Set theory in Topology

Alan Dow^{1}

Department of Mathematics

York University

North York, Ontario, Canada M3J 1P3

Version: January 31, 1997

1NSERC of Canada

1. Introduction

This article is intended to give a survey of the primarily set-theoretic techniques which, in the author’s view, have played a major role in the development of settheoretic topology over the past several years. The focus is on the techniques rather than the results themselves. We have made a selection of results which we feel are both interesting and which illustrate the techniques. However we have not tried to make this a survey of the many recent results which one might normally associate with the contents of this article. Of course, the interested reader will have found many of these results in the other articles contained in this volume.

Much of set-theoretic topology is focussed on the study of reflection – when we have it and when we do not. When we construct interesting examples it is often of a space which witnesses some kind of non-reflection. When we prove theorems we prove that some kind of reflection holds. It is even true that in many cases in which we produce examples of spaces to witness non-reflection we have to utilize reflection in the construction – as we will see in the second section in a beautiful construction of a non-collectionwise Hausdorff space by M.E.Rudin. In the third section, we use countable elementary submodels to investigate stationary set reflection and relate it to reflecting collectionwise Hausdorffness in locally small spaces. The fourth section explores the technique of factoring large iterations of forcing posets, again using reflection. The techniques developed in §4 are then applied in §5 by discussing the wonderful result of Balogh that there may be no locally compact normal noncollectionwise normal spaces. In §6 – PFA and OCA – we have tried to emphasize the Ramsey theoretic approach to applying these axioms as outlined in Todorčević’s book ( [Tod89]). We also include a proof of the underlying combinatorial result to Balogh’s solution of the Moore-Mrowka problem. The final section – Kurepa families – was kindly supplied by Stevo Todorčević. These structures should find many applications in the inductive constructions of interesting topological spaces.

2. A technique of Rudin

In this section we give a proof that there is a space which is normal but not collectionwise Hausdorff (hereafter CWH). The purpose is to discuss a prediction type principle formulated in [Rud84]. Many inductions are based on the fact that all the countable structures on c can be listed in length c. The point of this technique is that we actually use this list as a kind of oracle in the induction.

Let θ be a regular cardinal which is much larger than c.

Definition 2.1. Given a model (set) M, the Mostowskii collapse is a well-known isomorphism from {M, ∈} to its transitive collapse. Simply

M̂ = {m̂ : m ∈ M}

where, for m ∈ M,

m̂ = {x̂ : x ∈ M ∩ m} ,

It is easily checked that m ↦ m̂ is an isomorphism from (M, ∈) to (M̂, ∈).

The idea of the Mostowski collapse can be applied quite generally. In case we are building an object on c it is useful to define a similar kind of isomorphism (collapse), say F_{M} : M → F_{M}(M), which would be the identity on c. For simplicity we will send all ordinals to ordinals.

By induction on rank: if F_{M}(Y) has been defined for all Y ∈ M ∩ X, then

F_{M}(X) = \{ \begin{matrix}X & if X ∈ M ∩ c, \\ c ∪ \{F_{M}(Y) : Y ∈ M ∩ X\} & if X ∈ M ∩ [c, θ), \\ F_{M}(X) = \{F_{M}(Y) : Y ∈ M ∩ X\} & otherwise. \end{matrix}

Hence for ordinals α ∈ M − c, F_{M}(α) ∈ [c, c + ω_{1}).

Note that if X is any set (not necessarily in M), then F_{M}(X) is defined. Hence F_{M}(M) = {F_{M}(X) : X ∈ M}. If the set M is clear from the context, we will use F rather than F_{M}. Perhaps we should point out that although F is an isomorphism from the model (M, ∈) to (F_{M}(M), ∈) (provable by induction on rank), it does not preserve even the subset relation. For example, if X = c + 2 − {c}, then F_{M}(X) is simply the countable set {F_{M}(α) : α ∈ M ∩ X} and so does not contain the set c = F_{M}(c). Therefore we have that F_{M}(c) ⊄ F_{M}(X), but

(F_{M}(M), ∈) ⊧ (∀x)(x ∈ F_{M}(c) → x ∈ F_{M}(X)) ≡ (F_{M}(c) ⊂ F_{M}(X)) .

Since F (M) is uniquely determined by the Mostowskii collapse of M, M̂, which is in H_{ω_{1}}, and the countable set F (M) ∩ [0, c + ω_{1}), there are only c many such sets (or even ω-sequences of such sets). For this construction we want to predict both a topology τ on a set X containing c and possible separations, f, of c as a subspace. Let {⟨M_{α}, X_{α}, τ_{α}, f_{α}⟩ : α < c} enumerate the set

{⟨F_{M}(M), F_{M}(X), F_{M}(τ), F_{M}(f)⟩ :
{X, τ, f} ⊂ H(θ) and M ≺ H(θ) is countable} .

We may assume that M_{α} ∩ c ⊂ α. We could also restrict the list to those ⟨X, τ, f⟩ which have some special properties (e.g. being a topological space) and this would be reflected in M. The idea now is that we define the βth object by looking at M_{β} and we can also look ahead to all those M_{α} which contain β.

We intend to build a space which is not CWH. The ordinals in c will form the closed discrete set which cannot be separated. Any potential separation will be “trapped” by some M_{α} – i.e. after we are done we take a countable elementary submodel M which contains the separation (in fact, as in our list above, we add the separation as a distinguished predicate in the language). Some M_{α} will equal F (M) and since we added the predicate to the language we know what corresponds to the separation. We ensure that the point α will be a limit point of this separation by virtue of the points which are in M_{α}. Naturally this will do nothing to ensure normality. This is where the trick of making the list in advance comes in. The points in M_{α} will know that α is going to be used in this way.

Similar to the space in [Rud84], the structure of the space itself is quite simple. The space X will be c ∪ [c]^{2}. Each of the pairs {α, β} will be isolated in X. For an α ∈ c, we will specify the neighbourhood base at α, U(α, A) for A ⊂ c such that α ∈ A. We will ensure that for α ∈ A and β ∉ A, U(α, A) and U(β, c − A) are disjoint. If we do so, it is clear that X is normal and that c will be a closed discrete subset of X.

The simplest way to ensure that α will “ruin” the potential separation predicted by f_{α} will be to pick a sequence {β_{n}^{α}} ⊂ M_{α} such that α will be a limit point of the assigned neighbourhoods of {β_{n}^{α}}. The only way this is possible is to ensure that {β_{n}^{α}, α} is in each of the respective assigned neighbourhoods and to ensure that at least one of these pairs is in every neighbourhood of α – i.e. that the sequence {β_{n}^{α}, α} has α as a limit point.

The difficulty is with normality. Let A ⊂ c be such that α ∈ A; for each n such that β_{n}^{α} ∉ A we must choose between (Case 1) α ∉ U(β_{n}^{α}, c − A) and (Case 2) β_{n}^{α} ∉ U(α, A). We will use an ultrafilter to decide. Fix any free ultrafilter p_{α} on c such that {β_{n}^{α} : n ∈ ω} ∈ p_{α}. Now Case 1 will correspond to A ∈ p_{α} and Case 2 with A ∉ p_{α}. Hence the neighbourhood trace of α on {{β_{n}^{α}, α} : n ∈ ω} will be like p_{α}.

More formally, once we have chosen the sequences {β_{n}^{α} : n ∈ ω}, we define, for A ⊂ c and α ∈ c,

U(α, A) = {α} ∪ {{β_{n}^{α}, α} : n ∈ I_{α}(A)} ∪ {{α, γ} :
α < γ and either γ ∈ A or α ∉ {β_{n}^{γ}} or A ∩ {β_{n}^{γ}} ∉ p_{γ}}

where I_{α}(A) is the set {n : β_{n}^{α} ∈ A} if it is in p_{α}, otherwise it is ω. It can be shown that these sets form a base at α.

These sets are essentially the largest possible sets to accomplish the above normality-type separations. All that remains is to show that we can choose the sequences so as to ensure that X is not CWH. The tricky part now is that we must use our knowledge of what the restrictions to M_{α} of the neighbourhood assignment given by a potential separation f look like – U(β, A_{β}) for some A_{β} – in order to try to ensure that {β, α} will be in U(β, A_{β}) for “p_{α}-many” β. Notice that if A_{β} is the restriction of a set, say Ã_{β}, which contains α then {β, α} ∈ U(β, Ã_{β}). So in the situation when we actually have the restriction of a separation, it will be the case that α is not in Ã_{β} for all β in some member of the ultrafilter p_{α}. Therefore, choose, if possible an increasing sequence {β_{n}^{α} : n ∈ ω} cofinal in M_{α} ∩ α such that β_{n}^{α} ∉ ∪_{k<n}A_{k} (reflecting the case that α ∉ ∪_{k<n}A_{k}). If this is not possible, then as discussed above, M_{α} is not predicting a separation so the sequence is chosen to be any increasing cofinal sequence.

The proof that the space is not CWH is to simply note that for any potential separation f, we can choose a countable elementary submodel, M, of ⟨H_{θ}, f⟩. Then there is an α such that F (M) = M_{α} and F (f) = f_{α}. In this case, the choice of {β_{n}^{α} : n ∈ ω} was successfully carried out. That is, let I be the infinite set {n ∈ ω : {β_{n}^{α}, α} ∈ f(α)}. Fix any n ∈ I and note that since {β_{n}^{α}, α} is in f(α), it follows that {β_{n}^{α}, α} is not in f(β_{n}^{α}). Let f(β) = U(β, Ã_{β}) for all β. In keeping with the notation of the previous paragraph, F (f(β)) = U(β, A_{β}) where A_{β} = Ã_{β} ∩ M_{α}. Now, for each integer n ∈ I, α ∉ U(β_{n}^{α}, Ã_{β_{n}^{α}}) (hence α ∉ Ã_{β_{n}^{α}}). By the elementarity of M, c − ⋃{Ã_{β_{k}^{α}} : k ∈ I ∩ n} is cofinal in c for each integer n.

3. More reflection

Recall that if X is any set then there are notions of closed and unbounded (cub) and stationary for the structure [X]^{ω} ordered by ⊂. In this section we show how we can take advantage of the fact that the set of countable elementary submodels of H_{θ} is cub in [H(θ)]^{ω}. It was shown by Shelah that if one Levy collapses a strongly compact cardinal then every first countable locally separable space is CWH if it is ℵ_{1}-CWH. It was later noticed that any proper collapse will yield the same result. The idea behind these proofs is Fleissner’s lemma about the closure “going up” stationarily often.

Fix a space X which is locally separable and has countable tightness. Suppose that D is a closed discrete subset of X and that τ is the topology on X. Let

S = \{U ∈ [X]^{ω} : U ⊂ int_{X} \overline{U} and D ∩ \overline{U} ⊄ U\} .

Claim 1. D can be separated in X if and only if S is not stationary.

Proof. Suppose f: DMApstoτ is a separation of D. Let S̃ = {a ∈ [H(θ)]^{ω} : A∩X ∈ S}. By Claim 1.12 of [She82], S̃ is stationary in [H(θ)]^{ω} if and only if S is stationary in [X]^{ω}. Since the set, C, of countable elementary submodels M of H(θ) which contain the function f is a cub it suffices to show that C ∩ S̃ is empty. Fix any M ∈ C and d ∈ D − M. We claim that f(d) is a neighbourhood of d which misses M ∩ X. Indeed, if x ∈ M ∩ f(d), then by elementarity M ⊧ (∃y ∈ D)x ∈ f(y). Hence there is such a y ∈ D ∩ M which contradicts that f is a separation.

