A TREE π-BASE FOR ℝ^{∗} WITHOUT COFINAL BRANCHES.

FERNANDO HERNÁNDEZ-HERNÁNDEZ

Abstract. We prove an analogue to Dordal’s result in [Dor87]. He obtained a model of ZFC in which there is a tree π-base for ℕ^{∗} with no ω_{2} branches yet of height ω_{2}. We establish that this is also possible for ℝ^{∗} using a natural modification of Mathias forcing.

1. Introduction

Definition 1.1. Let X be a topological space. The Novák number, 𝔫(X), is the minimum number of dense open subsets whose intersection is empty. w𝔫(X) denotes the minimum number of dense open subsets of X whose intersection has empty interior and it is called the weak Novák number.

𝔫(X) is also known as the Baire number of X. J. Novák was among the first topologist who studied this characteristic for general topological spaces. J. van Mill and S. Williams in [vMW83] introduced the definition of the weak Novák number. B. Balcar, J. Pelant and P. Simon in [BPS80] studied the Novák number of ℕ^{∗} using that ℕ^{∗} always has a tree π-base. In general, a tree π-base for a topological space is a π-base which forms a tree when ordered by reverse inclusion. They also introduced the cardinal, now denoted by 𝔥 or 𝔥(ℕ^{∗}), as the minimum height of a tree π-base. In general, 𝔥(X) is the distributivity number of the Boolean algebra RO(X); equivalently, it is the minimum cardinal κ such that forcing with RO(X) does not add a new subset of κ. By results in [Wil82], for every locally compact noncompact metric space X, its Stone-Čech remainder, X^{∗}, always has a tree π-base, 𝔥(X^{∗}) = w𝔫(X^{∗}) and this cardinal coincides with the minimum height of a tree π-base.

Trivially, 𝔥(X^{λ}) ≥ 𝔥(X^{γ}) holds whenever the numbers are defined and λ ≤ γ. In fact, if {D_{α} : α < 𝔥(X^{λ})} is a family of dense open sets of X^{λ} whose intersection is not dense then letting D′_{α} = {f ∈ X^{γ} : f ↾ λ ∈ D_{α}} we get a family of dense open subsets of X^{γ} and their intersection has empty interior. We have been interested in the following questions appeared in [Dow98]:

2000 Mathematics Subject Classification. Primary 54G05, 54A35, 03E17, 06E15.

Key words and phrases. distributivity of Boolean algebras, cardinal invariants of the continuum, Stone-Čech compactification, tree π-base.

The author’s research was partially supported by CONACYT grant 122437 and PAPIIT grant IN106705.

2 FERNANDO HERNÁNDEZ-HERNÁNDEZ

Is 𝔥 ((ℝ^{∗})^{ω}) = 𝔥 (ℝ^{∗})?

Is it true that 𝔥(ℝ^{∗}) ≤ 𝔥(ℕ^{∗} × ℕ^{∗})?

Is it true 𝔫(ℝ^{∗}) = 𝔫(ℕ^{∗})?

Alan Dow has conjectured positive answers for the first two questions; the third one is due to E. K. van Douwen.

Following Dow, we denote by \overline{cpt} the ideal of regular open sets which are bounded, or equivalently, have compact closure and we also adopt the convention that a ^{∗} adorning a word or symbol will mean that it should be interpreted modulo the ideal \overline{cpt} in RO(ℝ); e.g. A ⊆^{∗} B ⊆ ℝ will mean that A \ B is bounded (it has compact closure). Then as he does, we represent RO(βℝ \ ℝ) as the completion of the quotient RO(ℝ)⧸\overline{cpt}, which from now on we will denote by ℜ^{∗}. Although the elements of ℜ^{∗} are equivalence classes of regular open sets, for convenience, we treat them as actual regular open sets.

In the Mathias model (i.e. the model obtained by a countable support iteration of Mathias forcing) P(ω)⧸fin has no dense tree of height ω_{1}, so 𝔥(ℕ^{∗}) = ℵ_{2} in the Mathias model. However, while there is no dense tree of height ω_{1}, P. L. Dordal showed [Dor87] that there must be a dense tree in which there are no branches of length ω_{2}. This means that a tree π-base for ℕ^{∗} in this model has barely height ω_{2}. Dordal’s model was actually constructed using a modified support for Mathias forcing but was easily adapted once Shelah showed [She84] that the “not filling towers” property —the key of Dordal’s proof— is preserved by countable support proper iterations.

Shelah and Spinas [SS00] showed that in the Mathias model, 𝔥(ℕ^{∗} × ℕ^{∗}) = ℵ_{1} (but 𝔥(ℕ^{∗}) = ℵ_{2} hold). It was known that 𝔥(ℝ^{∗}) ≤ 𝔥(ℕ^{∗}), and Alan Dow proved that in the Mathias model 𝔥(ℝ^{∗}) = ℵ_{1}. Later, armed with a better understanding of the cardinal 𝔥(ℝ^{∗}), B. Balcar and M. Hrušák proved the following result from which Dow’s result easily follows:

Theorem 1.2 ([BH04]). 𝔥(ℝ^{∗}) ≤ {𝔥, add (M)}, where M is the ideal of meager sets in ℝ, and add (M) its additivity number.

Dordal’s result implies that 𝔥(ℕ^{∗}) = 𝔫(ℕ^{∗}) = ℵ_{2} in the Mathias model. However it is shown in [BPS80] that 𝔫(ℝ^{∗}) is always at least ℵ_{2}, hence, in the Mathias model, 𝔥(ℝ^{∗}) < 𝔫(ℝ^{∗}). Nevertheless,

Theorem 1.3 ([Dow89]). There is a model for ZFC in which 𝔥(ℝ^{∗}) = 𝔫(ℝ^{∗}).

A TREE π-BASE FOR ℝ^{∗} WITHOUT COFINAL BRANCHES. 3

2. Modification of Mathias forcing

Our terminology is mostly standard. The bar over a set denotes the closure of it with respect to the ambient space understood from the context. Remember that u is a regular open set if int (\overline{u}) = u. We often use the first lowercase letters to denote open intervals in ℝ, while letters like s, t, u, v, etc. for regular open sets in ℝ. Letters like k, m, n, etc. denote positive integers. Unless otherwise explicitly stated, the first capital letters denote finite unions of open intervals with rational endpoints. For convenience we will sometimes confuse the elements of the algebra ℜ^{∗} with unbounded regular open subsets of the reals.

