular Hausdorff space is hereditarily separable, then it is hereditarily Lindelöf.

He also demonstrated that the existence of S and L spaces are different considerations.

Theorem 7.6. [21] It is consistent with MA_{ℵ_{1}} that there are no S spaces and yet that there is an L space.

Until now, however, it remained unclear whether Todorcevic’s methods could be used to prove that PFA implies there are no L spaces. I will now show that this is not the case — that there is an L space which can be constructed without appealing to additional axioms of set theory.

Definition 7.7. For each β < ω_{1}, define w_{β} : ω_{1} → 𝕋 by w_{β}(ξ) = o(ξ, η) if ξ < β and w_{β}(ξ) = 1 otherwise. Let ℒ be the collection {w_{β} : β < ω_{1}} viewed as a subspace of 𝕋^{ω_{1}}.

The “+1” in the exponent of o(ξ, β) is present so that w_{β}(ξ) = 1 iff ξ ≥ β. This will simplify the discussion at certain points. Notice that ℒ is non-separable.

If X ⊆ ω_{1}, then let ℒ_{X} be {w_{β} ↾ X : β ∈ X} viewed as a subspace of 𝕋^{X}. Since the map w_{β} ↦ w_{β} ↾ X defined on {w_{β} : β ∈ X} is a bijection, we will write w_{β} for w_{β} ↾ X when referring to elements of ℒ_{X}. It should be emphasized, however, that while the above bijection is continuous, its inverse may not be. Hence, even with this identification, ℒ_{X} should not be viewed as a subspace of ℒ.

We will now see that ℒ_{X} is Fréchet. Let 𝒰 be the collection of open arcs in 𝕋 whose endpoints are roots of 1 and which do not contain 1 as an element. Let ℱ be the collection of all sets of the form {w ∈ ℒ : w(α) ∈ U} where U is an element of 𝒰 and α is in X. Notice that ℱ is point countable and that the topology on ℒ_{X} is the smallest in which every element of ℱ is clopen. The following theorem shows that ℒ_{X} is always Fréchet.

Theorem 7.8. If X is a set and ℱ is a point countable point, separating family of subsets of X, then the topology on X defined by declaring elements of ℱ to be clopen is countably tight and has every countable subspace metrizable. In particular the topology is Fréchet.

Remark 7.9. I attribute this result to Zoltan Balogh. He once told me that “If there is an L space, then there is a countably tight one.” While I never saw his proof, I think it is reasonable to assume this he may have proceeded along these lines.

Proof. That countable subspaces are metrizable follows from the assumptions and the well known fact that regular Hausdorff second countable spaces are metrizable. To see that the space is countably tight, let A ⊆ X have an accumulation point x in X. Let M be a countable elementary submodel of H(θ) for θ regular and large enough so that X, ℱ, x, and A are all in M. It suffices to show that x is an accumulation point of A ∩ M.

Suppose this is not the case and let U_{i} (i < k) and V_{i} (i < l) be elements of ℱ such that

W = ⋂_{i<k} U_{i} \ ⋃_{j<l} V_{j}

contains x and is disjoint from A ∩ M. Without loss of generality, we may assume that the empty set is not in ℱ. The important observation is this: if V is in ℱ, then V ∩ M is non-empty iff V is in M. This is because {V ∈ ℱ : y ∈ V } is countable for all y in X and therefore a subset of M whenever y is in M (by Facts 5 and 7). Hence each U_{i} must be in M since x is in U_{i} and x is in M. Moreover, since we are only interested in having W be disjoint from A ∩ M, we may assume without loss of generality that each V_{j} is in M. But then W must be in M and, by elementarity of M, there must be an element of W ∩ A which is in M, a contradiction. □

As mentioned above, if X is uncountable, then ℒ_{X} can be shown to be an L space using Theorem 5.6 and the methods of [5]. I will prove the following theorem of independent interest and then derive this as a consequence.

Theorem 7.10. If X, Y ⊆ ω_{1} have countable intersection, then there is no continuous injection from any uncountable subspace of ℒ_{X} into ℒ_{Y}.