For the other direction, suppose that S is not stationary. We prove this (as in [Fle74]) by induction on κ = |D|. Let C be a cub in [X]^{ω} which misses S. Fix an elementary chain of elementary submodels {M_{α} : α < cf (κ)} such that S, D, C etc. are all in M_{0}, |M_{α}| < κ for all α and D ⊂ ∪M_{α}. Since X is locally separable,

D ∩ M_{α} ⊂ int_{X} \overline{M_{α} ∩ X} for all α .

Therefore, by the induction hypothesis, it suffices to show that if d ∈ D − M_{α}, then d ∉ \overline{M_{α} ∩ X}. To do so, let \{x_{n} : n ∈ ω\} enumerate any countable subset of M_{α} ∩ X. Recursively choose a_{n} ∈ C ∩ M_{α}, such that a_{n} ∪ {x_{n+1}} ⊂ a_{n+1}. Clearly, U = ∪a_{n} ∈ C and d ∉ U. Since U ∉ S, it follows that d ∉ \overline{U}. Since X has countable tightness, this completes the proof.

Definition 3.1. A subset S of [H_{θ}]^{ω} will be said to ∈-reflect to a set Y ⊂ H_{θ} if (S ∩ Y) ∩ [Y]^{ω}) is stationary in [Y]^{ω}.

The idea here is that many members of S ∩ [Y]^{ω} are also members of Y. This is probably stronger than demanding only that S ∩ [Y]^{ω} is stationary and Y is unbounded in [Y]^{ω}. The statement that stationary subsets of [H_{θ}]^{ω} reflect will mean that every stationary subset of [H_{θ}]^{ω} ∈-reflects to some size ℵ_{1} subset of H_{θ}.

Corollary 3.2. If stationary subsets of [H_{θ}]^{ω} reflect, then every ℵ_{1}-CWH locally separable countably tight space is θ-CWH.

Proof. Let X be a locally separable countably tight space and suppose that D ∈ [X]^{≤θ} is discrete. Since X is locally separable, there is a subspace X′ of X of cardinality no more than θ with the property that D ⊂ int_{X} \overline{X′}. We can assume that the base set of X′ is just the ordinal θ. As above, let

S = \{U ∈ [X′]^{ω} : U ⊂ int_{X} \overline{U} and D ∩ \overline{U} ⊄ U\} .

Now let κ be a large enough so that {X, D, S} is in H_{κ}. Let S̃ be all countable subsets of H_{θ} which are of the form M ∩H_{θ} for some M ≺ H_{κ} such that {X, D, S} ∈ M and such that M ∩ X′ is a member of S. Then S̃ is stationary if and only if S is.

Let A ⊂ H_{θ} be of size ω_{1} and witness that S̃ ∈-reflects. Consider s ∈ S̃∩A∩[A]^{ω}. We claim that \overline{s ∩ X} ∩ D − s meets A. Indeed, since s ∈ A, there is an a ∈ S̃ ∩ [A]^{ω} such that s ∈ a. It will actually suffice to know that there is a y ∈ a ∩ [A]^{ω} such that s ⊂ y. Fix M ≺ H_{κ} with X, D ∈ M such that M ∩ H_{θ} = a. Since s ∈ S̃, it follows that

M ⊧ \overline{s ∩ X′} ∩ D − s ≠ ∅,

hence this intersection meets A. Note that we can assume that a contains \overline{y ∩ X} ∩D since this set must be countable in order for X to be ℵ_{1}-CWH.

Proposition 3.3. The statement that every stationary subset reflects is equivalent to Fleissner’s axiom R.

Proof. Recall that axiom R is the statement that if S is a stationary subset of [κ]^{ω} and C is a t.u.b. subset of [κ]^{ω_{1}} (i.e. C is unbounded and closed under unions of strictly increasing ω_{1} chains), then there is a Y ∈ C such that S ∩[Y]^{ω} is stationary. Axiom R implies that every stationary set reflects since H_{θ} may be identified with some κ and C may be taken to be all those elementary submodels of H_{θ} which are equal to a union of an ∈-chain of countable elementary submodels.

For the other direction, assume that every stationary subset reflects. Let S and C be as in axiom R. Choose a large enough θ and let S̃ be the pull back of S to [H_{θ}]^{ω}. Let A be chosen to witness that S̃ will ∈-reflect (we may assume that we have shrunk S̃ to only those members which are elementary submodels of H_{θ} which also contain C as an element). Inductively choose members c_{α} of C ∩ A so as to form a cofinal increasing chain of length ω_{1}. Suppose that {c_{β} : β < α} has been chosen. There is a member, say a, of S̃ ∩ A containing this countable subset of A. By elementarity, there is a member c_{α} of C which contains the union of a ∩ C. We may also assume that c_{α} includes the α^{th} member of A ∩ κ under some ordering in order type ω_{1}. Therefore the union Y = ∪{c_{α} : α ∈ ω_{1}} will be a member of C and it will equal A ∩ κ. Now we must show that S ∩ [Y]^{ω} is stationary. This is direct. If D is cub in [Y]^{ω}, then D̃ = {c ⊂ [A]^{ω} : c ∩ Y ∈ D} is cub in [A]^{ω}. Fix an s̃ ∈ S̃ ∩ D̃. Now simply check that s̃ ∩ κ = s̃ ∩ Y hence s̃ ∩ κ is a member of S ∩ D.

4. A standard iteration lemma

Let us try to formulate some useful (cross between general and understandable) reflection lemmas for iterations.

Definition 4.1 ( [Kun80]). For a set M (usually a model of most of ZF) and a generic filter G of a poset P (often P ∈ M),

M[G] = { val_{G}(Ẋ) : Ẋ ∈ M is a P -name } .

Recall that a filter G is P -generic over M if (P ∩ M) ∩ D is not empty for each D ∈ M which is a dense open subset of P. A condition p ∈ P is said to be (M, P)- generic ( [She82]) if every generic filter which contain p is P -generic over M. We use “generic” to mean P -generic over V.

Theorem 4.2 ( [She82], III.211). Suppose that P is a poset in an elementary submodel M ≺ H_{θ} and suppose that G is P -generic over both V and M. Then M[G] ≺ H_{θ}[G].

Proof. Note that since P is in H_{θ}, it follows that |P | < θ. Therefore, for every Ẋ and p ∈ P such that p ⊩ Ẋ ∈ H_{θ}, there is a Ẏ ∈ H_{θ} with |Ẏ | < θ such that p ⊩ Ẋ = Ẏ. Hence the set of hereditarily < θ sets in V [G] is equal to (H̆_{θ})[G].

We use the Tarski-Vaught criteria to establish that M[G] ≺ H_{θ}[G]. To prove this, one simply uses elementarity of M in H_{θ} to assert that the necessary names reside in M (and the fact that the forcing relation is absolute between M and H_{θ}).

The important thing about M[G] in the previous result is that it is isomorphic to a model in V [G ∩ M] - namely its transitive collapse. It is also useful to consider a pseudo-collapse which is the identity on the ordinals similar to the one we used in Section 2. Let j denote the inverse of the Mostowskii collapsing map. Hence j is an elementary embedding of { \atop M[G]} into H_{θ}[G] which takes { \atop M[G]} to M[G]. In such a situation, it is customary to let j” be defined by

j”X = {j(x) : x ∈ X} .

Also define e recursively, as follows,

e(α) = j(α) for ordinals α ∈ M̂

and

e(X) = e”X = {e(x) : x ∈ X} for X ∈ M̂ − ON .

The pseudo-collapse, F, we are speaking of is just the composition of the Mostowskii collapse with e. That is, for α ∈ M, F (α) = e(α̂) = α and for other X ∈ M, F (X) = e(X̂) = {e(x̂) : x ∈ X ∩ M} .

Now, if we consider { \atop M[G]} (or F (M[G])), we have that for each member X̂ of { \atop M[G]} there is a P̂-name in M̂ whose valuation by Ĝ is equal to X̂. Furthermore pMApstop̂ is an isomorphism from the poset P ∩ M to P̂ which takes G ∩ (P ∩ M) to Ĝ. Since the Mostowski collapse is definable, (hence Ĝ ∈ V [G∩M]) and { \atop M[G]} = M̂[Ĝ] it follows that { \atop M[G]} is in V [G∩M]. An alternative way to view all this is that F = e ∘ is obtained by recursively re-interpreting the P -names of M as P ∩ Mnames of their restrictions to M[G] (using the fact that G is P -generic for M). Hence if we are interested in the properties in V [G] of a structure X ∈ M[G] then F (X) will be in V [G ∩ M] and should be an interesting substructure of X. Since F (M[G]) is isomorphic to { \atop M[G]} we can restrict our discussion to the transitive collapse.

So, { \atop M[G]} will be a submodel of H_{θ}[G∩M] (not elementary) which is elementarily embedded into H_{θ}[G] by the inverse of the Mostowskii collapse, j. So to show that the structure X with a certain property has a small substructure with that property we show that X̂ has the property. The proof splits into two steps. What happens when we pass from { \atop M[G]} to H_{θ}[G ∩ M] – i.e. is M “closed” enough to give enough absoluteness? Next, what happens when we pass from H_{θ}[G ∩ M] to H_{θ}[G] – i.e. is the forcing “mild” enough to preserve the property?

In summary, we think of M[G] as being equal to { \atop M[G]}, hence it is a member of V [G ∩ M] but is an elementary submodel of V [G].

Theorem 4.3 (CH). Suppose that P = ∪{P_{α} : α < ω_{2}} is a proper poset such that |P_{α}| ≤ ℵ_{1} for each α < ω_{2} and such that {λ < ω_{2} : P_{λ} is completely embedded in P } is stationary in ω_{2}. Let G be P -generic over V and suppose that X is a structure in H_{θ}[G]. Then there is a λ < ω_{2} and an M ∈ H_{θ}[G] of cardinality ℵ_{1} such that

M̂ ∈ V [G_{λ}] and V [G_{λ}] ⊧ M̂^{ω} ⊂ M̂

and

V [G] ⊧ ⟨M, X ∩ M, [X]^{ω} ∩ M, ϵ⟩ ≺ ⟨H_{θ}, X , [X]^{ω}, ϵ⟩ .

The usefulness of the previous result is that since M̂ is closed under ω-sequences in V [G_{λ}] there will be a lot of absoluteness between M̂ and V [G_{λ}]. The final model is obtained by forcing with the tail P/P_{λ}, hence the standard next step in an argument is to use the reflected properties of M (usually some maximality type property) to show that the tail will preserve this and contradict some property of X in V [G].

In many instances it is not necessary to worry about M̂ since the collapsing function may be the identity on a large part of M. That is the idea of the next two corollaries.

Corollary 4.4. Suppose that P is a poset in an elementary submodel M ≺ H_{θ} and that G is P -generic over both V and M. Let κ be the least cardinal such that M ∩ κ ∉ κ and assume (with no loss of generality) that the base set for P is an ordinal. Then

(M ∩ H̆_{κ})[G ∩ M] = (M ∩ H̆_{λ})[G ∩ M] ≺ H_{λ}[G]

where λ is the minimum member of (M ∪ {θ}) ∩ [κ, θ].

Proof. The Mostowskii collapse is just the identity function on M ∩ H_{κ}. If λ = θ, then we are just restating Proposition 4.2. If λ < θ, then λ ∈ M, hence H_{λ} ∈ M and M ∩ H_{λ} ≺ H_{λ}. Then again we have that (M ∩ H_{λ})[G] ≺ H_{λ}[G]. Finally we observe that, for any Ẋ ∈ M ∩ H_{λ}, val_{G∩M}(Ẋ) = val_{G}(Ẋ) .