We will use the following forcing notion which we will denote by 𝕄_{ℝ^{∗}}. Let us first remember that Mathias forcing 𝕄 is the set of all pairs (s, A) ∈ [ω]^{<ℵ_{0}} × [ω]^{ℵ_{0}} such that s ∩ A = ∅ with the ordering (s, A) ≤ (t, B) if and only if t ⊆ s ⊆ t ∪ B and A ⊆ B. Mathias forcing is the natural forcing to increase 𝔥(ℕ^{∗}) since it adds a branch to any given tree and at the same time it diagonalizes that branch. However, Mathias forcing does not increase 𝔥(ℝ^{∗}); actually, as we said before, after an ω_{2}-iteration of 𝕄 over a model of CH, 𝔥(ℝ^{∗}) remains ℵ_{1}. To increase 𝔥(ℝ^{∗}) we consider an analogue of 𝕄 for the regular open algebra on ℝ.

Definition 2.1. 𝕄_{ℝ^{∗}} will be the following forcing notion:

1) conditions are pairs (A, s) where A is a finite union of separated open intervals in ℝ with rational endpoints, s is an unbounded regular open set and \overline{A} ∩ \overline{s} = ∅,

2) the ordering is (A, s) ≤ (B, t) if and only if B ⊆ A ⊆ B ∪ t and s ⊆^{∗} t.

Like conditions in the Mathias forcing, in our forcing we have that (A, s) ⊥ (A, t) if and only if s ∩ t =^{∗} ∅. From this follows that 𝕄_{ℝ^{∗}} has the κ^{+}-cc, where κ = c(ℝ^{∗}) ≤ w(ℝ^{∗}) ≤ w(βℝ) ≤ 2^{d(ℝ)} = 𝔠; for example, 𝕄_{ℝ^{∗}} has the ℵ_{2}-cc under CH. If (A, s) and (B, t) are two conditions in 𝕄_{ℝ^{∗}}, we have that ¬ ((A, s) ⊥ (B, t)) if and only if 1) A \ B ⊆ s and B \ A ⊆ t, 2) s ∩ t ≠^{∗} ∅.

Like Mathias forcing, 𝕄_{ℝ^{∗}} can be factored as an ℵ_{1}-closed forcing followed by a σ-centred forcing.

Lemma 2.2. There is a two-stage iteration ℙ ∗ ℚ̇ such that ℙ is ℵ_{1}-closed, ⊩_{ℙ} “ℚ̇ is σ-centred”, and 𝕄_{ℝ^{∗}} can be densely embedded in ℙ ∗ ℚ̇.

Proof. ℙ is ⟨ℜ^{∗}, ⊆^{∗}⟩. A generic Ġ for ℙ is an ultrafilter over ℝ; we now define ℚ̇ in V^{ℙ} to be the σ-centred notion of forcing that adds an unbounded regular open x ⊆ ℝ almost contained in every y ∈ Ġ. Specifically, ℚ can be defined as the set of all pairs (A, s) with A a finite

4 FERNANDO HERNÁNDEZ-HERNÁNDEZ

union of separated open intervals with rational endpoints and s ∈ Ġ. As before, we say that (A′, s′) ≤ (A, s) if A ⊆ A′ ⊆ A ∪ s and s′ ⊆ s. It is possible to check that the image of 𝕄_{ℝ^{∗}} under the embedding (A, s) ↦ (s, (A, s)) is dense.

We will denote ℚ in the previous lemma by 𝕄_{ℝ^{∗}}(Ġ), where Ġ is the generic filter added by ℜ^{∗}. So, we can express 𝕄_{ℝ^{∗}} = ℜ^{∗} ∗ 𝕄_{ℝ^{∗}}(Ġ).

Corollary 2.3. The modification of Mathias forcing is a proper forcing, and hence, it does not collapse cardinals over a model of CH.

If G is ℜ^{∗}-generic, then it is a P -point on ℝ; that is, if {s_{n}}_{n∈ω} ⊆ G then there exists s ∈ G such that (∀n ∈ ω) (s ⊆^{∗} s_{n}). Indeed, this is a consequence of the fact that countably closed posets are ℵ_{1}-Baire. If {s_{n} : n ∈ ω} ⊆ G, and D_{n} = {t ∈ ℜ^{∗} : s_{n} ∩ t =^{∗} ∅ or s_{n} ⊇^{∗} t}, then D_{n} is open dense subset of ℜ^{∗}. Hence D = ⋂_{n∈ω} D_{n} is still dense. Thus, if s ∈ G ∩ D, then (∀n ∈ ω) (s ⊆^{∗} s_{n}). This diagonalization method will be used often in the sequel. Another trivial observation which will be important later is that for each (A, s) ∈ 𝕄_{ℝ^{∗}} we can find (A, s′) ≤ (A, s) , where s′ is a countable union of separated open intervals with rational endpoints.

Our next theorem shows that after an ω_{2}-iteration of 𝕄_{ℝ^{∗}} we get 𝔥 (ℝ^{∗}) = ℵ_{2}. The rest of the section will show that the forcing 𝕄_{ℝ^{∗}} preserves towers in ℜ^{∗} and in the next section we conclude that despite 𝔥 (ℝ^{∗}) = ℵ_{2} there is a tree π-base for ℝ^{∗} without cofinal branches.

In the following proof we use the concept of almost disjoint family. A family of unbounded regular open sets E is almost disjoint (with respect to the ideal \overline{cpt}) if U ∩ V ∈ \overline{cpt} whenever U and V are distinct elements of E. A maximal almost disjoint family of unbounded regular open sets is a maximal element in the collection of all the almost disjoint families with the containment order.

Theorem 2.4. Let V ⊨ CH, and let ℙ_{ω_{2}} be an iteration of modified Mathias forcing. Then V^{ℙ_{ω_{2}}} ⊨ 𝔥 (ℝ^{∗}) = ℵ_{2}.

Proof. It suffices to show that every collection {E_{γ} : γ < ω_{1}} of maximal almost disjoint families of unbounded regular open sets in V^{ℙ_{ω_{2}}}, there is an unbounded regular open set x almost contained in a member of each E_{γ}. Let {Ė_{γ} : γ < ω_{1}} be a sequence of names for such almost disjoint families and fix a p ∈ ℙ_{ω_{2}}. Let G = G_{ω_{2}} be ℙ_{ω_{2}}-generic, with G_{α} = G ↾ α ℙ_{α}-generic for all α < ω_{2}. Let B_{α} denote the α-th generic subset. We will find α < ω_{2} such that p ⊩ “B_{α} is almost contained in a member of each E_{γ}”.

