LOCALLY COMPACT, LOCALLY COUNTABLE SPACES AND RANDOM REALS

J. TATCH MOORE

Abstract. In this note I will present a proof that, assuming PFA, if R is a measure algebra then after forcing with R every uncountable locally compact locally countable cometrizable space contains an uncountable discrete set. The lemmas and techniques will be presented in a general form as they may be applicable to other problems.

1. Introduction

The purpose of this paper is to further the understanding of which topological and combinatorial consequences of MA_{ℵ_{1}} (and forcing axioms in general) can hold after forcing with a measure algebra. This program began with Laver’s result which states that, assuming MA_{ℵ_{1}}, all Aronszajn trees are special after forcing with any measure algebra[5]. This was used to establish the consistency of Suslin’s hypothesis with the continuum having arbitrary cardinality.^{1} Todorčević extended this result to show that, assuming MA_{ℵ_{1}}, after forcing with a measure algebra the conjecture (L) is true for any regular space with a countably tight compactification [11]. This paper can be considered as a continuation of [7] — which investigates the conjectures (S) and (L) for various classes of space in this context. The reader is assumed to have some familiarity with arguments involving MA_{ℵ_{1}} and forcing axioms.^{2}

In [9] Todorčević suggested that studying random forcing extensions of models of MA_{ℵ_{1}} might yield a better understanding of perfectly normal compacta and in particular a solution to Katětov’s problem. A corollary of his result in [11] is that in such forcing extensions all perfectly normal compacta are separable. Moreover he demonstrates in [9] that any counterexample providing a solution to Katětov’s problem in a forcing extension of MA_{ℵ_{1}} by a nonseparable measure algebra must have a square which is a compact S-space. Recently in [4] Todorčević and Larson showed that Katětov’s problem is independent of the usual axioms of set theory by using fragments of forcing axioms to analyze different types of generic extensions. Still, an analysis of random combinatorial objects using MA_{ℵ_{1}} and other forcing axioms is of interest in that it may yield a better understanding of Katětov’s problem and perfect compacta in general.

1Laver provided the last case in which ℝ has singular cardinality.

2[10] — chapters 7 and 8 in particular — is a useful reference, both for the results it contains and for its bibliography.

In [2] Gruenhage used CH to construct an example of compact space X with a hereditarily separable and hereditarily normal but nonmetric square. This and Nyikos’s example under MA_{ℵ_{1}} [2] are the only known consistent counterexamples to Katětov’s problem. Gruenhage’s construction closely followed a construction due to Kunen [3] of a locally countable locally compact strengthening of the topology on ℝ whose closure operator differs from the metric closure by a countable set — commonly known as a a Kunen line. In fact a Kunen line appears as a subspace of the square of Gruenhage’s example. Todorčević has constructed a Kunen line on any ω_{1}-sequence of reals using only the assumption that 𝔟 = ω_{1} [10]. In this paper I will prove the following.

Theorem 1.1. (PFA) After forcing with any measure algebra, every uncountable locally compact, locally countable cometrizable^{3} space contains an uncountable discrete space. In particular there are no Kunen lines in such forcing extensions.

This establishes the following consistency result which in particular suggests that the hypothesis 𝔟 = ω_{1} which Todorčević uses to construct a Kunen line is somewhat optimal.^{4}

Theorem 1.2. It is relatively consistent that there are no Kunen lines, there is a set of reals of size ℵ_{1} of positive measure, and the continuum is any cardinal greater than ℵ_{1} having uncountable cofinality.

It should be noted also that the hypotheses on the topological space in Theorem 1.1 are somewhat optimal since it is demonstrated in [7] that neither local compactness nor cometrizability can be removed if the measure algebra is nonseparable.

In addition to any interest in the theorem itself, the proof is of significance for two reasons. First, I will prove a set of ZFC results — Theorem 2.2 and Lemmas 3.4-3.7 — which govern the probabilistic behavior of random names for elements of separable metric spaces. Second, for the first time MA_{ℵ_{1}} did not seem sufficient for our analysis of the random graphs involved. Certainly there are consequence of PFA (such as the non-existence of Kurepa trees) which require PFA in the corresponding analysis of V^{R}. In our case, though, MA_{ℵ_{1}} is sufficient to imply that Kunen’s construction can’t be carried out in V yet the stronger PFA seems necessary when proving the analogous theorem in V^{R}.