Corollary 4.5. Suppose that P = ∪{P_{α} : α < ω_{2}} is a proper poset such that P_{α} ∈ H_{ω_{2}} for each α < ω_{2} and such that S = {λ < ω_{2} : P_{λ} is completely embedded in P } is stationary in ω_{2}. Let X be any P -name and let M be any ℵ_{1}-sized elementary submodel of H_{θ} (in V) such that {X , P } ⊂ M, λ = M ∩ ω_{2} ∈ S and such that M^{ω} ⊂ M. Then, (M ∩ H_{ω_{2}})[G_{λ}] ≺ H_{ω_{2}}[G] and (M ∩ H_{ω_{2}})[G_{λ}] is closed under ω-sequences.

The method outlined above is almost always used when establishing consistency results about βN. Typically X is some family of subsets of N which may be one of: an ultrafilter, a tower, a tree π-base, or a splitting family. As an example, let us prove that there are no towers of lenth ω_{2} in the Mathias model – the model obtained by the ω_{2}-lenth countable support iteration of the Mathias poset. This was established for an slightly modified iteration of Mathias forcings by Dordal [BD85]. The only reason that they modified the support is that they did not have at hand the very useful Lemma 1.13 of [She84]. Recall that Mathias forcing is the poset {(a, A) : a ∈ [ω]^{<ω} and A ∈ [ω]^{ω}} where (a, A) ≤ (b, B) if a ⊃ b and (a−b∪A) ⊂ B. Given a filterbase S ⊂ [ω]^{ω}, we say that Y ∈ [ω]^{ω} diagonalizes S if Y − S is finite for all S ∈ S. It is interesting that Mathias forcing adds a filter and then diagonalizes it ( [Mat77]) but that it preserves undiagonalizability for most existing filters (Dordal [BD85]). It cannot preserve this for all filters because of the following easily established fact.

Proposition 4.6. There is an undiagonalizable filter F on P(N) such that any forcing which adds a real also diagonalizes F.

We can see (e.g. from [She84]) that we have to strengthen the notions that we are trying to preserve to countably complete versions. So let us say that a filter base S is ω-diagonalizable if there is a countable family Y such that each member of S contains a member of Y.

Proposition 4.7. In the Mathias model, if a filter S ⊂ [ω]^{ω} cannot be ω-diagonalized, then there is a subset S′ ∈ [S]^{ω_{1}} which cannot be ω-diagonalized.

Proof. Let P_{λ} (for λ ≤ ω_{2}) be the countable support iteration of λ-many copies of Mathias forcing. In V [G], suppose that S = {S_{α} : α < ω_{2}} is a filter base of infinite subsets of ω which cannot be ω-diagonalized (e.g. a tower). Let λ, M and M̂ be as in Theorem 4.3. By elementarity, M̂ ⊧ Ŝ cannot be ω-diagonalized. For each α ∈ λ, Ŝ_{α} is equal to S_{α} – hence Ŝ = {S_{α} : α ∈ λ}. Since V [G_{λ}] ⊧ M̂^{ω} ⊂ M̂, it follows that Ŝ cannot be ω-diagonalized in V [G_{λ}]. We have shown that this property of S has reflected to V [G ∩ M] which completes the first step of a proof of this type. The next two lemmas handle the second step.

To complete the proof we reproduce the result of Baumgartner and Dordal and sketch the proof of this special case of Shelah’s powerful iteration technique developed in [She84].

Proposition 4.8. If S ⊂ [ω]^{ω} is not ω-diagonalizable then this remains true of S after Mathias forcing.

Proof. Suppose that {Ẏ_{n} : n ∈ ω} are Mathias names of a infinite subsets of ω. Let M be a countable elementary submodel containing {Ẏ_{n} : n ∈ ω} and A. Let B be an infinite subset of ω which will be chosen as a generic object for M over some forcing. Let S be any member of S which does not contain (mod finite) any infinite member of M. We want for all n ∈ ω, (a, B) ⊩ |Ẏ_{n} − S| = ω. It will simplify the proof if we just show this for Ẏ (any one of the Ẏ_{n}’s) and leave it to the reader to observe that since a hasn’t changed this can be done not only for all of the Ẏ_{n}’s but even for every name in M.

Given any (b, B) ∈ M, there is an B′ ∈ [B]^{ω} ∩ M and (b′, B′) ≤ (b, B) such that (b′, B′) ⊩ (∃k > max b) k ∈ Ẏ − S. This is simply because {k : (b, B) ⊮ k ∉ Ẏ } is an infinite member of M – hence S does not contain this set (mod finite). Next, in a usual Ramsey type argument, choose a descending sequence A_{k} for k ∈ ω and let j_{k} = min A_{k}. Choose A_{k+1} so that for every b ⊂ {j_{i} : i ≤ k}, if there is an A′′ ⊂ A_{k+1} such that (a ∪ b, A′′) forces some integer bigger than k + 1 to be in Ẏ − S, then already (a ∪ b, A_{k+1}) does this. It follows that if (b, B) is any extension of (a, {j_{k} : k ∈ ω}), where b ⊂ {j_{i} : i < k}, then (a ∪ b, B) ⊩ Ẏ − S ⊄ k.

There seems to be an additional difficulty in proving the iteration preserves the property. The approach in [She84] is to strengthen further the notion of non-ω-diagonalizability. We refer the interested reader there for the notion of a bounding set for a nice relation. The nearly equivalent alternative we take here is to demand a stronger preservation property of each poset of the iteration. For the sake of brevity we will be somewhat informal, the interested reader is referred to Steprāns [DoSt91] for a more complete treatment of a similar type of problem. Let us notice that in the previous proof, we showed that if M is a countable elementary submodel (containing the poset) and if S ∈ S does not contain any member of M, then for any condition p ∈ M, there is a q < p (which is (M, P)-generic) and which forces that S does not contain any member of M[G]. Let us say then that P strongly preserves that S is not ω-diagonalizable.

Corollary 4.9. If S ⊂ [ω]^{ω} is not ω-diagonalizable then this is strongly preserved by Mathias forcing.

Lemma 4.10. If P is the countable support iteration of proper posets each of which strongly preserves that S ⊂ [ω]^{ω} cannot be ω-diagonalized, then P strongly preserves that S cannot be ω-diagonalized.

Proof. Since P is proper and we are only interested in the existence of countably many subsets of N, we may assume that P is the iteration of countably many proper posets Q_{n}. Let P ∈ M be a countable elementary submodel of a suitably large H_{θ} and choose S ∈ S which does not contain any infinite member of M. Since we are assuming that P_{n} strongly preserves that S cannot be diagonalized, then for any condition p ∈ P_{n} ∩ M, there is an (M, P)-generic condition q < p such that if G_{n} ∋ q is P_{n}-generic over V then S does not contain any infinite member of M[G_{n}].

Let Ẏ be a P -name in M and suppose that p ∈ P ∩ M. We first show that we may suppose that p is such that n ∈ Ẏ is decided by conditions in P_{n} – that is, if q < p and q ⊩ n ∈ Ẏ then (q ↾ n ∧ p) ⊩ n ∈ Ẏ ( [She82]). For n ∈ ω, let P^{n} denote the P_{n}-name of the poset such that P = P_{n} ∗ P^{n} and D_{n} be the P_{n}-name of the dense open subset of P^{n} of conditions which decide the statement n ∈ Ẏ. Now inductively define p_{n} ∈ P so that

p_{n+1} ≤ p_{n} and p_{n+1} ↾ n = p_{n} ↾ n
p_{n} ↾ n ⊩_{P_{n}} p_{n+1} ↾ [n, ω] ∈ D_{n}

Now let p(n) = p_{n+1}(n) for each n, i.e. p = ⋀ p_{n}. It follows that p has the desired property.

Let {D_{n} : n ∈ ω} be an enumeration of the dense subsets of P which are in M. Fix n and now suppose that q_{n} has been chosen so that q_{n} ↾ n < p ↾ n is (M, P_{n})- generic and if q_{n} ↾ n ∈ G_{n} a P_{n}-generic filter, then val_{G_{n}}(q_{n}) ∈ M[G_{n}] ∩ D_{n}– i.e. q_{n} ↾ n ⊩ q_{n}^{−} = q_{n}/P_{n} = q′/P_{n} for some q′ in M ∩ D_{n}. So working in M[G_{n}] we can find a q_{n+1}^{−} < q_{n}^{−} which is in M[G_{n}] and which forces that some integer larger than n is in Ẏ − S (here we are using the hypothesis that P_{n} preserves that S does not contain any infinite subset of M[G_{n}]). Furthermore, since q_{n} is (M, P_{n})-generic, M[G_{n}] ⊧ val_{G_{n}}(D_{n}) = { val_{G_{n}}(q) : q ∈ D_{n}} is a dense subset of P/P_{n}. Therefore we suppose that q_{n+1}^{−} ∈ val_{G_{n}}(D_{n}). Next we can “strengthen” q_{n+1}^{−} on just the n^{th}-coordinate so that it is P_{n+1}/P_{n}-generic over M[G_{n}]. Since q_{n} ↾ n forces all this to happen, there is a P_{n}-name of q_{n+1}^{−} such that q_{n} ↾ n forces that it has the above properties. The condition q_{n+1} = q_{n} ↾ n ∗ q_{n+1}^{−}. It turns out that the fusion q of the q_{n}, i.e. q = ∪{q_{n} ↾ n : n ∈ ω}, is a condition which is (M, P)-generic and which forces that Ẏ −S is infinite. The main point is that q ↾ n ⊩ q/P_{n} ≤ q_{n+1}^{−}.

Now we turn to applications of the method to more topological questions. For example we show that if ω_{2} Cohen subsets of ω_{1} are added then we get CWN wrt ω_{1} copies of ω_{1} in first countable spaces – hence the hypothesis ♢^{+} in Rudin’s construction of a normal non-CWN manifold cannot be weakened to just ♢ ( [Kun80, VIII,J5]).

Lemma 4.11. If G is P = Fn(ω_{2}, 2, ω_{1})-generic over a model V of CH, then, in V [G], normal first countable spaces are collectionwise normal with respect to ω_{1} copies of ω_{1}.

Proof. In V [G], let X be a normal first countable space and, for each α ∈ ω_{1}, let {x_{α,β} : β ∈ ω_{1}} be a topological copy of ω_{1} in X such that the collection of all ω_{1} many copies of ω_{1} is discrete. Let X be the structure consisting of X, the ω_{1} copies of ω_{1}, and a countable neighbourhood assignment for each point. Let M and λ be given by the Lemma 4.3. By the following Proposition, the forcing does not introduce any new neighbourhoods of any of the copies of ω_{1} in the subspace X ∩ M. Therefore, if we collapse each of the copies of ω_{1} to a point, the quotient space of X ∩ M is a subspace of the quotient space of X (in V [G] under the same collapse). Furthermore, since X is normal the closed discrete set in the quotient space corresponding to the discrete family of copies of ω_{1} is normalized. Then Proposition 4.13 finishes the proof.

Proposition 4.12 (Fleissner). If ω_{1} is embedded in a first countable space, then the collection of ground model neighbourhoods of ω_{1} remains a local base for ω_{1} after any countably closed forcing.