A TREE π-BASE FOR ℝ^{∗} WITHOUT COFINAL BRANCHES. 5

Given a ℙ_{ω_{2}}-name ẋ and p ∈ ℙ_{ω_{2}} such that p ⊩ ẋ∈ℜ^{∗}, there is some α = α (x) < ω_{2} such that α < ω_{2} and p ⊩ ẋ ∈ V [G_{α}]. Let each Ė_{γ} be enumerated as a sequence of ℙ_{ω_{2}}-names {ẋ_{ξγ} : ξ < ω_{2}}, where we have assumed for convenience that

p ⊩ (∀γ < ω_{1}) (|Ė_{γ}| = ℵ_{2}) .

Define f : ω_{2} → ω_{2} by f(β) = sup {α(ẋ_{ξγ}) : ξ < β, γ < ω_{1}}. Thus for every β < ω_{2} and γ < ω_{1}, we have p ⊩ {ẋ_{ξγ} : ξ < β} ∈ V [G_{f(β)}]. Since V [G_{β}] ⊨ CH for β < ω_{2}, define g : ω_{2} → ω_{2} such that

p ⊩ (∃ξ < g(β)) (\overline{ẏ ∩ ẋ_{ξγ}} is not compact) ,

for every β < ω_{2}, every γ < ω_{1}, and every ℙ_{β}-name ẏ such that p ⊩ ẏ ∈ ℜ^{∗}.

Finally, choose α < ω_{2} so cf(α) = ω_{1} and for every β < α, f(β) < α and g(β) < α. We claim that α is as desired. First, the definition of f implies that p ⊩ {ẋ_{ξγ} : ξ < α} ∈ V [G_{α}] for all γ < ω_{1}. Secondly, the preservation of ω_{1} implies that every regular open subset in V [G_{α}] in fact lies in some V [G_{β}] with β < α. By definition of g, each {ẋ_{ξγ} : ξ < α} , γ < ω_{1}, is a maximal almost disjoint family in V [G_{α}]. Now, for each γ < ω_{1}, {ẋ_{ξγ} : ξ < α} is a maximal almost disjoint family in ℜ^{∗}, therefore one of its elements must be a member of Ġ, the ℜ^{∗}-generic filter. Therefore B_{α} must be contained in a member of each {ẋ_{ξγ} : ξ < α}, as we wanted to show.

Theorem 2.5. Let {x_{α} : α < κ} be a tower in ℜ^{∗}. Then it remains a tower in V^{𝕄_{ℝ^{∗}}}.

Proof. Suppose not, and let ẋ be a 𝕄_{ℝ^{∗}}-name so as (A_{0}, s_{0}) ⊩ “ẋ ∈ ℜ^{∗} and (∀α < κ) (ẋ ⊆^{∗} x_{α}) ” for some (A_{0}, s_{0}) ∈ 𝕄_{ℝ^{∗}}. Without loss of generality assume (A_{0}, s_{0}) ⊩ “ẋ ⊆ ℝ^{+} \ ℕ =∅ and ẋ ∩ [n, n + 1) is a finite union of separate open intervals with rational endpoints, for each n ∈ ℕ”. We will need some lemmas.

Lemma 2.6. Suppose D ⊆ 𝕄_{ℝ^{∗}} is a dense open subset and that (A, s) ∈ 𝕄_{ℝ^{∗}}. Then there exists some regular open t ⊆ s such that if (C, u) ≤ (A, t) and (C, u) ∈ D, then there is m ∈ ℕ such that (C, t \ [0, m]) ∈ D.

Proof. By induction construct sequences {b_{n} : n ∈ ω} and {t_{n} : n ∈ ω} of separate open intervals with rational endpoints and elements of ℜ^{∗}. Suppose that {b_{n} : n ≤ k} and t_{k} are already constructed. Let B_{n} = {B_{i} : i ∈ ω} be the family of all C extending A and such that C \ A ⊆ ⋃_{i=0}^{k} b_{i}. Construct regular open subsets u_{i+1} ⊆ u_{i} ⊆ t_{k} (i ∈ ω) as follows: Suppose the set u_{i} is given, if there exists v ⊆ u_{i} such that (B_{i+1}, v) ∈ D, then u_{i+1} = v; otherwise u_{i+1} = u_{i}.

6 FERNANDO HERNÁNDEZ-HERNÁNDEZ

Finally let t_{k+1} ∈ ℜ^{∗} such that t_{k+1} ⊆^{∗} u_{i}, for all i ∈ ω, and let b_{k+1} ⊆ ⋂_{i=0}^{k+1}t_{i} be an interval with rational endpoints. Then t = ⋃_{n∈ω} b_{n} has the required properties.

Indeed, suppose (C, u) ≤ (A, t) and (C, u) ∈ D. Let n ∈ ω be such that C \ A ⊆ ⋃_{i=0}^{n} b_{i}. Then C = B_{j+1}, for some j ∈ ω and B_{j+1} ∈ B_{n}. Note that u ⊆^{∗} t ⊆^{∗} t_{j+1} ⊆^{∗} u_{i}. Thus we can choose m ∈ ℕ such that v = u \ [0, m] ⊆ u_{i} and so that (C, t \ [0, m]) ∈ 𝕄_{ℝ^{∗}}. As (C, v) ∈ D we have chosen some u_{i+1} ⊆ u_{i} such that (C, u_{i+1}) ∈ D. It follows that (C, t \ [0, m]) ∈ D.

Previous lemma says that 𝕄_{ℝ^{∗}} has the pure decision property; that is to say: whenever φ is a sentence of the forcing language for 𝕄_{ℝ^{∗}} and (A, s) ∈ 𝕄_{ℝ^{∗}}, there is a regular open set t ⊆ s such that, if (C, u) ≤ (A, t) and (C, u) decides φ, then there is m ∈ ℕ such that (C, t \ [0, m]) also decides φ. Another consequence of Lemma 2.6 is the following.

Lemma 2.7. Let (A, s) ⊩ ẋ ∈ ℜ^{∗} be given. Then there is (A, t) ≤ (A, s) so that if (B, u) ≤ (A, t) and (B, u) decides that an open interval with rational endpoints b ⊆ ẋ, then there exists m ∈ ℕ so that (B, t \ [0, m]) decides b ⊆ ẋ.