3Recall that a space X is cometrizable if there is a metric topology on the underlying set such that every point has neighborhood base of sets which are closed in the metric topology.

4The covering number for the Lebesgue null ideal is the only cardinal in Cichoń’s diagram which is not either at least 𝔟 or at most non(N).

2. Some notation and background

When considering forcing extensions obtained by adjoining sequences of random reals to a ground model, we will take the Boolean algebraic approach and view these as forcing extensions by measure algebras. Here a measure algebra (R, μ) is defined to be a complete Boolean algebra R together with a strictly positive probability measure μ : R → [0, 1]. If there is no opportunity for confusion we will write R instead of (R, μ).

The prototypical examples of measure algebras are the Haar algebras (R_{θ}, μ). Here μ is the product measure on the Baire subsets of 2^{θ}, where 2 = {0, 1} is given the uniform probability measure. R_{θ} is then obtained by taking the quotient by the μ-null sets. By a deep result of Maharam [6], these are the only homogeneous measure algebras. A measure algebra is separable if it is completely generated by a countable set.

We will need the following theorem due essentially to Laver and isolated as a theorem unto itself in [8].

Theorem 2.1. (MA_{ℵ_{1}}) If R is a measure algebra and Ġ : [ω_{1}]^{2} → R is a R-name for a graph on ω_{1} then either

(1) There is a sequence R-names Ẋ_{n} : ω_{1} → R indexed by ω such that for all α < ω_{1}

⋁_{n<ω} Ẋ_{n}(α) = 1

and for all n and α, β < ω_{1}

Ẋ_{n}(α) ∧ Ẋ_{n}(β) ∧ Ġ(α, β) = 0

(i.e. Ġ is forced to be countably chromatic) or else

(2) there is a sequence F_{ξ} (ξ < ω_{1}) of disjoint finite subsets of ω_{1} and a δ > 0 such that for all ξ ≠ η

⋁_{α∈F_{ξ}} ⋁_{β∈F_{η}} Ġ(α, β)

has measure at least δ.

The proof of this theorem is carried out explicitly in [8]. The techniques of the proof already appear in [5] and the argument can readily be extracted from section 2 of [11]. The reader is encouraged to extract a proof from the techniques used to prove Lemma 4.5 below.

We will also need the following generalization of Theorem 3 from [11]. The proof is reproduced from [8] for completeness and due to its brevity.

Theorem 2.2. Suppose that R and S are homogeneous measure algebras such that R is a subalgebra of S. If S_{0} is a subalgebra of S which has character less than that of R then there is a measure preserving homomorphism h : S_{0} → R such that for all a in S_{0}, h(a) ≤ π_{R}(a).

Remark 2.3. Here π_{R}(a) is the projection of a in the subalgebra R — the meet of all b in R which satisfy a ≤ b. This result reduces to Todorčević’s when one considers the case that S is an ultrapower of R.

Proof. Let R_{0} be the image of S_{0} under the projection map. Define h_{0} so that it fixes the elements of R_{0}. Using a standard lemma from the proof of Maharam’s theorem (see, e.g., Lemma 3.4 of [1]), extend h_{0} to a measure preserving homomorphism h defined on the algebra generated by S_{0} ∪ R_{0}. Now observe that if a is in S_{0} then h(a) ≤ π_{R}(a) = h(π_{R(a)}). □

3. ω_{1}-sequences of random elements of [0, 1]^{ω}

Before we begin with the proof of the main result, it will be useful to prove a few lemmas which concern the behavior of sequences of R-names for elements of a separable metric space (Ẋ, ḋ) where (R, μ) is a measure algebra. First recall that every separable metric space is homeomorphic to a subspace of ([0, 1]^{ω}, d) where d is a metric compatible with the product topology on [0, 1]^{ω}. Hence we will concentrate on R-names for elements of [0, 1]^{ω}.