Proof. Let U(α, n) for n ∈ ω be a neighbourhood base for x_{α} in X. Let P be a countably closed forcing and let ḟ be a P -name of a function from ω_{1} into ω. Assume that for each g ∈ ^{ω_{1}}ω from V, there is a p and an α such that p ⊩ U(α, g(α)) − ⋃_{β} U(β, ḟ(β)) ≠ ∅. Let M be a countable elementary submodel and let δ = M ∩ ω_{1}. Suppose that p ∈ P ∩ M and n ∈ ω. Claim: there is a x ∈ M and a q < p in M such that q ⊩ x_{n} ∈ U(δ, n) − ⋃_{β<α} U(β, ḟ(β)) . Suppose otherwise, then there is a γ ∈ M (i.e. determined by [γ, δ] ⊂ U(δ, n)) such that for all α > γ there is an m such that p ⊩ U(α, m) ⊂ ⋃_{β} U(β, ḟ(β)); i.e. for each such α in M we can take m to be the minimum one such that U(α, m) ⊂ U(δ, n). This would determine a ground model g.

Therefore by a countable induction we can choose a descending sequence of conditions p_{n} and a sequence x_{n} converging to δ such that p_{n} ⊩ x_{n} ∉ ⋃_{β} U(β, ḟ(β))

– an obvious contradiction.

Proposition 4.13 (Tall). Let X is a space and let D be a closed discrete set of cardinality ω_{1}. If D is not separated, then D is not normalized in V [G], where G is F n(ω_{1}, 2, ω_{1})-generic.

The Lemma 4.3 also gives us the following factoring lemma.

Lemma 4.14 (CH). Suppose that P = {P_{α}, Q_{α} : α ∈ I} (I a set of ordinals) is a finite support iteration of ccc posets of size ℵ_{1}, and M^{ω} ⊂ M is an elementary submodel of ⟨H_{θ}, P, I, ∈⟩. Then P factors as P_{I} ∗ Q_{I}, where P_{I} is the iteration of the Q_{α} for α ∈ I ∩ M and Q_{I} (in V^{P_{I}}) is the iteration of those Q_{α} for α ∈ I − M.

Proof. We may suppose that each Q_{α} is simply an ordering on ω_{1}. Therefore we can replace P by its dense set of elements which are functions into ω_{1} and whose domain is a member of [I]^{<ω}. This is also true for P′ = P_{I} ∗ Q_{I} but for the sake of clarity let us represent P′ differently. Let I_{0} = I ∩ M and let I_{1} = I − I_{0}. Hence members of P_{I} are finite functions from I_{0} and, in V^{P_{I}}, members of Q_{I} are finite functions from I_{1} into ω_{1}. So P′ has a dense set of conditions of the form (p, q) where p ∈ P_{I} and q ∈ V is a finite function from I_{1} into ω_{1}. Again, replace P′ by this dense set of conditions. To finish the proof, we just check that (p, q)MApstop ∪ q is an isomorphism between P′ and P. An easy induction on α ∈ I shows that

(p ↾ α, q ↾ α) <_{P′} (p′ ↾ α, q′ ↾ α) iff (p ∪ q) ↾ α <_{P} (p′ ∪ q′) ↾ α

which completes the proof.

Since many properties (those obtained by P_{I}) are preserved by finite support iterations of ℵ_{1}-sized posets the above result has many applications.

Corollary 4.15. If we have an iteration P as in the previous theorem which adds an ω_{2}-tower or an unbounded family of size ω_{2} in ^{ω}ω or an (ω_{2}, λ)-gap in P(ω)/fin etc., then these objects keep these properties in the final model.

5. Supercompact

In this section we will discuss Balogh’s important result that if supercompact Cohen or Random reals are added then in the resulting model locally compact normal spaces are collectionwise normal.

Recall that an elementary embedding j: V ↦ W is called a λ-supercompact embedding with critical point κ if κ ≤ λ < j(κ), j is the identity on V_{κ} and W is closed under λ-sequences. Normally the inner model W is denoted by M but we intend to use M in the role it played in §4, also the j we introduced there was modelled after the j given here.

Another way to view this is that the model W has a very special submodel. That is, choose some large θ (and then choose λ much larger). Then M = {j(x) : x ∈ H_{θ}} is an elementary submodel of the interpretation of H_{j(θ)} in W. There are two very important properties of M. In the model W, M^{<θ} ⊂ M and the Mostowski collapse of M, M̂, is just equal to the H_{θ} of V. This is exactly the situation we described at the beginning of section 4. That is, we have a model M ≺ H_{j(θ)} (of W) whose Mostowskii collapse, H_{θ} (that lives in a different “universe” – V) and the elementary embedding j from the Mostowskii collapse back to M. The function e from Section 4 corresponds to recursively applying j” (except to ordinals). The additional thing we have now is that θ < j(κ) where κ is fixed (the supercompact) and λ can be chosen, in advance, to be as large as we like.

The first thing to realize is that, since V is elementarily equivalent to W (via j), anything we prove about j(X) in W, is also true of X in V. Then we use the above setup and results from § 4 to prove things like j(X) has a small substructures sharing its properties – e.g. by all the elementarity and closure available j(X) ∩ M = j”X – therefore X has such substructures.

The new idea of Balogh is to be make use of more general kinds of substructures than simply subspace or subset with a weaker topology. That is, when we have a space (X, T ) which is a member of an elementary submodel M we frequently take X_{M} = M ∩ X as a set of points and consider the possibly different topologies on X – that inherited as a subspace of X and that given by M ∩T . Perhaps in hindsight, it is not so surprising that one should consider a quotient space.

We do not expect to be able to find a small substructure on a subspace since the character can be large. However, there is no real reason to insist on keeping the same set of points. Instead we view a space as a collection of neighbourhood filters hence a substructure would be a smaller collection of smaller neighbourhood filters.

In case the space is zero-dimensional a natural object to take is a reflected subalgebra of the Boolean algebra of clopen sets. In case the space is Tychonoff a similar object is to take an embedding of the space into [0, 1]^{Λ} as in Tychonoff’s theorem and to then consider the projection of X (as a subspace) into the space [0, 1]^{M∩Λ}. This turns out to be very fruitful. This idea was also discussed by E. Bandlow during the Symposium.

Let us explore the approach a little. If X is a zero-dimensional space then CO(X) is an algebraic object. Stone’s duality makes it easy to understand the idea of a substructure — a subset of the topology (i.e. a subalgebra) corresponds canonically to a continuous image of X. More generally we could look at C^{∗}(X)

– the ring of continuous functions (or even any embedding of X into [0, 1]^{Λ} for some index set Λ). With this approach, we can regard a substructure as a subring of continuous functions which will yield a projection function, π, from [0, 1]^{Λ} onto [0, 1]^{M∩Λ}. There is also a well-developed duality between compact spaces and rings. A compact space X is equal to the space of maximal ideals of C^{∗}(X) (in general, closed subsets of X are determined by ideals, namely all functions which are constantly zero on the set). Note that π(X) is almost always disjoint from M since the members of π(X) are functions with domain M ∩ Λ which is not usually in M. However we can think of x ∈ X ∩ M as a representative of π(x) – but it is even still likely that π(X) − {π(x) : x ∈ X ∩ M} is non-empty (even large). For this reason we are better off keeping our focus on C^{∗}(X) ∩ M.

Definition 5.1. Given a Tychonoff space X, in an elementary submodel M, let π_{X,M} (or just π if X, M are clear from the context) denote the projection map from [0, 1]^{C^{∗}(X)} onto [0, 1]^{M∩C^{∗}(X)}. In this context, we will identify X with its homeomorphic image in [0, 1]^{C^{∗}(X)} via the evaluation map. If X is locally compact, let X_{M} denote the space π(\overline{X}) − π(\overline{X} − X). That is, since \overline{X} = βX, we are letting X_{M} = π(X) − π(βX − X).

Now we consider the problem of separating normalized collections of closed sets in locally compact spaces. We need some simple topological properties of quotient maps on locally compact spaces. An idea first used by Tall is that one can recognize when a family is discrete in a locally compact space just by looking at countable subcollections.

Lemma 5.2. Let X be a locally compact space and let C be a discrete collection of subsets of X. Suppose that X, C, C^{∗}(X) are in an elementary submodel M such that every countable subset of M is a member of some countable element of M. Then the collection C_{M} = {π(C) : C ∈ M ∩ C} is discrete in X_{M}.

Proof. Suppose that the collection is not discrete. Let C_{M} denote π(C) for each C ∈ M ∩ C. If it is not discrete then there is a compact set F ⊂ X_{M} which meets C_{M} for infinitely many C ∈ C ∩ M. Since every such countable set is contained in some countable element of M there is a countable C′ ⊂ C which is an element of M such that F meets infinitely many members of {C_{M} : C ∈ C′}. Since C in discrete in X, there is a function f ∈ C^{∗}(X) such that {f(C) : C ∈ C′} is a closed discrete subset of (0, 1] and f(x) > 0 for all x ∈ X. By elementarity, there is such a function in M. Obviously, f_{M} = f ∘ π^{−1} ∈ C^{∗}(X_{M}) for all f ∈ C^{∗}(X) ∩ M (indeed, f and f ∘ π^{−1} are just restrictions of the projection function at the f-th coordinate in [0, 1]^{C^{∗}(X)} and [0, 1]^{M∩C^{∗}(X)}). Since f_{M}(π(C)) = f(C) for C ∈ C′ and f_{M}(X_{M}) ⊂ f(X) ⊂ (0, 1], it follows that C′ is discrete in X_{M}.

Proposition 5.3. Let X be a normal locally compact space. Let π be a continuous function of βX onto a space Y. Let X_{M} = Y − π(βX − X).

1. If A ⊂ X_{M} and U is a neighbourhood of π^{−1}(A) in X, then π(U) is a neighbourhood of A in X_{M}.

2. If A, B ⊂ X_{M} are closed and are such that π^{−1}(A) and π^{−1}(B) have disjoint closures in X then A and B are separated in X_{M}.

Proof. 1. Let F be the compact set βX − U. Therefore π(F) is a compact subset of Y which is disjoint from A – hence A is contained in the open set Y − π(F) ⊂ π(U).

2. Since X is normal, choose disjoint neighbourhoods U and V of the closure (in X) of π^{−1}(A) and π^{−1}(B). By part (i), π(U) is a neighbourhood of A. Furthermore, the closure of π(V) is contained in the image of the closure of V in βX (which is disjoint from A). Therefore π(U) − \overline{π(V)} and π(V) form the required separation of A and B.

Corollary 5.4. Let X be a normal locally compact space and let C be a discrete collection of subsets of X. Suppose that X, C, C^{∗}(X) are in an elementary submodel M such that every countable subset of M is a member of some countable element of M. Then the family {π(C) : C ∈ C ∩ M} is normalized in X_{M}.

Proof. By Lemma 5.2, the family {π(C) : C ∈ C ∩ M} is discrete. By similar reasoning as in 5.2, we can check that the family {π^{−1}(π(C)) : C ∈ C ∩ M} is also discrete in X. Indeed, if C′ is a countable subset of C ∩M then we may assume that C′ ∈ M and choose a function f ∈ C^{∗}(X) ∩ M which witnesses the discreteness of C. Since f ∈ M, it follows that f(π^{−1}(π(x))) = f(x) for all x ∈ X. Therefore, f witnesses the discreteness of {π^{−1}(π(C)) : C ∈ C′}. Therefore if C′ ⊂ C ∩ M and A = ⋃{π(C) : C ∈ C′}, it follows that π^{−1}(A) is closed in X. The result now follows from 5.3.