We may now suppose that our original (A_{0}, s_{0}) has the property of the (A, t) in the conclusion of Lemma 2.7. We then make the following definition. Denote by I the family of all intervals with rational endpoints, and for all (A, s) ≤ (A_{0}, s_{0}), let

F_{A} = {b ∈ I : (∃m ∈ ℕ) ((A, s_{0} \ [0, m]) ⊩ b ⊆ ẋ)}
= {b ∈ I : (∃ (A, s) ≤ (A_{0}, s_{0})) ((A, s) ⊩ b ⊆ ẋ)} .

Our previous lemma implies that F_{A} only depends on A, it does not change as s_{0} becomes smaller.

Lemma 2.8. There is (A, s) ≤ (A_{0}, s_{0}) such that ⋃ F_{B} is bounded for every (B, t) ≤ (A, s) .

Proof. Otherwise we would have that for every (A, s) ≤ (A_{0}, s_{0}) there is some (B, t) ≤ (A, s) such that ⋃ F_{B} is unbounded. There are only countably many F_{B}’s so we may find x_{α} such that ⋃ F_{B} ⊈^{∗} x_{α} for all possible B for which ⋃ F_{B} is unbounded. Now choose (A, s) ≤ (A_{0}, s_{0}) such that (A, s) ⊩ “ẋ\[0, n] ⊆ x_{α} for some n ∈ ℕ. Also choose (B, t) ≤ (A, s) so as ⋃ F_{B} is unbounded, and select an interval b ∈ I with

b ⊆ ⋃ F_{B} \ ([0, n] ∪ x_{α}) .

Then (A, s \ [0, m]) ⊩ “b ⊆ ẋ \ [0, n] and b ⊈ x_{α}”, a contradiction.

A TREE π-BASE FOR ℝ^{∗} WITHOUT COFINAL BRANCHES. 7

Simplify again by assuming that our original (A_{0}, s_{0}) has the property of (A, s) in the statement of Lemma 2.8. Moreover, we have assumed that (A_{0}, s_{0}) forces that ẋ ∩ [0, n] is a finite union of separate open intervals. To simplify even more, without loss of generality, let us assume further that (A_{0}, s_{0}) forces that ẋ ∩ [n, n + 1) is at most one interval for every n ∈ ℕ. Then, under these assumptions, we have that ⋃F_{B} is a finite union of separate open intervals, for every (B, t) ≤ (A_{0}, s_{0}). Indeed, let m ∈ ℕ be the least natural such that ⋃F_{B} ⊆ [0, m] and fix 1 ≤ k ≤ m. Put q_{0}^{k} = inf ⋃F_{B} ∩ [k − 1, k] and q_{1}^{k} = ⋃F_{B} ∩ [k − 1, k]. We claim that the interval (q_{0}^{k}, q_{1}^{k}) is equal to ⋃F_{B} ∩ [k − 1, k]; in other words that (q_{0}^{k}, q_{1}^{k}) ∈ F_{B}. If not, there is no unbounded regular open set s ⊆ s_{0} such that (B, s) ⊩ (q_{0}^{k}, q_{1}^{k}) ⊆ ẋ. Thus given s ⊆ s_{0}, there is an extension (C, t) of (B, s) and some open interval b′ ⊆ (q_{0}^{k}, q_{1}^{k}) such that (C, t) ⊩ b′ ∩ ẋ = ∅.

On the other hand, by the choice of q_{0}^{k} and q_{1}^{k} there are intervals b_{0}, b_{1} ∈ F_{B} such that q_{0}^{k} is less than every member of b_{0}, every member of b_{0} is less than every member of b′, every member of b′ is less than every member of b_{1} and every member of b_{1} is less than q_{1}^{k}. For b_{0} and b_{1}, there are n_{0}, n_{1} ∈ ℕ such that (B, s_{0} \ [0, n_{0}]) ⊩ b_{0} ⊆ ẋ and (B, s_{0} \ [0, n_{1}]) ⊩ b_{1} ⊆ ẋ. Letting n = max {n_{0}, n_{1}} we get that (C, t \ [0, n]) forces that ẋ ∩ [k − 1, k] is the union of at least two disjoint intervals, contradicting the fact that (A_{0}, s_{0}) forces that ẋ ∩ [k − 1, k] is at most one interval.

Lemma 2.9. Suppose (A_{0}, s_{0}) is as assumed. Then there is (A_{0}, t) ≤ (A_{0}, s) such that for all (B, u) ≤ (A_{0}, t), there is some m ∈ ℕ such that (B, t \ [0, m]) ⊩ ⋃ F_{B} ⊆ ẋ. Thus,

F_{B} = {b ∈ I : (∃m ∈ ℕ) ((B, t \ [0, m]) ⊩ b ⊆ ẋ)} .

Proof. We will construct a sequence of open intervals with rational endpoints a_{0}, a_{1}, …, a_{n}, … and a ⊆^{∗}-decreasing sequence of unbounded regular open sets s_{n} in such a way that if t is the union of the a_{i}’s then t satisfies the conclusion of the lemma.

Suppose we have succeed constructing a_{0}, a_{1}, …, a_{n−1} and an unbounded regular open set s_{n} such that for every (B, u) with A_{0} ⊆ B ⊆ A_{0} ∪_{⋃i<n}a_{i} and u ⊆^{∗} s_{n} we have that (B, s_{n} \ [0, m]) ⊩ ⋃F_{B} ⊆ ẋ, for some m ∈ ℕ. Let a_{n} be an open interval with rational endpoints contained in ⋂i≤ns_{i} \ [0, sup a_{n−1} + 1]. Now let {B_{i}}_{i∈ω} be an enumeration of all possible finite unions of open intervals with rational endpoints B such that A_{0} ⊆ B ⊆ A_{0} ∪_{⋃i≤n}a_{i}. We are going to construct s_{n}^{B_{i}} ⊆ s_{n} \ [0, sup a_{n} + 1] such that (B_{i}, s_{n}^{B_{i}}) ⊩ ⋃F_{B_{i}} ⊆ ẋ.