Proof. Suppose for contradiction that such an injection g does exist. Then g is of the form w_{β} ↦ w_{f(β)} where f : X_{0} → Y is an injection for some uncountable X_{0} ⊆ X which is, without loss of generality, disjoint from Y. For each ξ < ω_{1}, let β_{ξ} and ζ_{ξ} be elements of X_{0} and Y respectively such that f(β_{ξ}) > ζ_{ξ} and if ξ < ξ′ then β_{ξ} < ζ_{ξ′}. Let Ξ ⊆ ω_{1} be uncountable such that for some open neighborhood V in 𝕋, g(w_{β_{ξ}})(ζ_{ξ}) is not in the closure of V whenever ξ is in Ξ.

Applying continuity of g at w_{β_{ξ}} to

W_{ξ} = {w ∈ ℒ_{Y} : w(ζ_{ξ}) ∉ V̄ },

there is a basic open neighborhood U_{ξ} of w_{β_{ξ}} such that U_{ξ} ⊆ g^{−1}W_{ξ}. Applying the Δ-system lemma and the second countability of 𝕋, there is an uncountable Ξ′ ⊆ Ξ, open neighborhoods U_{i} (i < k) in 𝕋, and a_{ξ} in [X]^{k} such that for all ξ in Ξ′:

(1) {a_{ξ} : ξ ∈ Ξ′} is a Δ-system with root a.

(2) The set

{w ∈ ℒ_{X} : ∀i < k(w(a_{ξ}(i)) ∈ U_{i})}

has w_{β_{ξ}} as an element and is a subset of U_{ξ}.

(3) The inequality β_{ξ} < f(β_{ξ}) does not depend on ξ.

(4) |ζ_{ξ} ∩ a_{ξ}| does not depend on ξ. Let 𝒜 be the collection of all a_{ξ} ∪{ζ_{ξ}}\a and ℬ be the collection of all {β_{ξ}, f(β_{ξ})}. Applying Theorem 5.6, it is possible to find ξ < ξ′ in Ξ′ such that for all i < k

a_{ξ} ∪ {ζ_{ξ}} < min(β_{ξ′}, f(β_{ξ′})),
w_{β_{ξ′}} (a_{ξ}(i)) = o(a_{ξ}(i), β_{ξ′}) ∈ U_{i},
g(w_{β_{ξ′}}) = w_{f(β_{ξ′})}(ζ_{ξ}) = o(ζ_{ξ}, f(β_{ξ′})) ∈ V.

But now w_{β_{ξ′}} is an element of U_{ξ} even though g(w_{β_{ξ′}}) is not in W_{ξ}, contradicting our choice of U_{ξ}. This finishes the proof. □

Corollary 7.11. For every X, ℒ_{X} is hereditarily Lindelöf.

Proof. If not, then ℒ_{X} would contain an uncountable discrete subspace. It is then possible to find disjoint Y, Z ⊆ X such that ℒ_{Y} and ℒ_{Z} each contain uncountable discrete subspaces. But any function from a discrete space to another discrete space is continuous and we can then easily contradict Theorem 7.10. □

Kunen has shown, however, that under MA_{ℵ_{1}} every L space contains an uncountable discrete subspace in one of its finite powers [9]. As might be expected, this happens at the first possible instance in our example.

Theorem 7.12. For every uncountable X ⊆ ω_{1}, ℒ_{X} ^{2} contains an uncountable discrete subspace.

Proof. This essentially follows from the following observation which is of independent interest.

Proposition 7.13. The tree

T (o) = {o(·, β) ↾ α : α ≤ β < ω_{1}}

is Aronszajn.

Proof. Theorem 5.6 implies that T (o) does not contain an uncountable branch. It suffices to prove that all levels of T (o) are countable. For this, it is sufficient to show that T (ϱ_{1}) and T (L) have countable levels. In the case of T (ϱ_{1}), this is the content of Remark 2.8. In order to see that T (L) has countable levels, let α < ω_{1} be given. Let β < ω_{1} be greater than α. Using the compactness of α + 1 and Fact 2, select a finite set F_{β} ⊆ α + 1 containing 0 and α so that if γ_{0} < γ are consecutive elements of F_{β}, then L(γ, β) < L(ξ, γ) whenever γ_{0} < ξ < γ. It suffices to show that if

L(·, β) ↾ F_{β} = L(·, β′) ↾ F_{β′},

then

L(·, β) ↾ α = L(·, β′) ↾ α.