Let κ be supercompact and let P be the forcing for adding either κ many Cohen reals or κ many random reals. Fix P -names of X, C and C^{∗}(X) such that p forces that X is a locally compact space and that C is a discrete normalized collection. We will show that p does not force that C is unseparated (and since p was arbitrary, p forces that C is separated). Let θ be a cardinal much larger than the cardinality of the transitive collapse of (the names of) X and C^{∗}(X). Since θ is so large, it suffices to show that for some generic filter G with p ∈ G, H_{θ}[G] ⊧ C is separated.

Assume, for convenience of notation, that [H_{θ}]^{<θ} ⊂ H_{θ} and let j be a λsupercompact embedding for κ into a model W with λ > |H_{θ}|. As above let M be the image of H_{θ} under j. Now in W, let G be a j(P)-generic filter such that j(p) ∈ G. Naturally j(P) is just the forcing for adding j(κ) many Cohen reals. Therefore j(P) ∩ M is completely embedded in j(P) (hence G is j(P)-generic over M) and j(P)/(j(P) ∩ M) is isomorphic to j(P). Furthermore, j(P) ∩ M is taken to P by the Mostowskii collapsing function and since M^{ω} ⊂ M (it is easily checked that) Ĝ = { \atop G ∩ M} is P -generic over V (so is G ∩ H_{κ} but these are different generic filters).

¿From our results of § 4 we have.

Proposition 5.5. A summary of observations:

1. In W [G], M[G] is an elementary submodel whose Mostowskii collapse is H̆_{θ}[Ĝ].

2. H̆_{θ}[Ĝ] is a submodel of W [G ∩ M].

3. In W [G ∩ M], H̆_{θ}[Ĝ] is closed under <θ-sequences.

4. In W [G ∩ M], C^{∗}(j(X)) ∩ M (also taking into account the intersection of M with the ideal giving βX − X) corresponds to the subspace of [0, 1]^{M∩C^{∗}(j(X))} which is equal to j(X)_{M} ∩ W [G ∩ M].

5. Both X and C are in the model H̆_{θ}[Ĝ].

6. The image of C^{∗}(j(X)) ∈ M under the Mostowskii collapse is C^{∗}(X).

7. The image of j(C) by the Mostowskii collapse is C in the sense that, for each C ∈ C, the ideal {f ∈ C^{∗}(j(X)) : f(j(C)) = 0} is sent to {f ∈ C^{∗}(X) : f(C) = 0}.

Condition (iv) is not as complicated as it looks. For example, notice that even if j(X) is just [0, 1]^{ω} then j(X)_{M} = j(X). Then j(X)_{M} ∩ W [G ∩ M] will indeed be [0, 1]^{ω} but as determined in the inner model, more reals will be added.

Corollary 5.6. In W [G∩M], the space j(X)_{M} ∩W [G∩M] has a discrete collection {π(j(C)) ∩ W [G ∩ M] : C ∈ C} which is separated in j(X)_{M} ∩ W [G ∩ M] if and only if C is separated in X.

As discussed above, the final ingredient is taken from the Normal Moore Space investigations (see Fleissner [DTW90, Tal84, Fle84]).

Proposition 5.7. The family {π(j(C)) ∩ W [G ∩ M] : C ∈ C} is separated in j(X)_{M} ∩ W [G ∩ M] if and only if it is normalized after adding (enough) Cohen reals.

Therefore, to finish the proof of Balogh’s theorem, it now suffices to prove that {π(j(C)) : C ∈ C} is normalized in W [G]. But this follows immediately from Proposition 5.2 and Corollary 5.4.

6. PFA and OCA

Inspired by Todorčević’s book [Tod89], we take Ramsey-theoretic approach to the proper forcing axiom, PFA. Consider the following Ramsey-type axiom: Suppose that [X]^{2} = K_{0} ∪ K_{1}, then either X is the union of countably many 1homogeneous sets (i.e. X = ∪_{n}X_{n} such that [X_{n}]^{2} ⊂ K_{1}) or there is an uncountable 0-homogeneous set. We know that this axiom is false as it stands but it is a highly informative prototype. That is, if one sees an application of it then there is a natural poset (described below) which forces the axiom to hold for the specific case (where “uncountable” means size ω̆_{1}). If the partition is reasonably nice then the poset (or some modification of condition 4) is proper and we have a consequence of PFA. Of course, OCA is the (consistent) statement(Corollary 6.3) obtained by restricting to the case when X is a set of reals and K_{0} is an open set in the upper half plane.

Definition 6.1. Given a partition [X]^{2} = K_{0} ∪ K_{1}, we define p ∈ P if there are X_{p} and M_{p} such that

1. X_{p} is a finite 0-homogeneous subset of X,

2. M_{p} is a finite ∈-chain of countable elementary submodels of some H_{θ} (for a suitably large θ),

3. X_{p} is separated (ordered) by M_{p}, i.e. for x ≠ y ∈ X_{p}, either x comes before y in the sense that there is an M ∈ M_{p} such that either x ∈ M and y ∉ M or vice versa,

4. for each M ∈ M_{p} and x ∈ X_{p} − M, x is not in any 1-homogenous set which is a member of M.

Proposition 6.2. If X is a set of reals and K_{0} is an open subset of X^{2}, then either X is a countable union of 1-homogeneous sets or the poset of Definition 6.1 is proper.

Proof. Let P be a member of M ( a countable elementary submodel of some large H_{λ}) and fix any p ∈ P. We show that p′ = (X_{p}, {M ∩ H_{θ}} ∪ M_{p}) is the desired M-generic condition. Fix any D ∈ M which is a dense subset of P and choose q ∈ D which is below p′. Since X_{q} = {z_{1}, …, z_{m}, x_{1}, …, x_{n}} is ordered by M_{q}, {M_{0}, …, M_{n}} ⊇ M_{q} so that M_{0} = M ∩H_{θ}, {z_{1}, …, z_{m}} ⊂ M_{0}, and for i < j ≤ n, x_{j} ∈ M_{j} −M_{i}. Since K_{0} is open there are rational intervals B_{i} for i = 1, …, n such that {z_{1}, …, z_{m}} × ∪_{i≤n}B_{i} ⊂ K_{0} and for 0 < i < j ≤ n B_{i} × B_{j} ⊂ K_{0}. We will inductively choose, for i = 1, …, n, y_{i} ∈ M_{0}∩B_{i} so that {y_{n}, …, y_{m}}∪{x_{1} …, x_{m}} is 0-homogeneous and so that there is a q′ ∈ D∩M_{n} such that X_{q′} = {z_{1}, …, z_{m}}∪ {y_{1}, …, y_{m}}. Let us first recall that simply by choosing y_{i} ∈ B_{i} ∩ M_{n} we obtain that {y_{i}, x_{j}} ∈ K_{0} for all i ≠ j ≤ m. This is unusually simple but we encounter the more typical difficulties with the i = j case – here is where we use the fact that x_{i} is not in any 1-homogeneous set which is a member of M_{i−1}. Notice, for example, that for each open rational interval B the set

{x ∈ B : {x} × B ⊂ K_{1}}

is a 1-homogeneous set which is a member of M_{0} = M ∩ H_{θ} – hence does not contain x_{i}. Therefore, there are plenty of y_{i} in B_{i} such that {y_{i}, x_{i}} ∈ K_{0}, but of course we need enough so that we can get q′ to be in D.

Here is a standard technique for dealing with this. Although, D is not in H_{θ} the set

{X_{r} − {z_{1}, …, z_{m}} : r ∈ D and {z_{1}, …, z_{m}} ⊂ X_{r}} is.

We first consider only

t ∈ {X_{r} − {z_{1}, …, z_{m}} : r ∈ D and {z_{1}, …, z_{m}} ⊂ X_{r}}

which have size n and we consider them to be n-tuples in Π_{i=1}^{n}B_{i}. Then we let T′ be the tree of all ≤n-tuples generated from this set. Then we prune T′ to a tree T such that the non-maximal (in T′) nodes of T have a “large” number of immediate successors. The notion of large we use should for an i-tuple t, should guarantee that there is a y such that {y, x_{i+1}} ∈ K_{0} and t ∪ {y} ∈ T′ and (of course) if t ∪ {y} is not maximal it also should have a large number of immediate successors. It will also be important that the tree T is a member of M_{0} (note that D itself is not).

For i = 1, …, n, let t_{i} = {x_{j} : j < i}. Now t_{n} ∈ M_{n} and with t = t_{n}, T′_{t} = {y : t ∪ {y} ∈ T′} is a member of M_{n}. Furthermore,

{x ∈ T_{t} : {x} × T_{t} − {x} ⊂ K_{1}}

is a 1-homogeneous set and so does not contain x_{n}. Therefore, there is a w ∈ T_{t} − {x_{n}} ⊂ B_{n} such that {x_{n}, w} ∈ K_{0}. Choose basic open B′_{n}, C_{n} ⊂ B_{n} such that x_{n} ∈ B′_{n}, w ∈ C_{n} and B′_{n} × C_{n} ⊂ K_{0}. Let T^{n} be the subtree of T′ obtained by taking the downward closure of those nodes whose n^{th} point is in C_{n}. Note that t = t_{n−1} ∈ T^{n}. Also

{x ∈ T_{t}^{n} : {x} × T_{t}^{n} − {x} ⊂ K_{1}}

is a 1-homogeneous set which is a member of M_{n−1}. Therefore we can choose basic open B′_{n−1}, C_{n−1} ⊂ B_{n−1} such that

x_{n−1} ∈ B′_{n−1} B′_{n−1} × C_{n−1} ⊂ K_{0} and T_{t_{n−1}}^{n} ∩ C_{n−1} ≠ ∅ .

Then let T^{n−1} = {t ∈ T^{n} : (∃s ∈ T_{t}^{n}) s(n−1) ∈ C_{n−1}}. In this way we define C_{n}, …, C_{1} and T^{n}, …, T^{1}. The desired tree T is T^{1}. It turns out that finding the tree T has already done all the work – there is no reason to inductively choose y_{1}, …, y_{n} since any member of T will work. First of all, T is not empty since, by induction, t_{i} ∈ T^{i} – hence ∅ = t_{1} ∈ T.

To check that this works, let r ∈ D be such that {z_{1}, …, z_{m}} ⊂ X_{r} and X_{r} − {z_{1}, …, z_{m}} = {y_{1}, …, y_{n}} ∈ T. By definition of T^{i}, it follows that y_{i} ∈ C_{i} ⊂ B_{i}. It is easily checked that X_{r} ∪ X_{q} is 0-homogeneous.

Corollary 6.3 ( (PFA) implies (OCA)). Assume PFA. Suppose that X is a separable metric space and [X]^{2} = K_{0} ∪ K_{1} with K_{0} open in X^{2}. Then either there is an uncountable K_{0}-homogeneous set or X is a countable union of K_{1}-homogenous sets.

Here is a well-known application of OCA. Recall that Kunen proved that there are no (c, c)-gaps in [ω]^{ω} if PFA holds ( [Bau84]). A (λ, μ)-gap is a pair of chains of subsets of [ω]^{ω}, ({a_{α} : α < λ}, {b_{β} : β < μ}, such that each a_{α} is almost disjoint from each b_{β} and with the property that if a ^{∗}⊃ a_{α} for all α < λ, then there are β < μ such that a ∩ b_{β} is infinite.

Proposition 6.4 (OCA). If ω_{1} ≤ λ < μ, then there are no (λ, μ)-gaps in [ω]^{ω}.