Suppose we have s_{n}^{B_{i}} and we want to get s_{n}^{B_{i+1}}. Firstly, we know that ⋃F_{B_{i+1}} is bounded; let m_{B_{i+1}} ∈ ℕ such that ⋃F_{B_{i+1}} ⊆ [0, m_{B_{i+1}}]. By the remarks before the lemma, there is an

8 FERNANDO HERNÁNDEZ-HERNÁNDEZ

extension (B_{i+1}, v) of (B_{i+1}, s_{n}^{B_{i}}) such that (B_{i+1}, v) ⊩ ẋ∩[0, m_{B_{i+1}}] = d_{1} ∪⋯∪d_{m_{B_{i+1}}} , for some open intervals with rational endpoints. Since d_{j} ∈ F_{B_{i+1}}, for each j ≤ m_{B_{i+1}}, there is some n_{j} ∈ ℕ such that (B_{i+1}, s_{n}^{B_{i}} \ [0, n_{i}]) ⊩ d_{j} ⊆ ẋ. Then let n_{B_{i+1}} = sup a_{n} + max{n_{j} : 1 ≤ j ≤ m_{B_{i+1}}} + 1. Then

(B_{i+1}, s_{n}^{Bi} \ [0, n_{B_{i+1}}]) ⊩ ⋃F_{B_{i+1}} ⊆ ẋ.

Letting s_{n}^{B_{i+1}} = v ∩ s_{n}^{Bi} \ [0, n_{B_{i+1}}] we finish with the inductive step in the construction of the s_{n}^{B_{i}}’s. Now let s_{n+1} ⊆^{∗} s_{n}^{B_{i}}, for every i ∈ ω. This completes the inductive construction of the s_{n}’s.

Finally let t = _{⋃n∈ω}a_{n}. Then t is clearly an unbounded regular open set. To verify that t satisfies the conclusion of the lemma, let (B, u) ≤ (A_{0}, t) and let n be the least natural number such that A_{0} ⊆ B ⊆ A ∪_{⋃i≤n}a_{i}. Then considering the enumeration of all possible B’s at stage n of our construction we have that B = B_{i} for some i ∈ ω. Then (B_{i}, s_{n+1}) ⊩ ⋃F_{B_{i}} ⊆ ẋ and since t ⊆^{∗} s_{n+1} there is some m ∈ ℕ such that (B_{i}, t \ [0, m]) ⊩ F_{B} ⊆ ẋ.

Now we will be able to finish the proof of Theorem 2.5. Assume (A_{0}, s_{0}) satisfies the conclusion of Lemmas 2.8 and 2.9, so that if (B, s) ≤ (A_{0}, s_{0}) then ⋃ F_{B} is bounded and (B, s) ⊩ ⋃ F_{B} ⊆ ẋ.

First build a condition (A_{0}, s) ≤ (A_{0}, s_{0}) such that, for all (B, t) ≤ (A_{0}, s), one of the following two cases holds:

(i) (\overline{B} ∩ \overline{a} = ∅ ⇒ F_{B∪a} = F_{B}) , for all intervals with rational endpoints a ⊆ s, or (ii) there exists α < κ such that for all j ∈ ℕ there is some m ∈ ℕ such that ⋃ F_{B∪a} \

(x_{α} ∪ [0, j]) ≠ ∅, for all intervals with rational endpoints a ⊆ s \ [0, m] . For this construction use again recursion as before. Suppose that, at stage n, we have defined

a_{0}, …, a_{n−1} and s_{n} ∈ ℜ^{∗} such that either (i) or (ii) holds for every (B, t) ≤ (A_{0}, s_{n}) with B \A_{0} ⊆ ⋃ {a_{0}, …, a_{n−1}}. Take one interval from s_{n} and call it a_{n}, and consider sequentially all possible B with B \ A_{0} ⊆ ⋃ {a_{0}, …, a_{n−1}, a_{n}}. Let {B_{i}}_{i∈ω} be an enumeration of all the possible B’s. We will construct a regular open sets s_{n}^{B_{i}} ⊇ s_{n}^{B_{i+1}} for every i ∈ ω. Suppose we have taken care of B_{i}. For B_{i+1}, one of the following holds:

Case 1. ⋃ {⋃ F_{B_{i+1}∪a} : a ⊆ s_{n}^{B_{i}}} is bounded. In this case we select an unbounded s_{n}^{B_{i+1}} ⊆ s_{n}^{B_{i}} such that

⋃ {F_{B_{i+1}∪a} : a ⊆ s_{n}^{B_{i+1}}} = F_{B_{i+1}},

to make (i) hold for B_{i+1}. First, it follows from 2.9 that F_{B_{i+1}} ⊆ F_{B_{i+1}∪a} for a ⊆ s_{n}^{B_{i}}. Now suppose ⋃ ⋃ {F_{B_{i+1}∪a} : a ⊆ s_{n}^{B_{i}}} = d_{0} ∪ ⋯ ∪ d_{k}. For each j ≤ k we have that either there exists s′ ⊆ s_{n}^{B_{i}} such that d_{j} ∈ F_{B_{i+1}∪a} if and only if a ⊆ s_{n}^{B_{i}} \ s′ and s_{n}^{B_{i}} \ s′ is bounded in which case

A TREE π-BASE FOR ℝ^{∗} WITHOUT COFINAL BRANCHES. 9

just replace s_{n}^{B_{i}} by s′, or else v = ⋃ {a ⊆ s_{n}^{B_{i}} : b ∈ F_{B_{i+1}∪a}} is unbounded. In this latter case

(B_{i+1} ∪ a, v \ [0, sup a]) ⊩ d_{j} ⊆ ẋ

for every a ⊆ v, and so (B_{i+1}, v) ⊩ d_{j} ⊆ ẋ; that is, d_{j} ∈ F_{B_{i+1}}. Thus we replace s_{n}^{B_{i}} by v to take care of d_{j}. In summary, after k + 1 replacements we can get s_{n}^{B_{i+1}} with the desired property.

Case 2. ⋃ {⋃ F_{B_{i+1}∪a} : a ⊆ s_{n}^{B_{i}}} is unbounded. Arrange for (ii) to hold. Choose α < κ so as that set is not almost contained in x_{α}. Now choose s_{n}^{B_{i+1}} ⊆ s_{n}^{B_{i}} so that for every j ∈ ℕ, if a ⊆ s_{n}^{B_{i+1}} then there is an interval disjoint from [0, j] which is contained in ⋃ F_{B_{i+1}∪a} \ x_{α}. Note that if (B, t) ≤ (B, s_{n}^{B_{i+1}}) and j ∈ ℕ, there is an extension of (B, t) which forces some interval b disjoint from [0, j] into ẋ \ x_{α}.

Let s_{n+1} be almost contained in these s_{n}^{B_{i}}, with i ∈ ω, and then let s = ⋃_{n∈ω} a_{n}. This completes the construction. It can be checked that either (i) or (ii) holds.

