THE REGULAR OPEN ALGEBRA OF βℝ \ ℝ IS NOT EQUAL TO THE COMPLETION OF P(ω)/fin

ALAN DOW

Abstract. Two compact spaces are co-absolute if their respective regular open algebras are isomorphic (i.e. homeomorphic Gleason covers). We prove that it is consistent that βω \ ω and βℝ \ ℝ are not co-absolute.

1. Introduction

While it is rather apparent that βℕ \ ℕ and βℝ \ ℝ are quite distinct they also have many similarities. Comfort and Negrepontis showed that if the Continuum Hypothesis holds then they even have homeomorphic dense subsets. Following that result, the investigation of the property of two spaces being co-absolute was of some interest. The absolute of a regular space (the Gleason cover in the case of compact spaces) is the unique extremally disconnected space which maps onto the space by a perfect irreducible map. The Boolean algebra of clopen subsets of the absolute is isomorphic to the regular open algebra, r.o.(X), of a space X (the absolute is the subspace of ultrafilters on r.o.(X) which converge to a point of X). Since the regular open algebra of a space is isomorphic to the regular open algebra of any dense subspace it is clear how this investigation grew out of the Comfort-Negrepontis result. In the case of a zero-dimensional space, the clopen algebra of the absolute is just the completion of the algebra of clopen sets of the original space, thus the completion of P(ℕ)/fin is isomorphic to r.o.(βℕ \ ℕ).

The interest in these regular open algebras was heightened following the remarkable discovery of Balcar et al. ([1]) that many of them had a dense subset which formed a tree (by reverse inclusion). Topologically, a dense subset is called a π-base for the space. Thus two spaces are co-absolute if and only if they share isomorphic (under set-theoretic inclusion) π-bases. The question of whether the remainders of ℕ and of ℝ are co-absolute has been asked in several forms many times in the literature. Van Douwen observed in [12] that 𝔭 = 𝔠 implied they were co-absolute.

Our representation of r.o.(βℝ \ ℝ) will be as the completion of a quotient of r.o.(ℝ). It is a basic fact of Čech-Stone remainders that the relative interiors of the closures of open subsets of the original space form a base for the open sets in the remainder. From this it easily follows that r.o.(ℝ)/\overline{cpt} = r.o.(ℝ)/bounded is isomorphic to a dense subset of r.o.(βℝ\ℝ), where \overline{cpt} = bounded denotes the ideal of regular open sets which are bounded, or equivalently, have compact closure.

For convenience we will adopt, by analogy with the standard notation for subsets of ℕ, the convention that a ^{∗} adorning a word or symbol will mean that it should be interpreted modulo the ideal \overline{cpt} in r.o.(ℝ), e.g. W ⊂^{∗} O ⊂ ℝ will mean that W \ O is bounded and has compact closure. When we speak of a tree^{∗} in r.o.(ℝ) we shall really mean a tree in the quotient algebra (and similarly for dense^{∗} and similarly for P(ℕ)).

Our technique of differentiating between these two algebras has additional interest because we show there is a model in which r.o.(ℝ) has a dense^{∗} tree of height ω_{1} while P(ℕ) does not (it is known that the reverse is not possible). The model is the standard Mathias model and it is immediate that, in this model, P(ℕ) has no dense^{∗} tree of height ω_{1}. This fact is essentially due to Mathias, since if r ⊂ ℕ is a Mathias real over V, then {a ⊂ ℕ : r ⊂^{∗} a} is a (P(ℕ)/fin, ⊂^{∗})-generic filter (and thus contains a cofinal branch of each dense^{∗} tree).

There are already several interesting results concerning dense^{∗} subtrees of P(ℕ) shown to hold in the Mathias model. While there is no dense tree of height ω_{1}, Dordal showed [3] that there is a dense tree in which there are no branches of length ω_{2}. The key to Dordal’s result is the result that single stage Mathias forcing did not fill towers (due to Baumgartner). Dordal’s model was actually constructed using a modified support for Mathias forcing but was easily adapted once Shelah showed [9] that the “not filling towers” property is preserved by countable support proper iterations (see also [5]) .

Finally, the most compelling evidence comes from the recent result of Shelah and Spinas ([10]), that the square of ω^{∗} does have a tree π-base of height ω_{1}. This is perhaps only the author’s view but r.o.(ℝ) seems to have a two-dimensional aspect to it, i.e., for w ∈ r.o.(ℝ) we have {n : w ∩ (n, n + 1) ≠ ∅} and, e.g., {{q_{1}, q_{2}} ∈ ℚ^{2} : (∃n ∈ ℕ)(n + q_{1}, n + q_{2}) ∈ w ∩ (n, n + 1)} .