To see this, let ξ be an arbitrary element of α \ F_{β} and pick γ_{0} < γ in F_{β} so that γ_{0} < ξ < γ. Applying Fact 1 we have

L(ξ, β) = L(ξ, γ) ∪ L(γ, β) = L(ξ, γ) ∪ L(γ, β′) = L(ξ, β′).
□

To finish the proof of Theorem 7.12, select a sequence (β_{ξ}^{0}, β_{ξ}^{1}) indexed by an uncountable set Ξ ⊆ ω_{1} such that for each ξ in Ξ the following conditions are satisfied:

(22) β_{ξ}^{0} < β_{ξ}^{1} are both elements of X.

(23) There is a fixed ε > 0 such that

|w_{β_{ξ}^{0}}(β_{ξ}^{0}) − w_{β_{ξ}^{1}}(β_{ξ}^{0})| = |1 − w_{β_{ξ}^{1}}(β_{ξ}^{0})| ≥ ε.

(24) If η < ξ is in Ξ, then β_{η}^{0} < ξ.

(25) w_{β_{ξ}^{0}} ↾ ξ = w_{β_{ξ}^{1}} ↾ ξ.

This is possible since T (o) has countable levels. Now consider the open neighborhoods

U_{ξ} = {(u, v) ∈ ℒ_{X} : |w_{β_{ξ}^{0}}(β_{ξ}^{0}) − u(β_{ξ}^{0})| + |v(β_{ξ}^{0}) − w_{β_{ξ}^{1}}(β_{ξ}^{0})| < ε}

of (w_{β_{ξ}^{0}}, w_{β_{ξ}^{1}}) in ℒ_{X} ^{2} for ξ in Ξ. If η < ξ are in Ξ, then by (23) (w_{β_{η}^{0}}, w_{β_{η}^{1}}) is not in U_{ξ} since w_{β_{η}^{1}}(β_{ξ}^{0}) = 1. If ξ < η are in Ξ, then by (24) β_{ξ}^{0} < η and by (25)

w_{β_{η}^{0}}(β_{ξ}^{0}) = w_{β_{η}^{1}}(β_{ξ}^{0}).

Consequently (w_{β_{η}^{0}}, w_{β_{η}^{1}}) being in U_{ξ} would contradict (23) via the triangle inequality. Hence (w_{β_{η}^{0}}, w_{β_{η}^{1}}) is in U_{ξ} iff η = ξ and therefore {(w_{β_{ξ}^{0}}, w_{β_{ξ}^{1}}) : ξ ∈ Ξ} is an

uncountable discrete subspace of ℒ_{X} ^{2} as desired. □

At this point, I will make the following conjecture.

Conjecture 4. (PFA) If X is a non-separable, regular Hausdorff space, then X^{2} contains an uncountable discrete subspace.

While ℒ is 0-dimensional, its closure in 𝕋^{ω_{1}} is very far from being 0-dimensional.

Theorem 7.14. If E ⊆ ℒ is uncountable, then the closure of E in 𝕋^{ω_{1}} contains a set of the form {f ∈ 𝕋^{ω_{1}} : f ↾ α = τ} for some α < ω_{1} and τ in 𝕋^{α}. In particular, cl(E) is not 0-dimensional.

Remark 7.15. This example may be informative in studying the following problem of M. E. Rudin: Is it consistent that every locally connected perfect compact space is metrizable? See [4].

Proof. Let E ⊆ ℒ be uncountable and put T (E) be the set of all τ in T (o) such that for uncountably many β in E, τ is a restriction of w_{β}. If τ is in T (E), put E_{τ} to be the set of all w_{β} in E such that τ is a restriction of w_{β}.