Proof. Suppose that ({a_{α} : α < λ}, {b_{β} : β < μ}), is a gap. Let S be the set of (a, b) ∈ ([ω]^{ω})^{2} such that a ∩ b = ∅ and there are α < λ and β < μ such that a − a_{α} and b − b_{β} are both finite. Define a partition K_{0} ∪ K_{1} of S by

⟨(a, b), (c, d)⟩ ∈ K_{0} iff (a ∪ c) ∩ (b ∪ d) is not empty .

It is easily checked that if S is identified (canonically) with a subspace of ω^{ω}, then K_{0} is relatively open. Furthermore an uncountable 0-homogeneous set, {(c_{α}, d_{α}) : α < ω_{1}}, results in a Hausdorff-Lusin style gap. That is, suppose that A ⊂ ω is such that c_{α} − A and d_{α} ∩ A are finite for all α < ω_{1}. Fix n ∈ ω and uncountable I ⊂ ω_{1} such that c_{α} − A and d_{α} ∩ A are contained in n for all α ∈ I. By further refining I, we may suppose that d_{α} ∩ n is the same for all α ∈ I. But now it follows that c_{α} ∩ d_{β} is empty for any α and β in I – contradicting 0-homogeneity. But this contradicts the existence of A = ω − b_{ξ} where ξ < μ is chosen such that d_{α} < b_{ξ} for all α < ω_{1}.

Therefore it suffices to show that S cannot be expressed as a countable union of 1-homogeneous sets. Here is a nice argument for this. Suppose that S = ∪_{n}J_{n} and [J_{n}]^{2} ⊂ K_{1} for each n. Let M be a countable elementary submodel, with {J_{n} : n ∈ ω}, S in M, of some large H_{θ}. Let α_{0} and β_{0} be the supremum of M ∩ λ and M ∩ μ respectively. Let a = a_{α_{0}} and b = b_{β_{0}} − a. Fix n such that (a, b) ∈ J_{n}. Then, by virtue of the existence of (a, b),

M ⊧ (∀α ∈ λ) (∀β ∈ μ) (∃(a, b) ∈ J_{n}) such that a_{α} ⊂^{∗} a and b_{β} ⊂^{∗} b .

Therefore,

A = ⋃{a′ : (∃b′) (a′, b′) ∈ J_{n}}

contains, mod finite, every a_{α}, and

B = ⋃{b′ : (∃a′) (a′, b′) ∈ J_{n}}

contains, mod finite, every b_{β}. Furthermore, A and B are disjoint since [J_{n}]^{2} ⊂ K_{1}.

A recent very important result using this general technique (in hindsight(?)) is Balogh’s solution of the Moore-Mrowka problem (see [Juh92, 3.1]). It depends on showing that countably compact regular spaces of countable tightness contain uncountable free sequences. Let us check if the Ramsey axiom is relevant.

Suppose that X is a regular space and that F is a countably complete maximal filter of closed subsets of X. Fix, for each point x ∈ X, a pair, W, U of neighbourhoods of x such that \overline{W} ⊂ U and X − U ∈ F.

Define a partition of [X]^{2} as follows:

{x, y} ∈ K_{0} iff W_{x} ∩ W_{y} contains exactly one of {x, y} .

Claim: If some member of F is a countable union of 1-homogeneous sets, then X contains an uncountable closed discrete set.

Proof. Suppose that {I_{n} : n ∈ ω} are 1-homogeneous sets such that ∪I_{n} ∈ F. For each n choose, by the maximality of F, F_{n} ∈ F such that either F_{n} ∩ I′_{n} = ∅ or F_{n} ⊂ I′_{n}. Let F = ∩F_{n} ∈ F. Pick n such that I_{n} ∈ F^{+} and inductively choose F_{α} ∈ F, x_{α} ∈ I_{n} ∈ F_{α}, and and an increasing chain of countable elementary submodels M_{α} for α < ω_{1} as follows. Having defined {F_{β}, x_{β}, M_{β} : β < α}, let M_{α} be a countable elementary submodel containing this sequence together with X, F. Let F_{α} = ⋂{F : F ∈ F ∩ M_{α}} . Pick any x_{α} ∈ F_{α} ∩ I_{n}.

By countable tightness, it suffices to show that {x_{β} : β < α} is closed. Suppose that x ∈ F is a limit point and fix the minimum β < α such that x_{β} ∈ W_{x}. Choose m such that x ∈ I_{m}. Since x ∈ F_{1} and F is countably complete it follows that I_{m} ∩ F is uncountable. If x ∉ I′_{m}, then I′_{m} ∩ F = ∅ and I_{m} ∩ F would be an uncountable closed discrete set. Therefore, F ⊂ I′_{m}, in particular, x_{β} ∈ I′_{m}. By countable tightness, there is a y ∈ M_{β} ∩ I_{m} such that y ∈ W_{x} ∩ W_{x_{β}}. By the minimality of β, x is in \overline{\{x_{γ} : β < γ < α\}} ⊂ F_{β+1}. Since X − W_{y} ∈ F ∩ M_{β+1}, F_{β+1} ∩ W_{y} = ∅, and it follows that y ∈ W_{x} ∩ W_{y}, but x ∉ W_{y}. But this means that {x, y} ∈ K_{0} contradicting that x, y ∈ I_{m}.

Similarly, suppose that X contains a 0-homogeneous set, I, which meets every member of F. Inductively choose, {M_{α}, F_{α}} and x_{α} ∈ F_{α} ∩ I for α < ω_{1} just as above. By the construction x_{α} ∉ W_{x_{β}} for β < α. Since I is 0-homogenous, it follows that \{x_{β} : β < α\} ⊂ W_{α} ⊂ \overline{W_{α}} ⊂ U_{α}. Furthermore, F_{α+1} ∩ U_{α} = ∅ and {x_{γ} : γ > α} ⊂ F_{α+1}. Therefore {x_{α+1} : α < ω_{1}} is a free sequence.

If X is not a countable union of 1-homogeneous sets, then we can try the prototype poset P as given in Definition 6.1 for adding an uncountable 0-homogeneous set. Since this uncountable set will be added generically, we could even ensure that, if X had character at most ℵ_{1}, then the sequence {x_{α} : α < ω_{1}} would be a topological copy of ω_{1}. To do so we would simply have to include side conditions consisting of a finite sequence of neighborhoods of the members of X_{p} and restrict the definition of <_{P} in a rather obvious manner. There is no essential change to the following proof in order to obtain the stronger result.

Theorem 6.5 (PFA). Let X be a regular space. Suppose that Y is a subspace which has countable tightness and that F is a free countably complete maximal filter of closed subsets of Y. Furthermore, let f be any function with domain Y such that \overline{f^{−1}(f(y))} ∉ F for all y ∈ Y. Suppose that Y has no infinite closed discrete subspace on which f is one-to-one. Then X contains, {x_{α} : α < ω_{1}}, an uncountable free sequence which is a subset of Y (and on which, f is one-to-one). In addition, if {W_{y} : y ∈ Y } is any neighbourhood assignment, then {x_{α} : α ∈ ω_{1}} can be chosen so that for each α < ω_{1}, there is a β < α such that {x_{γ} : β < γ ≤ α} ⊂ W_{x_{α}}.

Proof. For each y ∈ Y, fix an X-neighbourhood W_{y} and F_{y} ∈ F so that \overline{W_{y}} ∩ Y − U_{y} = ∅ (i.e. U_{y} = X − \overline{F_{y}} in the discussion above). We make two changes to the definition of P from definition 6.1. Replace conditions 4 and 1 by

1’ X_{p} is a finite subset of Y,

4’ If M ∈ M_{p} and x is the minimal point of X_{p} −M (i.e. M ∈ M′ ∈ M_{p} implies

x ∈ M′), then x ∈ \overline{F ∩ M} for all F ∈ F ∩ M. Also the elementary submodels in M_{p} should include X with its topology, Y, F and the functions f, {⟨y, (W_{y}, F_{y})⟩ : y ∈ Y }.

We define p < q if X_{p} ⊃ X_{q}, M_{p} ⊃ M_{q}, and for each M ∈ M_{q} if y is the minimal point of X_{q} − M, then for each y′ ∈ X_{q} such that y ∈ W′_{y}

X_{p} ∩ M ⊂ W_{y′} ∪ ⋃{W_{z} : z ∈ X_{q} ∩ M} .

This condition is slightly weaker than demanding that X_{p} is a 0-homogeneous set where K_{0} is defined as in the discussion above. It is easier to view it as an approximation to a copy of ω_{1}. Indeed, given y as above, let z be the largest element of M ∩X_{q}. Then W′ = ∩ {W_{y′} : y ∈ W_{y′} and y′ ∈ X_{q}} is a neighbourhood of y and q will force that W′ will contain all members of X_{p} which are between z and y (with p < q). Furthermore, since the generic uncountable set will have a natural well-ordering in order type ω_{1}, this will guarantee that the existence of a stationary 0-homogenous set can be deduced from the pressing down lemma.

We now show that P is proper and that if Y has no infinite closed discrete subspace on which f is one-to-one then ∪{X_{p} : p ∈ G} is uncountable for any P -generic filter G.

Suppose that p ∈ P ∩ M for some countable elementary submodel M ∋ P of a sufficiently large H_{λ}. Define p_{M} = ⟨X_{p}, M_{p} ∪ (M ∩ H_{θ})⟩.

Fix any dense open subset D ⊂ P with D ∈ M and choose q ≤ p_{M} with q ∈ D. Let n = |X_{q} −M| and X′_{q} = X_{q} ∩M. Let T′ be the set of n element sets, t, for which there is an r ≤ p_{M} in D with t = X_{r} − X′_{q}. Just as in the proof of Proposition 6.2, T′ is in M ∩ H_{θ} although D is not.

First suppose that n = 0. Let M′_{q} = M_{q} ∩ M ∈ M. Choose, by elementarity, any r ∈ D ∩ M such thaX_{r} = X′_{q} and M′_{q} ⊂ M′ for each M′ ∈ M_{r} − M′_{q}. Such an r will be a member of D ∩ M which is compatible with q - as required.

Now suppose that n > 0. Choose an increasing sequence {M_{1}, …, M_{n−1}} ⊂ M_{q} (with M_{1} = M ∩ H_{θ}) to witness that X_{q} − M is separated. And let X_{q} = {x_{1}, …, x_{n}} be enumerated such that x_{i} ∉ M_{i}. The main difficulty in finding an r ∈ D ∩ M which is compatible with q is in ensuring that X_{r} − X′_{q} ⊂ W′, where W′ = ∩ {W_{y′} : y ∈ W_{y′} and y′ ∈ X_{q}}. Here is where we use condition 4’.

As in proposition 6.2, we prune T′ to a subset T so that, in this case, for each non-maximal t ∈ T,

\overline{\{y ∈ Y : t ∪ \{y\} ∈ T \}} ∈ F

This is done by a reverse induction on the cardinality of t ∈ T′. Again, by induction, we show that {x_{1}, …, x_{k}} remains in the T^{k} ∈ M which is obtained after n−k steps. Hence, after n steps we have our set T with ∅ ∈ T = T^{0} (in this case {x_{1}, …, x_{n}} is also in T).

Suppose that we have T^{k+1} (where T^{n} = T′) and that {x_{1}, …, x_{k}} ∈ T^{k+1}. Let

\begin{matrix}T^{k} = \{t ∈ T^{k+1} : \\ (∃s) (s ∪ t ∈ T^{k+1} , |s| = k and \overline{\{y ∈ Y : s ∪ \{y\} ∈ T^{k+1}\}} ∈ F\}. \end{matrix}

To see that {x_{1}, …, x_{k−1}} is in T^{k}, notice that

\overline{\{y ∈ Y : \{x_{1}, …, x_{k−1}\} ∪ \{y\} ∈ T^{k+1}\}}

is in M_{k} and contains x_{k}. If it were not in F then, by the maximality of F, there would be a member of F ∩ M_{k} (containing x_{k}) which missed it.