There is an interesting subtlety at work here. As mentioned above, Mathias forcing does fill a “new” tower but, from Baumgartner and Dordal’s result, it does not fill any old tower. Analyzing which new towers it fills (or does not fill) is the major component of Shelah and Spinas’ paper [10]. Working with r.o.(ℝ), our major contribution is to discover the relevance of remote towers (or filters); i.e., Mathias forcing does not fill such towers. We also must construct a dense^{∗} tree in r.o.(ℝ) so that each ω_{1}-branch generates a remote filter at a certain critical inner model. For this latter task we use what we feel is a quite interesting enumeration principle on ω_{2}.

There are two interesting questions we have not been able to resolve. Following [1] and [13], for a space X, let 𝔥(X) denote the minimum number of dense open subsets of X whose intersection has empty interior (this is just the distributivity degree of the forcing Boolean algebra r.o.(X)). We also let 𝔫(X) (for Novak number) denote the minimum number of dense open sets whose intersection is empty. It is shown in [1] that if r.o.(X) has a dense tree, then 𝔥(X) is the minimum height of such a tree. Shelah and Spinas were answering (affirmatively) the question of whether 𝔥(ℕ^{∗}) can be larger than 𝔥(ℕ^{∗} × ℕ^{∗}). We conjecture that 𝔥(ℝ^{∗}) ≤ 𝔥(ℕ^{∗} × ℕ^{∗}) and that 𝔥(ℝ^{∗}) = 𝔥(ℝ^{∗} × ℝ^{∗}) . Van Douwen has asked if 𝔫(ℝ^{∗}) = 𝔫(ℕ^{∗})

– we do not know. Finally, let us remark that Dordal’s result above implies that 𝔥(ℕ^{∗}) = 𝔫(ℕ^{∗}) = ω_{2} in the Mathias model. However it is shown in [1] that 𝔫(ℝ^{∗}) is always at least ω_{2}, hence, in the Mathias model, 𝔥(ℝ^{∗}) < 𝔫(ℝ^{∗}). However 𝔥(ℕ^{∗}) = 𝔫(ℕ^{∗}) was established in a different model in [4], and the corresponding result for ℝ^{∗} also holds there.

Definition 1.1. A filter or filter base F on a space X is said to be a remote filter if for each dense open U ⊂ X, there is an F ∈ F such that the closure of F is contained^{∗} in U. If V is a set (or even class), we will say that F is V -remote if for each dense open U ⊂ X which is in V, there is an F in F as above.

THE REGULAR OPEN ALGEBRA OF βℝ 3

2. Laver property and Remote filters

The Mathias poset, 𝕄, is very well known. It consists of ordered pairs (a, A) ∈ [ω]^{<ω} × [ω]^{ω} (where a ∩ A = ∅) and a condition (a, A) is below (or stronger) than (b, B) providing b ⊂ a, A ⊂ B, and a ⊂ b ∪ B. Because of the Ramsey theoretic properties of [ω]^{ω}, 𝕄 has the Prikry property 2.1 and satisfies an important fusion Lemma (e.g. [2]).

Proposition 2.1 (Prikry property). If τ is an 𝕄 name, F any finite set, and (a, A) is any condition, then there is a B ⊂ A such that either (a, B) ⊩ τ ∉ F, or there is an f ∈ F such that (a, B) ⊩ τ = f.

Let us recall that a forcing notion P has the Laver property if it is (f, g)–bounding for every increasing function f ∈ ^{ω}ω, where g(n) = 2^{n} (in fact, g may be replaced by any other fixed increasing function in ^{ω}ω). A proper forcing notion P is (f, g)– bounding (for some increasing f, g ∈ ^{ω}ω) if

⊩_{P} (∀x ∈ ∏_{i∈ω} f(i))(∃S ∈ V ∩ ∏_{i∈ω}[ω]<ω)(∀i ∈ ω)(|S(i)| ≤ g(i) & x(i) ∈ S(i)).

Lemma 2.2. [8] The countable support iteration of the Mathias poset has the Laver property.

We will not prove this lemma, it is quite well-known, but here are the key ingredients. The Prikry property combined with the notion of fusion can be used to easily prove that Mathias forcing has the Laver property. In addition, Goldstern [7, 6.33] has shown that the Laver property is preserved by countable support proper iterations. This can also can be proved directly for Mathias forcing by using the standard <_{u,n}-relation and fusion.