Let (S, μ) be a measure algebra, R be a complete subalgebra of S, and ẋ be an S-name such that it is forced that ẋ is in [0, 1]^{ω}. For concreteness we will fix the following definitions.

Definition 3.1. ẋ is an R-name if for every i in ω and rational q

[[ẋ(i) < q]]

is in R. Equivalently, there is an R-name ẏ such that 1 forces ẋ = ẏ.

Definition 3.2. An element A of S decides ẋ to be an R-name if for every i in ω and rational q

[[ẋ(i) < q]]

is the meet of A with an element of R. Equivalently, there is a R-name ẏ such that A forces ẋ = ẏ.

Definition 3.3. ẋ is forced not to be an R-name if there is no positive A in S which decides ẋ to be an R-name.

The following lemma provides a useful equivalence to being forced not to be an R-name.

Lemma 3.4. Suppose that S, R, and ẋ are as above. The following are equivalent:

(1) For every ε > 0 there is a δ > 0 such that for every R-name ẏ for an element of [0, 1]^{ω}

μ([[d(ẋ, ẏ) ≥ δ]]) > 1 − ε

(2) ẋ is forced not to be an R-name.

Proof. The implication “1 implies 2” is trivial since if ẏ is an R-name and A is a positive element of S which forces ẋ = ẏ, no δ > 0 can be found for ε = 1 − μ(A).

To see that 2 implies 1, suppose that 1 fails and, for some ε > 0, pick a sequence of R-names ẏ_{n} and elements B_{n} of S such that μ(B_{n}) ≥ ε and B_{n} forces d(ẋ, ẏ_{n}) < 2^{−n}. Define an S-name Ė for a subset of ω by [[k ∈ Ė]] = B_{k}. The condition B = ⋀_{n=0}^{∞} ⋁_{k=n}^{∞} B_{k} forces that Ė is infinite and that {ẏ_{k} : k ∈ Ė} converges to ẋ. It is enough to prove that for some positive A ≤ B and some R-name ẏ the condition A forces {ẏ_{k} : k ∈ Ė} converges to ẏ.

Form a pair of homogeneous measure algebras R^{∗} ⊆ S^{∗} such that

(1) S^{∗} contains S as a subalgebra,

(2) S^{∗} is generated by R^{∗} ∪ S,

(3) R^{∗} is non-separable, and

(4) for all A ∈ S and B ∈ R^{∗}, A and B are independent given π_{R}(A) ∧ π_{R}(B).

Let S_{0} be a separable subalgebra of S^{∗} generated by the sequence {B_{n}}_{n=0}^{∞}. Fix a measure preserving homomorphism h : S_{0} → R^{∗} such that h(B) ≤ π_{R}(B) = π_{R^{∗}}(B) for all B in S_{0}. Define an R^{∗}-name Ė^{∗} for a subset of ω by putting [[k ∈ Ė^{∗}]] = h(B_{k}).

Set

C = ⋀_{n=0}^{∞} ⋁_{k=n}^{∞} (B_{k} ∧ h(B_{k})) .

First note that, by independence of the events B_{k} and h(B_{k}) below π_{R}(B_{k}) = π_{R}(h(B_{k})),

μ(B_{k} ∧ h(B_{k})) = \frac{μ(B_{k}) · μ(h(B_{k}))}{μ(π_{R}(B_{k}))} ≥ ε^{2} > 0

and hence C is positive. Similarly if A ≤ B is a positive element of S then C ∧ A is positive. Furthermore C forces that Ė ∩ Ė^{∗} is infinite and that {ẏ_{n} : n ∈ Ė^{∗}} converges to ż for some R^{∗}-name ż. The only way for this to happen is if there is no A ≤ B which forces that ẋ is not a R-name (if G is S^{∗}-generic then V [G ∩ R^{∗}] ∩ V [G ∩ S] = V [G ∩ R]). Hence there must be a positive A ≤ B and a R-name ẏ such that A forces {ẏ_{n} : n ∈ Ė} converges to ẏ. □

We will need the following definition.