Claim 7.16. For each τ in T (E) there is an α_{τ} < ω_{1} such that the projection map π : 𝕋^{ω_{1}} → 𝕋^{ω_{1}\α_{τ}} sends the closure of E_{τ} onto 𝕋^{ω_{1}\α_{τ}}.

Proof. Suppose for contradiction that this is not the case. For each ξ < ω_{1}, select a basic open neighborhood W_{ξ} ⊆ 𝕋^{ω_{1}} such that W_{ξ} ∩E_{τ} is empty and membership to W_{ξ} depends only on coordinates greater than ξ. By refining the sequence W_{ξ} (ξ < ω_{1}) it is possible to find open neighborhoods U_{i} (i < k) in 𝕋 and an uncountable pairwise disjoint 𝒜 ⊆ [ω_{1}]^{k} such that if a is in 𝒜, then there is no w_{β} in E_{τ} with

w_{β}(a(i)) ∈ U_{i}

for all i < k. Pairing 𝒜 with ℬ = {{β} : w_{β} ∈ E_{τ} }, we have a contradiction to Theorem 5.6. □

Let δ < ω_{1} be such that if τ is an element of T (E) of height less than δ, then α_{τ} < δ. Since T (o) has countable levels, such a δ exists. Since T (E) is uncountable and downwards closed, there is a τ in T (E) of height δ. Put

F = ⋂_{ξ<δ} cl(E_{τ↾ξ}).

Notice that the projection of each cl(E_{τ↾ξ}) onto the set of coordinates ω_{1} \ δ is onto 𝕋^{ω_{1}\δ}. Since the fibers of this map are compact and since the intersection is decreasing, F must project onto 𝕋^{ω_{1}\δ}. Since every element of F is an extension of τ,

F = {f ∈ 𝕋^{ω_{1}} : f extends τ}

is the desired subset of cl(E). □

The proof of Theorem 7.3 can be used to prove the following.

Theorem 7.17. For every X ⊆ ω_{1}, no uncountable subspace of ℒ_{X} is first countable.

Proof. The proof of Theorem 7.3 actually shows that if an L space is first countable, then there is a c.c.c. forcing which destroys it. It is easily checked that every uncountable subspace of ℒ_{X} is an L space in every c.c.c. forcing extension. □

We also have the following consequence of Proposition 7.13.

Theorem 7.18. If f is a continuous function from a subspace of ℒ into a metric space, then the range of f is countable.

Proof. Since ℒ ⊆ 𝕋^{ω_{1}} is hereditarily Lindelöf and f is continuous, f depends only on a countable set of coordinates. The conclusion of the theorem now follows from Proposition 7.13. □

Theorem 7.19. Any basis for the uncountable regular Hausdorff spaces must have cardinality greater than ℵ_{1}.

Proof. The first conclusion follows from Theorem 7.10 and the observation that there is an almost disjoint family of uncountable subsets of ω_{1} of cardinality ℵ_{2}. □

I will close this paper by referring the reader to [4] where a number of open problems related to this line of research are collected and discussed.

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Abstract. In this paper I will construct a non-separable hereditarily Lindelöf space (L space) without any additional axiomatic assumptions. The constructed space ℒ is a subspace of 𝕋^{ω_{1}} where 𝕋 is the unit circle. It is shown to have a number of properties which may be of additional interest. For instance it is shown that the closure in 𝕋^{ω_{1}} of any uncountable subset of ℒ contains a canonical copy of 𝕋^{ω_{1}}.

I will also show that there is a function f : [ω_{1}]^{2} → ω_{1} such that if A, B ⊆ ω_{1} are uncountable and ξ < ω_{1}, then there are α < β in A and B respectively with f(α, β) = ξ. Previously it was unknown whether such a function existed even if ω_{1} was replaced by 2. Finally, I will prove that there is no basis for the uncountable regular Hausdorff spaces of cardinality ℵ_{1}.

The results all stem from the analysis of oscillations of coherent sequences ⟨e_{β} : β < ω_{1}⟩ of finite-to-one functions. I expect that the methods presented will have other applications as well.

Department of Mathematics, Boise State University, Boise, ID 83725

   E-mail address: justin@math.boisestate.edu