Now choose α < κ larger than any required for (ii). Fix (A, t) ≤ (A_{0}, s) and j ∈ ℕ such that (A, t) ⊩ ẋ \ [0, j] ⊆ x_{α}. We again have two cases.

Case (a). For all (B, u) ≤ (A, t), (i) holds. Then F_{B∪b} = F_{B} for all b ⊆ t. By induction on the length of the union forming B, we have F_{B} = F_{A} for all (B, u) ≤ (A, t). Hence (A, t) ⊩ ẋ ⊆ ⋃ F_{A} contradicting the assumption that ẋ was forced to be unbounded.

Case (b). There is (B, u) ≤ (A, t) so that (ii) holds. Then we can find b ⊆ t so that there is some b ∈ F_{B∪b} which is disjoint from x_{α} ∪ [0, j] and hence (B, t) has an extension which forces b ⊆ ẋ \ (x_{α} ∪ [0, j]) contradicting the choice of α, j and (A, t).

3. Countable support preservation

Now we prove that a countable support iteration of length ω_{2} of 𝕄_{ℝ^{∗}} does not fill towers. When Baumgartner and Dordal show this for Mathias forcing they modified the support since they did not have at hand the very useful Lemma 1.13 of Shelah [She84]. Shelah’s lemma is very general and applies to multiple situations.

Definition 3.1. Let R be a binary relation on ω^{ω}. Then F ⊆ ω^{ω} will be said to be R-bounding if and only if for every g ∈ ω^{ω} there is f ∈ F such that g R f.

Typical examples of R and F are: (1) f R g holds if and only if

|{n ∈ ω : f(n) ≥ g(n)}| < ℵ_{0}.

10 FERNANDO HERNÁNDEZ-HERNÁNDEZ

In this case a family F ⊆ ω^{ω} is R-bounding if and only if it is dominating. (2) f R g holds if and only if |{n ∈ ω : f(n) < g(n)}| < ℵ_{0}. In this case a family F ⊆ ω^{ω} is R-bounding if and only if it is unbounded.

Definition 3.2. Let R be a binary relation on ω^{ω}, F ⊆ ω^{ω}, M ⊆ ω^{ω}, and f ∈ F. Then G(M, R, f) is the game of length ω with the following rules:

(1) at stage n of the game Player I chooses g_{n} ∈ M such that g_{n} ↾ k_{n−1} = g_{n−1} ↾ k_{n−1} (where k_{−1} = 0).

(2) at stage n of the game Player II then chooses k_{n} ∈ ω.

Player II is declared to be the winner of the game if

⋃ {g_{n} ↾ k_{n} : n ∈ ω} R f.

The idea is best understood where R is the usual ≮^{∗} relation and the g_{n}’s are thought of as approximations to a generic g. Player II is demanding a decision from the forcing out to k_{n+1} where this is chosen large enough so that f is bigger at least one more time. There are difficulties in general with the meaning of R and of Player II’s strategy when passing from one model of set theory to a larger one.

Definition 3.3. If R is a binary relation on ω^{ω}, let R(·, f) = {g : g R f}. The pair (F, R) will be said to be nice if and only if

(1) F is R-bounding.

(2) For every M ∈ [F]^{ℵ_{0}} there is some f ∈ F such that Player II has a winning strategy in the game G(⋃ {R(·, g) : g ∈ M} , R, f).

(3) Player II’s strategy remains a winning strategy in any extension of the universe in which F remains R-bounding.

The following lemma of Shelah (see Lemma 1.13 in [She84, p. 189] for the original proof or Lemma 3.13 of [She98, p. 313]) shows that if (F, R) is nice then in order to show that a countable support iteration of proper forcing notions preserves that F is R-bounding, one need only be concerned with the successor stages of the construction. The case of limit stages of uncountable cofinality are easily handed by Claim III 4.1B(2) of [She98, p. 120].

Lemma 3.4. Let ℙ be the countable support iteration of proper forcing notions ℙ_{n} and suppose that (F, R) is nice and that ℙ_{n} ⊩ “F is R-bounding” for each n ∈ ω. Then ℙ ⊩ “F is R-bounding”..

A TREE π-BASE FOR ℝ^{∗} WITHOUT COFINAL BRANCHES. 11

Now we are going to apply this lemma for our purposes. For each n ∈ ℕ, let

U_{n} = {(p, q) ⊆ ℝ : p, q ∈ ℚ, n < p < q} ∪ {∅} .

We agree that {I_{m}^{n} : m ∈ ω} enumerates U_{n}. ∏_{n∈ℕ} U_{n} generates a subalgebra A of ℜ^{∗}. Basically A is build up from countable unions of finite unions of separate intervals from each U_{n}. To each f ∈ ω^{≤ω} we can assign an element O(f) of A given by

O(f) = int \overline{⋃ \{I_{f(n)}^{n} : n ∈ dom f\}}.

Then define R ⊆ ω^{ω} × ω^{ω} by f R g if and only if O(f) ⊈^{∗} O(g). Any family F ⊆ ω^{ω} which generates a tower in A (or ℜ^{∗}) is R-bounding: if g ∈ ω^{ω}, there must be f ∈ F such that O(g) ⊈^{∗} O(f). Moreover, the pair (F, R) is nice. Given M ∈ [F]^{ℵ_{0}}, we can find f ∈ F so that O(f) ⊆^{∗} O(g) for all g ∈ M. Now, a winning strategy for Player II in the game

G(⋃ {R(·, g) : g ∈ M} , R, f)

is as follows: At stage n, Player I chooses h_{n} ∈ ω^{ω} so that h_{n} ↾ k_{n−1} = h_{n−1} ↾ k_{n−1} and h_{n} R g_{n} for some g_{n} ∈ M; then Player II only needs to choose k_{n} ∈ ω large enough so that

O(h_{n} ↾ k_{n}) \ O(f) ⊈ [0, n] .

That Player II’s strategy remains a winning strategy in any extension of the universe in which F remains R-bounding is clear.

On the other hand, it is not difficult to convince that any tower in A can be represented by a family of functions. We already know that our forcing preserves towers from A. Thus, we have the following conclusion.

Lemma 3.5. If ℙ_{δ} is a countable support iteration (of length δ) of 𝕄_{ℝ^{∗}}, then ℙ_{δ} preserves towers; that is, if 𝕄_{ℝ^{∗}} ⊩ “F is R-bounding” and (F, R) is nice, then ℙ_{δ} ⊩ “F is R-bounding”.