Definition 3.5. An increasing chain R_{α} (α < ω_{1}) of measure algebras is said to be continuous if for every limit ordinal α, R_{α} is completely generated by ⋃_{γ<α} R_{γ}.

For the next two lemmas R_{α} (α < ω_{1}) will be an increasing continuous chain of separable measure algebras, each of which are subalgebras of R. For each α < ω_{1}, ẋ_{α} is an R_{α}-name for an element of [0, 1]^{ω}. We will need the following two lemmas.

Lemma 3.6. For every stationary S ⊆ ω_{1} and every ε > 0 there is a stationary set S′ ⊆ S such that for all α, β in S′

μ([[d(ẋ_{α}, ẋ_{β}) < ε]]) > 1 − ε.

Lemma 3.7. If S, T ⊆ ω_{1} are stationary sets then for every ε > 0 there are stationary sets S′ ⊆ S and T′ ⊆ T and a δ > 0 such that for all α in S′ and β in T′

μ([[d(ẋ_{α}, ẋ_{β}) ≥ δ]]) > 1 − ε.

Proof. (of Lemma 3.6) For each γ < ω_{1} pick a countable set Z_{γ} of R_{γ}-names for elements of [0, 1]^{ω} such that if ẋ is an R_{γ}-name for an element of [0, 1]^{ω} and ε_{0} > 0 then there is a ż in Z_{γ} such that

μ([[d(ẋ, ż) < ε_{0}]]) > 1 − ε_{0}.

Now let ε > 0 be given. Notice that if α < ω_{1} is a limit ordinal then there is a γ < α and a ż_{α} in Z_{γ} with

μ([[ḋ(ẋ_{α}, ż_{α}) < ε/2]]) > 1 − ε/2.

Now, by applying the pressing down lemma, it is possible to find a single γ < ω_{1} and ż in Z_{γ} such that

S′ = {α ∈ S : ż_{α} = ż}

is stationary. It is now easily checked that the set S′ satisfies the conclusion of the lemma. □

Proof. (of Lemma 3.7) Use Lemma 3.6 to select a decreasing sequence S_{k} (k < ω) of stationary subsets of S such that for each k < ω and α, α′ in S_{k}

μ([[d(ẋ_{α}, ẋ_{α′}) < 2^{−k}]]) > 1 − 2^{−k}.

Let ż be the R_{ω_{1}}-name for the element of [0, 1]^{ω} to which the sets

{ẋ_{α} : α ∈ S_{k}}

converge. Now pick a δ > 0 such that

T′ = {β ∈ T : μ([[d(ż, ẋ_{β}) ≥ 2δ]]) > 1 − ε/2}

is stationary. Now let S′ = S_{k} where k is large enough so that

2^{−k} < min(δ, ε/2).

It is now easily checked that δ, S′, and T′ satisfy the conclusion of the theorem. □

4. The main result

The focus of this section will be to prove the following result.

Theorem 4.1. (PFA) If (R, μ) is any measure algebra and K̇ is an R-name for an uncountable locally compact, locally countable, cometrizable space then K̇ is forced to contain an uncountable discrete set.

Let (R, μ) and K̇ be as in the hypothesis of the theorem. Observe that K̇ contains a subspace of size ℵ_{1} which is also locally compact, locally countable, and cometrizable. Hence we may assume without loss of generality that K̇ is forced to have size ℵ_{1}. Since K̇ is forced to be locally compact, it must be a strengthening of the metric topology. Also, since we are clearly finished if K̇ refines a nonseparable metric topology, we will assume that 1 forces that K̇ refines the metric topology on a subspace of [0, 1]^{ω}. Let ẋ_{α} (α < ω_{1}) be a sequence of R-names which is forced to be an enumeration (without repetition) of the elements of K̇. Our proof will break into cases depending on the nature of Ẋ = {ẋ_{α} : α < ω_{1}}. These are handled by Lemmas 4.2 and 4.5. Let Ė_{α} be an R-name for a countable compact subset of Ẋ such that ẋ_{α} is forced to be in Ė_{α} and Ė_{α} is a neighborhood of ẋ_{α} in K̇. Fix an increasing continuous chain of complete separable subalgebras R_{α} (α < ω_{1}) such that if γ < α then both ẋ_{γ} and Ė_{γ} are added by R_{α}.