Next, we work in M and choose inductively, {y_{1}, …, y_{n}} ⊂ W′ ∩ M so that {y_{1}, …, y_{k}} is in T for each k. To see that y_{k+1} can be chosen recall that

x_{1} ∈ \overline{\overline{\{y ∈ Y : \{y_{1}, …, y_{k}\} ∪ \{y\} ∈ T \}} ∩ M}

and that, by countable tightness and elementarity,

\overline{\overline{\{y ∈ Y : \{y_{1}, …, y_{k}\} ∪ \{y\} ∈ T \}} ∩ M} = \overline{\{y ∈ Y : \{y_{1}, …, y_{k}\} ∪ \{y\} ∈ T \} ∩ M} .

Now suppose that Y does not contain an infinite closed set on which f is oneto-one. Fix any α ∈ ω_{1} and define

D_{α} = {p ∈ P : (∃M ∈ M_{p}) (α ∈ M and X_{p} − M ≠ ∅} .

It suffices to show that D_{α} is dense. Fix any p ∈ P and choose a countable elementary submodel M ≺ H_{λ} containing p and α. It is immediate that p_{M} (as defined above) is in P. Fix a descending base {F_{n} : n ∈ ω} for F ∩ M. Inductively choose y_{n} ∈ F_{n} ∩ M, so that y_{n} ∉ \overline{f^{−1}(f[\{y_{i} : i < n\}])}. Clearly f is one-to-one on \{y_{n} : n ∈ ω\}, so pick a limit point x. Note that x ∈ \overline{F_{n} ∩ M} for each n ∈ ω. Furthermore, x ∉ M since M ∩ ⋂{F_{n} : n ∈ ω} is empty. It follows, then that (X_{p} ∪ {x}, M ∪ {M ∩ H_{θ}}) is in D_{α} and is below p.

See [Juh92, 3.2,3.3] and [BDFN88], for other applications of the above theorem.

Corollary 6.6 (PFA). If a locally compact space X maps onto ω_{1} by a closed map, then X contains a perfect preimage of ω_{1}. If, in addition, the character of X is at most ω_{1}, then X contains a topological copy of ω_{1}.

There are many applications of this method to the structure of P(ω_{1}) and βω_{1} (see [Tod89, Ch. 8] and [Dow88]). Here is a very strong failure of the principle ♣ which was announced in [Tod89a].

Theorem 6.7 (PFA). If I ⊂ [ω_{1}]^{ω} is an ℵ_{1}-generated ideal then either

(0) there is an uncountable S ⊂ ω_{1} such that which is almost disjoint from each a ∈ I, or

(1) there are S_{n} such that ω_{1} = ⋃_{n} S_{n} and [S_{n}]^{ω} ⊂ I for each n.

Proof. Assume that there is no family {S_{n} : n ∈ ω} as in (1); we will produce an S as in (0). Let I_{0} be a generating set for I of cardinality ℵ_{1}. Also, let Ĩ denote the ideal {S ⊂ ω_{1} : [S]^{ω} ⊂ I}. Hence we are assuming that ω_{1} − ∪S is uncountable for any countable S ⊂ Ĩ. The rough idea is to get an uncountable 0-homogeneous set where (α, β) ∈ K_{0} might mean something like α is not in the β^{th} member of I_{0}.

Fix an appropriately large θ and define p ∈ P iff

p = A_{p} ∪ I_{p} ∪ M_{p} where \{ \begin{matrix}A_{p} ⊂ ω_{1} & is finite, \\ I_{p} ⊂ I_{0} & is finite \\ M_{p} & is a finite ϵ-chain separating A_{p}, \\ α ∉ ⋃(M ∩ Ĩ) & for any M ∈ M_{p} and α ∈ A_{p} − M, \\ each M ∈ M_{p} & is countable and M ≺ H_{θ} \end{matrix} ,

and p < q if p ⊃ q and A_{p} − A_{q} is disjoint from ∪I_{q}.

As usual, let M ≺ H_{κ} be countable, for a suitable κ and let p ∈ M ∩ P. We show that p ∪ {M ∩ H_{θ}} is (M, P)-generic. Let D ∈ M be a dense open subset of P and let r ∈ D be less than p ∪ {M ∩ H_{θ}}. Let {α_{0}, …, α_{n−1}} list A_{r} − M in increasing order. Also, let r_{0} = r ∩ M – note that r_{0} ∈ P. As usual, we consider

{A : (∃q ∈ D) q < r_{0} , |A_{q} − A_{r_{0}}| = n and A ⊂ A_{q} − A_{r_{0}}} .

Next, we can show, by considering {α_{0}, …, α_{n−1}}, that this has a definable subset T in M ∩ H_{θ} such that for each t ∈ T with |t| < n, the set T_{t} = {ξ : t ∪ {ξ} ∈ T } is not covered by countably many members of Ĩ.

Therefore, if we let I = ∪I_{r}, we have, by elementarity, that (M ∩ T_{t}) − I is not empty for each t ∈ T ∩ M. Since ∅ ∈ T ∩ M, we can recursively choose ξ_{i} ∈ a so that {ξ_{0}, …, ξ_{n−1}} is in T. Finally by elementarity, there is a q ∈ M ∩ D so that q < r_{0} and A_{q} − A_{r_{0}} = {ξ_{0}, …, x_{n−1}}. It follows that q is compatible with r, completing the proof that P is proper.

Recall that U(ω_{1}) ⊂ βω_{1} is the set of uniform ultrafilters on ω_{1}. Here is one interesting topological consequence of the previous result. It is also just a reformulation of a result in [Tod89] (see Proposition 6.9).

Corollary 6.8 (PFA). If H and K are disjoint closed subsets of ω_{1}^{∗} − U(ω_{1}), then there is an uncountable set I ⊂ ω_{1} such that one of I^{∗} ∩ H or I^{∗} ∩ K is empty.

Proof. For each α < ω_{1}, fix a_{α} ⊂ α such that

a_{α}^{∗} ∩ K = ∅ and H ∩ α^{∗} ⊂ a_{α}^{∗} .

Let I be the ideal generated by {a_{α} : α < ω_{1}} and apply Theorem 6.7.

Proposition 6.9. The following are equivalent.

1. If X and Y are Lindelöf countably tight spaces each with a unique non-isolated point, then X × Y is countably tight.

2. If H and K are disjoint closed subsets of ω_{1}^{∗} − U(ω_{1}), then there is an uncountable set I ⊂ ω_{1} such that one of I^{∗} ∩ H or I^{∗} ∩ K is empty.

Proof. Let X and Y be spaces as in a) with non-isolated points x and y respectively. Suppose that A ⊂ X × Y is uncountable and no countable subset of A has a limit point. It follows then that A ∩ (X × {y′}) and A ∩ ({x′} × Y is countable for each x′ ∈ X and y′ ∈ Y. Choose {(x_{α}, y_{α}) : α < ω_{1}} ⊂ A such that α < β implies x_{α} ≠ x_{β} and y_{α} ≠ y_{β}. Then b) will fail for H, K defined according to

H ∩ α^{∗} = ⋂\{S^{∗} : S ⊂ α and x ∉ \overline{\{x_{β} : β ∈ S\}}\}

and

K ∩ α^{∗} = ⋂\{S^{∗} : S ⊂ α and y ∉ \overline{\{y_{β} : β ∈ S\}}\}

for all α < ω_{1}.

Similarly, if H and K witness a failure for b), then X = ω_{1} ∪ {x} is obtained by taking the neighbourhood traces on ω_{1} of \overline{H} as the neighbourhood traces of x. Similarly for Y.

Remark 6.1. The following consequence of PFA is given in [Dow90] (via the above translation). If H is a closed subset of ω_{1}^{∗} − U(ω_{1}) with the property that the closure of H (in ω_{1}^{∗}) has character at most ω_{1} and contains U(ω_{1}) then there is an uncountable I ⊂ ω_{1} such that I^{∗} − U(ω_{1}) ⊂ H. If the character restriction could be removed this would be a significant strengthening of Theorem 6.8. However, it also shown in [Dow90], that the character restriction cannot be dropped.

We close this section with a method of getting other models of MA from a model of PFA. The first example of this in the literature is [Vlc90]. To demonstrate the technique we will again use gaps in [ω]^{ω}. Recall that a (λ, μ)-gap, ({a_{α} : α < λ}, {b_{β} : β < μ}, is said to be strong ( [vDKvM89]) if there is a unique (limit) ultrafilter p such that ω − (a_{α} ∪ b_{β}) ∈ p for all α < λ and β < μ.

Theorem 6.10. It is consistent with MA + c = ω_{2} that there is a strong (c, c)-gap.

Proof. Let V be a model of PFA. We define a poset P by

p = (a_{p}, b_{p}) ∈ P iff \{ \begin{matrix}a_{p} ∪ b_{p} ⊂ ω, \\ a_{p} and b_{p} & are disjoint, \\ ω − (a_{p} ∪ b_{p}) & is infinite. \end{matrix}

Define p ≤ q iff a_{p} ^{∗}⊃ a_{q} and b_{p} ^{∗}⊃ b_{q}.

Now P is σ-closed and so does not add any new subsets of ω.

Let A = {a_{p} : p ∈ G} and B = {b_{p} : p ∈ G} for some generic filter G. A standard density argument shows that,

{ω − (a ∪ b) : a ∈ A and b ∈ B} ∈ ω^{∗} .

And if P is ω_{1}-Baire, (i.e. the intersection of ℵ_{1} dense open sets will again be dense open - hence no new subsets of ω_{1} will be added) then A′ and B′ are separated for any A′ ∈ [A]^{ω_{1}} and B′ ∈ [B]^{ω_{1}}. Furthermore, MA + c=ω_{2} will hold in V [G] since all posets of size ℵ_{1} and ℵ_{1}-sized families of dense subsets will be from the ground model V. Lemma. P is ω_{1}-Baire.

To prove this we define an Amoeba forcing – i.e. a forcing whose generic object will result in a new condition in P.

Let A(P) = {(p, n) : p ∈ P and n ∈ ω} and define

(p, n) < (q, m) if \{ \begin{matrix}n ≥ m, \\ a_{p} ⊃ a_{q} & and a_{p} ∩ m = a_{q} ∩ m, \\ b_{p} ⊃ b_{q} & and b_{p} ∩ m = b_{q} ∩ m. \end{matrix}

We show that A(P) is proper. Let M be a countable elementary submodel of some large H_{θ} and let (p_{0}, m_{0}) ∈ P ∩ H_{θ}. Let {D_{n} : n ∈ ω} enumerate the dense open subsets of A(P) which are members of M.

Fact: For any D_{n} and p ∈ P and m ∈ ω, there is a q ∈ P and an m′ such that (q, m) < (p, m) and (6.10-1) if there is any (r, k) ∈ D_{n} with (r, 0) < (q, 0) then

( (a_{q} ∪ (a_{r} ∩ m), b_{q} ∪ (b_{r} ∩ m)), m) is in D_{n}.