Theorem 3.6. V^{ℙ_{ω_{2}}} ⊨ “there is no ω_{2}-towers in ℜ^{∗} = RO(ℝ)/\overline{cpt}”.

Proof. Suppose p ⊩ “ ⟨x_{ξ} : ξ < ω_{2}⟩ is a tower” for some p ∈ ℙ_{ω_{2}}. We may find an f : ω_{2} → ω_{2} so that no element in ℜ^{∗} ∩ V^{ℙ_{β}} lies below the sequence ⟨x_{ξ} : ξ < f(β)⟩, and so that the sequence ⟨x_{ξ} : ξ < β⟩ always lies in V^{ℙ_{f(β)}}. Choose a limit α < ω_{2} of cofinality ω_{1} so that f(β) < α whenever β < α. Then p ⊩ “ ⟨x_{ξ} : ξ < α⟩ is a tower in V^{ℙ_{α}}”. We may now show by induction on β ≥ α using our preservation theorems for single stages and iterations that ⟨x_{ξ} : ξ < α⟩ must remain a tower in V^{ℙ_{β}} for all β ≤ ω_{2}. In particular ⟨x_{ξ} : ξ < α⟩ remains a tower in ℙ_{ω_{2}} in contradiction to the fact that x_{α} must lie below all these x_{ξ}.

Corollary 3.7. There is a tree π-base for ℝ^{∗} which has no branch of length ω_{2}, yet it has height ω_{2}.

4. Other properties of 𝕄_{ℝ^{∗}}

Our investigation of 𝕄_{ℝ^{∗}} was motivated by the question of whether it is possible to have 𝔥(ℝ^{∗} × ℝ^{∗}) < 𝔥(ℝ^{∗}). By the result in the last section 𝕄_{ℝ^{∗}} seemed hopeful to use it with the methods of Shelah and Spinas [SS00]. Some time later we read their second part [SS98] where they claim that the important properties of Mathias forcing which are essential to their proof are:

(1) Mathias forcing factors into a ℵ_{1}-closed and a σ-centred forcing.

(2) Mathias forcing is Souslin-proper.

(3) Every infinite subset of a Mathias real is also a Mathias real.

(4) Mathias forcing does not change the cofinality of any cardinal from above 𝔥 to below 𝔥.

(5) Mathias forcing has the pure decision property and it has the Laver property.

In this section we show that not all of those properties are shared by 𝕄_{ℝ^{∗}}. The first property is Lemma 2.2 and the pure decision property is shown in Lemma 2.6.

Lemma 4.1. 𝕄_{ℝ^{∗}} does not change the cofinality of any cardinal above 𝔥(ℝ^{∗}) to below 𝔥(ℝ^{∗}) over a model of GCH.

Proof. Let λ be a cardinal with cf(λ) ≥ 𝔥(ℝ^{∗}) and let κ < 𝔥(ℝ^{∗}) be a cardinal, and let us consider a 𝕄_{ℝ^{∗}}-name ḟ for a function from κ to λ. Working in V and using properness, for all α < κ we can construct a maximal antichain {p_{β}^{α} = (A_{β}^{α}, s_{β}^{α}) : β < 𝔠} in 𝕄_{ℝ^{∗}} and {X_{β}^{α} : β < 𝔠} such that for all β < 𝔠, A_{β}^{α} = ∅, X_{β}^{α} ∈ [V]^{ℵ_{0}} ∩ V and p_{β}^{α} ⊩ ḟ(α) ∈ X_{β}^{α}. Then clearly S_{α} = {s_{β}^{α} : β < 𝔠} is a maximal antichain in ℜ^{∗}. By κ < 𝔥(ℝ^{∗}), {S_{α}}_{α∈κ} has a refinement, say S. Choose s ∈ S. For α < κ there is β(α) such that s ⊆^{∗} s_{β(α)}^{α}. Then clearly (∅, s) ⊩ ran(ḟ) ⊆ ⋃ {X_{β(α)}^{α} : α < κ}. Thus f cannot be cofinal in λ.

Definition 4.2. A forcing ℙ is said to have the Laver property if for every ℙ-name ḟ for a member of ω^{ω}, g ∈ ω^{ω} ∩ V and p ∈ ℙ, if

p ⊩ (∀n ∈ ω) (ḟ(n) < g(n)) ,

then there exists H : ω → [ω]^{<ℵ_{0}} and q ∈ ℙ such that H ∈ V, (∀n ∈ ω) (|H(n)| ≤ 2^{n}) ,

A TREE π-BASE FOR ℝ^{∗} WITHOUT COFINAL BRANCHES. 13

q ≤ p and

q ⊩ (∀n ∈ ω) (ḟ(n) ∈ H(n)) .

M. Goldstern [Gol93, 6.33] showed that the Laver property is preserved by countable support proper iterations. It is also known that if a forcing has the Laver property then it does not add any Cohen reals (see [BJ95]). In [BH04], B. Balcar and M. Hrušák show that 𝔥 (ℝ^{∗}) ≤ min {𝔥, add (M)}. Since forcing with 𝕄_{ℝ^{∗}} we have 𝔥 (ℝ^{∗}) = ℵ_{2} then our forcing 𝕄_{ℝ^{∗}} must add Cohen reals and therefore it does not have the Laver property.

The most used property of Mathias forcing in [SS00] is that of “every infinite subset of a Mathias real is also a Mathias real”; that translated to 𝕄_{ℝ^{∗}} would be: if u ∈ ℜ^{∗} is 𝕄_{ℝ^{∗}}-generic and v ∈ ℜ^{∗} is such that v ⊂ u, then v is 𝕄_{ℝ^{∗}}-generic as well. But in the case of ℝ there is too much room to choose subsets. For example, if is easy to see that the set of conditions (A, s) ∈ 𝕄_{ℝ^{∗}} where the last interval forming A has dyadic endpoints is dense in 𝕄_{ℝ^{∗}}, and therefore any 𝕄_{ℝ^{∗}}-generic subset of ℝ must contain intervals with dyadic endpoints. Of course, it trivially contain regular open subsets that are unions of intervals of exclusively irrational endpoints.

We conclude the section showing that the forcing 𝕄_{ℝ^{∗}} is not only proper but it also satisfies a more restrictive axiom which usually demands more structure over the partial order.