Lemma 4.2. (MA_{ℵ_{1}}) If there is a stationary set of υ < ω_{1} such that for all β ≥ υ ẋ_{β} is forced not to be an R_{υ}-name then K̇ is forced to be σ-discrete.

Remark 4.3. Notice that if Ẋ is an ω_{1}-sequence of random reals then it satisfies the hypothesis of this lemma.

Proof. By Theorem 2.1 it suffices to prove the following claim.

Claim 4.4. If F_{ξ} (ξ < ω_{1}) is a sequence of disjoint finite subsets of ω_{1} and ε > 0, there is a pair ξ < η such that if α is in F_{ξ} and β is in F_{η} then

μ([[ẋ_{α} ∈ Ė_{β}]]) < ε.

Proof. Suppose that υ < ω_{1} satisfies (1) for all β > υ it is forced that ẋ_{β} is not an R_{υ}-name and

(2) if γ < υ then there is a ξ such that γ < F_{ξ} < υ and if α ∈ F_{ξ} then ẋ_{α} is forced not to be an Ṙ_{γ}-name^{5}.

Now let η < ω_{1} be arbitrary such that υ < F_{η}. Define Ė to be the R-name for the union of all Ė_{β} such that β is in F_{η}. Let ζ_{i} (i < ω) enumerate all ordinals ζ ≥ ν with the property that

[[ẋ_{ζ} ∈ Ė]] ≠ 0

(since R is c.c.c. and Ė is forced to be countable, the set of such ζ’s is countable). Using 1 and Lemma 3.4, pick a sequence δ_{i} (i < ω) such that for all α < υ

μ([[d(ẋ_{α}, ẋ_{ζ_{i}}) ≥ δ_{i}]]) > 1 − ε/2^{i+2}.

Define U̇ to be R-name for the open set consisting of all ẏ such that for some i it is forced that d(ẏ, ẋ_{ζ_{i}}) < δ_{i}. Notice that for all α < υ

μ([[ẋ_{α} ∈ U̇]]) < ε/2.

Also, since Ė \ U̇ is forced to be compact and contained in Ẋ, by 2 it is forced for be contained in

{ẋ_{α} : α < γ̇}

for some R-name γ̇ for an ordinal less than υ. Now find a ξ < ω_{1} such that

μ([[γ̇ < F_{ξ} < υ]]) > 1 − ε/2.

It is now easily verified that ξ < η are as desired. □

□

5Here γ < F_{ξ} < ν abbreviates γ < min F_{ξ} and max F_{ξ} < ν.

Lemma 4.5. (PFA) Suppose that for a closed unbounded set C of υ < ω_{1} there is a β_{υ} ≥ υ and a condition B_{υ} in R which decides ẋ_{β_{υ}} to be an R_{υ}-name. Then there is a positive element of R which forces K̇ ∩ {ẋ_{β_{υ}} : υ ∈ C} to contain an uncountable discrete set.

Proof. For simplicity we will reenumerate our set of ẋ_{β_{υ}}’s, letting ẏ_{υ} = ẋ_{β_{υ}}. Also, set Ḟ_{υ} = Ė_{β_{υ}}. Let P be the collection of all (N , ρ, B) which satisfy

(1) N is a finite ∈-chain of countable elementary submodels of H(θ) for θ sufficiently large so that R, C, ⟨ẏ_{α} : α ∈ C⟩, ⟨Ḟ_{α} : α ∈ C⟩, and ⟨B_{α} : α ∈ C⟩ are in every member of N.

(2) ρ is a map from N into the rationals in (0, 1).

(3) B is a map from N into R such that μ(B(N)) > ρ(N) and B(N) ≤ B_{N∩ω_{1}} for all N in N.

(4) If N is in N then the restriction of B to N is in N.

(5) If N_{1}, N_{2} are in N with N_{1} ∈ N_{2} then B(N_{1}) ∧ B(N_{2}) forces that ẏ_{N_{1}∩ω_{1}} is not a member of Ḟ_{N_{2}∩ω_{1}}.