To prove this fact, simply list all pairs of disjoint subsets of m as {(c_{ℓ}, d_{ℓ})}. Define a descending sequence (q_{ℓ}, m) with q_{0} = p together with a sequence m′_{ℓ} of integers, such that, if possible, ( (a_{q_{ℓ+1}} ∪ c_{ℓ}, b_{q_{ℓ+1}} ∪ d_{ℓ}), m′_{ℓ}) is in D_{n}. Then q is the last q_{ℓ} and m′ is the maximum of the m′_{ℓ}.

Therefore, by applying the above fact for each n, there is a strictly decreasing sequence {(q_{n}, m_{n}) : n ∈ ω} with (q_{0}, m_{0}) = (q_{0}, n_{0}) < (p, n_{0}) and {m_{n} : n ∈ ω} strictly increasing, such that (6.10-1) holds (for each n) with q = q_{n}, m = m_{n} and m_{n+1} is larger than the m′ given by the Fact. Now let a = ∪{a_{q_{n}} : n ∈ ω} and b = ∪{b_{q_{n}} : n ∈ ω}. It is immediate from the definition of (q_{n+1}, m_{n+1}) < (q_{n}, m_{n}) that a and b are disjoint. However, it is not immediate that ω−(a∪b) is infinite. To obtain this, we simply ensure that m_{n} is chosen large enough so that |m_{n} − (a_{n} ∪ b_{n})| > n. Since ( (a, b), m_{n}) < (q_{n}, m_{n}) this ensures that |m_{n} − (a ∪ b)| > n for all n.

Claim: (q, n_{0}) = ( (a, b), n_{0}) < (p_{0}, n_{0}) is (M, P)-generic. Indeed, suppose that (r, k) ∈ D_{n} and (r, k) < (q, n_{0}). Clearly, (r, k) < (q_{n}, 0), hence by (6.10-1), it follows that ( (a_{q_{n}} ∪ (a_{r} ∩ m_{n+1}), b_{q_{n}} ∪ (b_{r} ∩ m_{n+1})), m_{n+1}) is in D_{n} ∩ M. It is also the case that this condition is below (q, n_{0}).

To finish the proof of the Theorem, we let D be a collection of ℵ_{1} dense open subsets of P. We shall show that ⋂ D is dense by showing that if G is suitably generic over A(P), then

(a_{G}, b_{G}) ∈ ⋂ D where a_{G} = ⋃_{p∈G} a_{p} and b_{G} = ⋃_{p∈G} b_{b} .

First of all, it is a simple matter to check that if D ⊂ P is dense open, then the set {(p, n) : p ∈ D and n ∈ ω} is a dense open subset of A(P). Furthermore, there is a countable family E of dense open subsets of A(P) such that if G ⊂ A(P) is E-generic, then (a_{G}, b_{G}) ∈ P.

7. Kurepa families

A Kurepa family is a family K of countable sets such that for every countable set A, the family K ↾ A = {K ∩ A : K ∈ K} is countable ( [Dev84, p.320]). K is cofinal iff for every countable A ⊂ ⋃ K, there is a K ∈ K such that A ⊂ K. Note that the cardinality of a cofinal Kurepa family is determined by the cardinality of its union, i.e., for | ⋃ K| > ω,

|K| = cf [⋃ K]^{ℵ_{0}}

Note also that if there is a cofinal Kurepa family with union θ, then there is such a family K which is well-founded with respect to ⊂. This fact is useful in inductive constructions that involve the existence of a cofinal Kurepa family. Note also the difference between cofinal Kurepa families and “global” ones, i.e., if A is a subset of ⋃ K of cardinality θ such that θ > cf θ = ω, then K ↾ A has size \underline{bigger} than |A|. This means that any construction of a cofinal Kurepa family on the cardinal θ must assume the existence of such a family on any cardinal less than θ.

To give a typical application of cofinal Kurepa families, let us say that X = X_{0}∪X_{1} is a Bernstein decomposition of a space X if for every C ⊆ X homeomorphic to the Cantor set, C ∩ X_{i} ≠ ∅ for each i < 2.

Proposition 7.1. Let X be the Σ-product of the unit interval [0, 1] over an index set I and suppose there is a cofinal Kurepa family on I. Then X admits a Bernstein decomposition.

Proof. For K in K let X_{K} = X_{K}^{0} ˙∪ X_{K}^{1} be a fixed Bernstein decomposition of

X_{K} = {f ∈ X : {i ∈ I : f(i) ≠ 0} ⊆ K} .

For f ∈ X, let K be the minimal element of K that contains {i ∈ I : f(i) ≠ 0}, and put f in X_{i} if f is an element of X_{K}^{i}. The fact that K is a Kurepa family easily gives that any Cantor set must intersect both X_{0} and X_{1}.

The existence of a cofinal Kurepa family on θ for any cardinal θ follows from an appropriate form of Jensen’s gap-one two-cardinal theorem in L ( [Jen72]). In the case of an ordinary Kurepa family an analysis of Jensen-Silver proof showed (see [Tod84, Theorem 8.7]) that we actually have a sharper result in the sense that only one □-sequence is needed and that, moreover, no form of GCH is necessary. As we have already indicated above, so sharp a result cannot be expected in the case of cofinal Kurepa families.

We say that a Kurepa family L extends a Kurepa family K if K ⊂ L and for all K ∈ K and L ∈ L, K ∩ L ∈ K. We use K ⊲ L to denote this relation. Note that the union of any ⊲-chain of cofinal Kurepa families is a Kurepa family which is, moreover, cofinal if the cofinality of the chain is uncountable.

Proposition 7.2. If every Kurepa family of size at most θ extends to a cofinal Kurepa family, then the same is true about θ^{+}.

Proof. Recursively construct a ⊲-chain of Kurepa families.

This shows that cofinal Kurepa families exist on any ω_{n} for n ∈ ω. Unfortunately, this is all one can prove in ZFC. To show that such a family exists, in the constructible universe, on any θ, we need only show (by copying the Jensen-Silver proof) the following.

Proposition 7.3 (V=L). Suppose ω = cf θ < θ and that any Kurepa family of size <θ can be extended to a cofinal Kurepa family. Then the same is true about Kurepa families of size θ.

Proof. We shall concentrate on proving the existence of a cofinal Kurepa family in [θ]^{ℵ_{0}} assuming □_{θ} and cf [θ]^{ℵ_{0}} = θ^{+}. The part of extending an arbitrary Kurepa family of size θ is obtained by a trivial modification of this proof.

Let ⟨C_{α} : α < θ^{+}⟩ be a fixed □_{θ}-sequence and for α < θ^{+}, let S_{α} be the set of all limit points of C_{α}. Let {A_{α} : α < θ^{+}} be a cofinal subset of [θ]^{ℵ_{0}} such that A_{α} ⊂ α for all α. The family K will be obtained as the union of an increasing ⊲-chain {K_{α} : α < θ^{+}} of Kurepa families of size θ such that α ⊂ ⋃ K_{α} for all α. We shall also have that K_{α} is closed under finite unions, intersections and differences. Moreover, we shall have an r_{α}: K_{α} ↦ ω respecting ⊆ (i.e., A ⊆ B implies r_{α}(A) ≤ r_{α}(B)) such that if K_{α,n} = {K ∈ K_{α} : r_{α}(K) ≤ n}, then:

1. K_{α,n} ⊲ K_{α,n+1} for n < ω,

2. K_{α,n} is a cofinal Kurepa family of size <θ for all n < ω,

3. K − ⋃ K_{α,n} ∈ K_{α,n+1} for all K ∈ K_{α,n+1} and n < ω,

4. α ∈ S_{β} implies r_{α} = r_{β} ↾ K_{α}. To handle the successor step, find K_{α} ⊇ A_{α} ∪ {α} such that K_{α} ∩ K ∈ K_{α} for all K in K_{α} and let K_{α+1} be generated by K_{α} ∪ {K_{α}}. Let r_{α+1}(K_{α}) = 0 and extend r_{α+1} on the rest of K_{α+1} using r_{α} and respecting ⊆. Note that r_{α+1} does not extend r_{α} since we must have that r_{α}(K) = 0 for all K ⊂ K_{α}.

As for limit steps, the only non-trivial case is when cf α = ω. For each ξ ∈ S_{α}, since K_{ξ,n} is a cofinal Kurepa family and r_{ξ} preserves ⊂, it follows that ⋃ K_{ξ,n} does not contain any member of K_{ξ,n+1} − K_{ξ,n+1}. In fact, since the r_{ξ} for ξ ∈ S_{α} cohere, it follows that ⋃ K_{ξ,n} does not contain any K such that r̄_{α}(K) > n where r̄_{α} = ∪_{ξ∈S_{α}}r_{ξ}. For each n, let K̄_{α,n} = r̄_{α}^{−1}([0, n]) = ⋃_{ξ∈S_{α}} K_{ξ,n}. In case S_{α} is cofinal in α we may now simply extend the K̄_{α,n}’s to cofinal Kurepa families K_{α,n}’s (using the inductive hypothesis of Proposition 7.3).

In case γ = sup S_{α} is not cofinal in α, we instead define K̄ to be K_{γ,n} ∪ ⋃_{ξ∈S_{α}} K_{ξ,n} − K_{γ}. Again extend making sure conditions 1-4 remain true.

Remark 7.1. Note that the above proof shows that the only assumption needed to construct a cofinal Kurepa family in any [κ]^{ℵ_{0}} is the following:

(∀θ) (ω = cf θ < θ → □_{θ} ∧ cf [θ]^{ℵ_{0}} = θ^{+} ).

There are many other places in the literature where the idea of the Jensen-Silver proof has been applied. We mention the closely related construction of Juhász-Nagy-Weiss [JNW79] of a regular locally countable countably compact monolithic (≡ closure of every countable set is second countable) space of any size of cofinality ≠ ω assuming the above hypothesis. The following fact explains this connection.

Proposition 7.4. There is a cofinal Kurepa family of size θ which is well-founded and closed under finite intersections iff there is a regular locally countable countably compact monolithic space of size θ.

Proof. Let K be a cofinal Kurepa family of size θ which is well-founded and closed under intersections. We may assume that K contains {x} for every x in ⋃ K. For K in K, let (., K] denote the set {L ∈ Kurepa : L ⊆ K}. Let σ be the topology on K determined by {(., K] \ (., L] : K, L ∈ K , L ⊂ K} as a subbasis. Then (K, σ) is a locally countable Hausdorff (because K is closed under intersections) space which is countably compact (because K is cofinal and well-founded) and first countable (because K is Kurepa). It follows that (K, σ) is also regular and monolithic.

Conversely, let X be a locally countable, countably compact monolithic space of size θ. Let K_{X} be the family of all compact clopen subsets of X. Then it is easily checked that K_{X} is a cofinal Kurepa family of size θ. (The fact that K_{X} is a Kurepa family was first observed by W. Weiss in a slightly different context).

Another closely related object from the literature is the simplified morass of [Vlm84]. (Clearly every simplified (ω_{1}, 1)-morass is a cofinal Kurepa family.) This suggests some natural strengthenings of the notion of a cofinal Kurepa family. Namely, we may require the family K to be, moreover, stationary in [⋃ K]^{ℵ_{0}} or even stationarycoding ( [Zwi84]) in the sense that K ⊊ L ∈ K implies that sup K ∈ L. It is, therefore, natural to expect that stationary Kurepa families (or stationary-coding Kurepa families) might have applications where morasses cannot be used because of the cardinality restrictions.

Remark 7.2. This entire section was simply copied from a preprint of [Tod90] except this sentence which was inserted into the following remark.

The sole purpose of this note is to unify several ideas scattered throughout the literature. All non-attributed facts mentioned in this note should be considered as parts of the folklore of this subject.

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