Definition 4.3. ℙ satisfies Axiom A if there is a sequence of orderings {≤_{n}}_{n∈ω} such that for all p, q ∈ ℙ :

1) p ≤_{0} q iff p ≤ q

2) p ≤_{n+1} q ⇒ p ≤_{n} q

3) if {p_{n} : n ∈ ω} is such that p_{n+1} ≤ p_{n} for all n ∈ ω, then

(∃q ∈ ℙ) (∀n ∈ ω) (q ≤_{n} p_{n})

4) if I is pairwise incompatible subset of ℙ, then

(∀p ∈ ℙ) (∀n ∈ ω) (∃q ∈ ℙ) (q ≤_{n} p)

and {r ∈ I : ¬ (q ⊥ r)} is countable.

It is not hard to see that 4) is equivalent to : 4’) (∀p ∈ ℙ) (∀n ∈ ω) (p ⊩ ȧ ∈ V ⇒ (∃x ∈ [V]^{ℵ_{0}}) (∃q ≤_{n} p) (q ⊩ ȧ ∈ x))

Proposition 4.4. The modification of Mathias forcing satisfies Axiom A.

14 FERNANDO HERNÁNDEZ-HERNÁNDEZ

Proof. For n ≥ 1, let (A, s) ≤_{n} (B, t) if and only if (A, s) ≤ (B, t) , A = B and (−n, n) ∩ s = (−n, n) ∩ t. Of course, (A, s) ≤_{0} (B, t) if and only if (A, s) ≤ (B, t). If {(A_{n}, s_{n}) : n ∈ ω} is a sequence such that (A_{n+1}, s_{n+1}) ≤_{n} (A_{n}, s_{n}) for all n ∈ ω, then setting

s = ⋂_{n∈ω} s_{n} = int(\overline{⋂_{n∈ω} s_{n}}),

we can see that s ∈ ℜ^{∗} and (A_{1}, s) ≤_{n} (A_{n}, s_{n}) for all n ∈ ω. s ∈ ℜ^{∗} because s is open, so s = int(s) ⊆ int (\overline{A}). If x ∈ int (\overline{s}), then x ∈ (−n, n) ∩ int (\overline{s}), for some n ∈ ω. Thus we can find an open set u ⊆ (−n, n) ∩ int (\overline{s}) such that x ∈ u. For any y ∈ u we have y ∈ \overline{s} ⊆ \overline{s}_{k}, for any k ∈ ω. In particular, for 0 ≤ k ≤ n, u ⊆ \overline{s}_{k}, which implies x ∈ int (\overline{s}_{k}) = s_{k}. So x ∈ ⋂_{k=0}^{n} s_{k}. And for k > n, since s_{k} ∩ (−n, n) = s_{n} ∩ (−n, n), x ∈ s_{k}. Thus x ∈ ⋂_{k∈ω} s_{k} = s.

On the other hand, we know that if ϕ is a sentence of the forcing language for 𝕄_{n} and (A, s) ∈ 𝕄_{ℝ^{∗}}, then there is a regular open set t ⊆ s so that, if (B, u) ≤ (A, t) and (B, u) decides ϕ, then there are rational numbers q_{0}, q_{1} such that (B, t \ [q_{0}, q_{1}]) decides ϕ too. Also, if p ∈ 𝕄_{ℝ^{∗}} is such that p ⊩ ȧ ∈ V we can find q ≤ p and a countable set x in V such that q ⊩ ȧ ∈ x —𝕄_{ℝ^{∗}} is proper. Thus, we can show that if p ∈ 𝕄_{ℝ^{∗}}, n ∈ ω and p ⊩ ȧ ∈ V, then there are x ∈ [V]^{ℵ_{0}} and q ≤_{n} p such that q ⊩ ȧ ∈ x.

Acknowledgements: The research contained in this paper was initiated when the author was a student at York University under the advice of Professor Paul J. Szeptycki. The research was suggested to the author by Professor Alan Dow when he was at York University. The author wish to thank them for helpful suggestions and hours of stimulating conversations.

References

[BH04] Bohuslav Balcar and Michael Hrušák. Distributivity of the algebra of regular open subsets of βℝ \ ℝ. To appear in Top. Appl., 2004.

[BJ95] Tomek Bartoszyński and Haim Judah. Set theory. A K Peters Ltd., Wellesley, MA, 1995. On the structure of the real line.

[BPS80] Bohuslav Balcar, Jan Pelant, and Petr Simon. The space of ultrafilters on N covered by nowhere dense sets. Fund. Math., 110(1):11–24, 1980.

[Dor87] Peter Lars Dordal. A model in which the base-matrix tree cannot have cofinal branches. J. Symbolic Logic, 52(3):651–664, 1987.

[Dow89] Alan Dow. Tree π-bases for βN − N in various models. Topology Appl., 33(1):3–19, 1989.

[Dow98] Alan Dow. The regular open algebra of βR \ R is not equal to the completion of P(ω)/fin. Fund. Math., 157(1):33–41, 1998.

[Gol93] Martin Goldstern. Tools for your forcing construction. In Set theory of the reals (Ramat Gan, 1991), volume 6 of Israel Math. Conf. Proc., pages 305–360. Bar-Ilan Univ., Ramat Gan, 1993.

[She84] Saharon Shelah. On cardinal invariants of the continuum. In Axiomatic set theory (Boulder, Colo., 1983), volume 31 of Contemp. Math., pages 183–207. Amer. Math. Soc., Providence, RI, 1984.

A TREE π-BASE FOR ℝ^{∗} WITHOUT COFINAL BRANCHES. 15

[She98] Saharon Shelah. Proper and improper forcing. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, second edition, 1998.

[SS98] Saharon Shelah and Otmar Spinas. The distributivity numbers of finite products of P(ω)/fin. Fund. Math., 158(1):81–93, 1998.

[SS00] Saharon Shelah and Otmar Spinas. The distributivity numbers of P(ω)/fin and its square. Trans. Amer. Math. Soc., 352(5):2023–2047 (electronic), 2000.

[vMW83] Jan van Mill and Scott W. Williams. A compact F -space not co-absolute with βN − N. Topology Appl., 15(1):59–64, 1983.

[Wil82] Scott W. Williams. Trees, Gleason spaces, and coabsolutes of βN ∼ N. Trans. Amer. Math. Soc., 271(1):83–100, 1982.

Instituto de Matemáticas, UNAM (Morelia), Apartado postal 61-3 (Xangari), Morelia Michoacán, México, 58089.

   E-mail address: fernando@matmor.unam.mx