Define an order ≤ on P by (N_{p}, ρ_{p}, B_{p}) ≤ (N_{q}, ρ_{q}, B_{q}) iff

(1) N_{p} contains N_{q},

(2) the restriction of ρ_{p} to N_{q} is ρ_{q}, and

(3) if N is in N_{q} then B_{p}(N) ≤ B_{q}(N).

We will now show that P is proper and in the process see that D_{α} = {p ∈ P : α ∈ ∪N_{p}} is dense for all α < ω_{1}. This is sufficient since if G meets D_{α} for all α < ω_{1}, set N to be the union of all N_{p} for p in G and

B̄_{ν} = ⋂{B_{p}(N) : p ∈ G and N ∈ N_{p} and ν = N ∩ ω_{1}}

for ν < ω_{1} of the form N ∩ ω_{1} for some N in N. Each B̄_{ν} is a positive element of R and if Ȧ is defined by putting [[ν ∈ Ȧ]] = B̄_{ν} then Ȧ is forced by some condition to be uncountable and satisfy {ẏ_{ν} : ν ∈ Ȧ} is discrete in K̇.

Claim 4.6. (P, ≤) is a proper partial order.

Proof. Let M be an elementary submodel of some large enough H(λ) containing all of the objects mention thus far in the proof and let p = (N_{p}, B_{p}) be a condition in P ∩ M. By extending p if necessary, we may assume that

μ ( ⋁_{N∈N_{p}} B(N)) < 1.

Set p̄ to be (N_{p}∪{M ∩H(θ)}, ρp̄, Bp̄) where ρp̄ ↾ M = ρ_{p}, Bp̄ ↾ M = B_{p},

0 < ρp̄(M ∩ H(θ)) ≤ \frac{1 − μ (⋁_{N∈N_{p}} B(N))}{2}

Bp̄(M ∩ H(θ)) = 1 − ⋁_{N∈N_{p}} B(N).

It is easily checked that p̄ is a condition in P. We will now see that p̄ is (M, P)-generic. To this end, let D ⊆ P be a dense open set in M and let r be an extension of p̄ which is in D. Define T_{0} to be the set of all elements s = (N_{s}, ρ_{s}, B_{s}) of P

(1) there is a condition s̄ in D which extends s such that |Ns̄| = |N_{r}|,

(2) N_{r} ∩ M is an initial part of N_{s},

(3) ρ_{r} and ρ_{s} agree on N_{r} ∩ M, and

(4) B_{r} and B_{s} agree on the N_{r} ∩ M. We will consider T_{0} as a tree when ordered by end extension on all three coordinates. Notice that, by an elementarity argument, T_{0} contains a subtree T such that if s is a nonterminal node of T then there are stationarily many ν < ω_{1} such that for some immediate successor s̄ of s in T

ν = max(Ns̄) ∩ ω_{1}.

The following subclaim will be key to our argument.

Subclaim 4.7. Suppose that S ⊆ ω_{1} is a stationary set in M. Then there is a sequence α_{n} in M ∩ S which converges to υ = M ∩ ω_{1} such that ẏ_{α_{n}} is forced to converge to an element of [0, 1]^{ω} \ Ẋ.

Proof. Fix a sequence ξ_{n} (n < ω) which is cofinal in υ. Using Lemmas 3.6 and 3.7, build a sequence of stationary sets S_{σ} (σ ∈ 2^{<ω}) which are elements of M and positive rationals δ_{k} > 0 such that:

(1) S = S_{⟨⟩}.

(2) If σ is an initial part of τ then S_{σ} ⊇ S_{τ}.

(3) If σ and τ are incomparable then S_{σ} and S_{τ} are disjoint.

(4) If |σ| = k and α, α′ are in S_{σ} then

μ([[d(ẋ_{α}, ẋ_{α′}) < δ_{−k−1}]]) > 1 − 2^{−k−1}.

(5) δ_{k} < 2^{−k}.

(6) If |σ| = k and α is in S_{σˆ 0} and β is in S_{σˆ 1} then

μ([[d(ẋ_{α}, ẋ_{β}) ≥ 2δ_{k}]]) > 1 − 2^{−k−1}.

(7) If |σ| = k, min(S_{σ}) > ξ_{k}.

For a given b in 2^{ω}, let α_{k}(b) be the least element of S_{b↾k}. Notice that for each b, it is forces that {ẏ_{α_{k}(b)}}_{k=0}^{∞} converges and that for distinct b and b′, these sequences are forced to converge to different elements of [0, 1]^{ω}. Since PFA implies that 2^{ℵ_{0}} > ℵ_{1}, there must be a b in 2^{ω} such that {ẏ_{α_{k}(b)}}_{k=0}^{∞} is forced to converge to something outside of Ẋ. □

We are finished once we prove the following subclaim.

Subclaim 4.8. If s is a nonterminal node of T which is compatible with r, then there is an immediate successor s̄ of s in T such that s̄ is compatible with r.

Proof. Let q be an extension of s and r. Pick an ε > 0 such that for all N in N_{r} \ M, μ(B_{q}(N)) > ρ_{q}(N) + ε. Let S be the set of all ν such that for some immediate successor s̄ of s in T

ν = max(Ns̄) ∩ ω_{1}.

By the definition of T, S is stationary. Therefore it is possible, using Subclaim 4.7, to find a sequence s̄_{n} (n < ω) of immediate successors of s in T such that

max(Ns̄_{n}) ∩ ω_{1} → υ and
y_{max(Ns̄n)} → ż

where ż is forced to be outside of Ẋ. Let Ḟ be the R-name for the union of all Ḟ_{N∩ω_{1}} for N in N_{r}. Since Ḟ is forced to be compact and contained in Ẋ, there is an R-name ṁ for an element of ω such that ẏ_{max(Ns̄ṅ)} is forced not to be in Ḟ for any ṅ forced to be larger than ṁ. Now find an n such that

μ([[ň > ṁ]]) > 1 − ε.

Finally, define Nq̄ = Ns̄ ∪ N_{r} and ρq̄ = ρ_{q} ∪ ρ_{r}. Define Bq̄ piecewise. If N is in Ns̄, then Bq̄(N) = Bs̄(N). If N is in N_{r} \ M, define

Bq̄(N) = B_{r}(N) \ [[ẏ_{max(Ns̄n)} ∉ Ḟ]].

Since for all N in N_{r} \ M the measure of [[ẏ_{max(Ns̄n)} ∉ Ḟ]] is less than μ(B_{r}(N)) − ρ(N), q̄ is a condition in P. Then q̄ is a condition in P and an extension of r and s̄_{n} as required. □

□

□

References

[1] David H. Fremlin. Measure algebras. In Handbook of Boolean algebras, Vol. 3, pages 877–980. North-Holland, Amsterdam, 1989.

[2] Gary Gruenhage and Peter Nyikos. Normality in X^{2} for compact X. Trans. Amer. Math. Soc., 340(2):563–586, 1993.

[3] Istvan Juhász, Kenneth Kunen, and Mary Ellen Rudin. Two more hereditarily separable non-Lindelöf spaces. Canad. J. Math., 28(5):998–1005, 1976.

[4] Paul Larson and Stevo Todorčević. Katětov’s problem. Trans. Amer. Math. Soc., 354:1783–1792, 2002.

[5] Richard Laver. Random reals and Souslin trees. Proc. Amer. Math. Soc., 100:531–534, 1987.

[6] Dorothy Maharam. On homogeneous measure algebras. Proc. Nat. Acad. Sci. U. S. A., 28:108–111, 1942.

[7] J. Tatch Moore. Random forcing and (S) and (L). Topology Appl., 131(2):139– 148, 2003.

[8] Justin Tatch Moore. Ramsey theory on sets of real numbers. PhD thesis, University of Toronto, 2000.

[9] Stevo Todorcevic. Chain-condition methods in topology. Topology Appl., 101(1):45–82, 2000.

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

[11] Stevo Todorčević. Random set mappings and separability of compacta. Topology and its Applications, 74:265–274, 1996.