It is common to let (a, A) <_{0} (b, B) denote the relation that (a, A) < (b, B) and a = b. More generally, we let (a, A) <_{n} (b, B) denote the relation (a, A) <_{0} (b, B) and the first n elements of B are also members of A . Similarly, for a countable support iteration, P_{δ}, of 𝕄 and p, q ∈ P_{δ}, u ∈ [δ]^{<ω}, and n ∈ ω, one defines p <_{u,n} q to mean that, for each α ∈ u, p ↾ α ⊩ p(α) <_{n} q(α) .

Theorem 2.3. If a partial order P satisfies the Laver property then it does not diagonalize any V -remote filter.

First we prove a Lemma.

Lemma 2.4. Suppose P is a poset that satisfies the Laver property and that G is P -generic. Let W ∈ V [G] be an unbounded subset of ℝ. Then there is a dense open U ∈ V of ℝ such that W \ U is unbounded.

Proof of Lemma 2.4. By shrinking W we may assume that for each n, there is at most one point w_{n} ∈ W ∩ (n, n + 1) (and there is no loss to assuming that W ∩ ℕ is empty). For each n such that W ∩ (n, n + 1) is empty, choose any w_{n} ∈ (n, n + 1).

Now define x ∈ Π_{i}(Π_{n<i} {(n + j/2^{i}, n + (j + 1)/2^{i} ) : j < 2^{i}}) so that for n < i, w_{n} ∈ x(i)(n) .

Apply the Laver property to find a sequence S = {S(i) : i ∈ ω} ∈ V of sets, |S(i)| ≤ i, so that for each i, x(i) ∈ S(i). We can assume that for each y ∈ S(i), y is a member of Π_{n<i} {(n + j/2^{i}, n + (j + 1)/2^{i} ) : j < 2^{i}})

From S we can define a dense open set U ∈ V. Simply, U is the union of all intervals of the form (n + ℓ/2^{i}, n + (ℓ + 1)/2^{i}) (with n < i and ℓ < 2^{i}) such that it is disjoint from s(n) for each s ∈ S(i) . It is easy to see that the density of U follows from the small branching size of S, i.e., |S(i)| ≤ i and for each s ∈ S and each n, s(n) is an interval of length at most 1/2^{i}. To see this, let J = (j/2^{k}, (j + 1)/2^{k}) be arbitrary. Fix an integer i so large that (j + 1)/2^{k} < i and i < 2^{i−k}. Clearly, one of the 2^{i−k} dyadic sub-intervals of length 1/2^{i} is disjoint from the at most i such subintervals which are contained in ⋃{s(n) : s ∈ S(i)}.

Now we check that for each n, w_{n} ∉ U. Suppose that J = (n+ℓ/2^{i}, n+(ℓ+1)/2^{i}) (with n < i) is a subset of U. There is an s ∈ S(i) such that x(i) = s, hence w_{n} ∈ s(n). Since J is disjoint from s(n), it follows that w_{n} ∉ J.

Proof of Theorem 2.3. Suppose that P satisfies the Laver property and that G is a P -generic filter over V. In V [G], suppose that F ⊂ V ∩ r.o.(ℝ) is a V -remote filter. Finally assume that W ∈ r.o.(ℝ) is unbounded. Apply Lemma 2.4 to find a dense open subset U ∈ V of ℝ such that W \ U is unbounded. Since F is V -remote, there is an F ∈ F such that the closure, \overline{F}, of F is a subset^{∗} of U. Therefore W \ \overline{F} is unbounded.

3. Building the tree for ℝ

For each λ ≤ ω_{2}, let P_{λ} denote the usual countable support iteration of the Mathias poset. It is most convenient to regard P_{λ} as a subet of P_{μ} for λ ≤ μ. Therefore if G is a P_{ω_{2}}-generic filter we can just take G_{λ} = G ∩ P_{λ} and we have V [G_{λ}] sitting as a natural submodel of V [G]. We will need the following well-known consequence of properness (and the fact that each member of r.o.(ℝ) is determined by a countable set).

Proposition 3.1. For each λ < ω_{2} of uncountable cofinality, each member of r.o.(ℝ) ∩ V [G_{λ}] is a member of V [G_{α}] for some α < λ .

Our plan is to build, in V [G], a set T which is to be our dense^{∗} subtree of r.o.(ℝ). However it will be easier to build a tree which is simply shattering and refer to the following Lemma which is proven exactly as the proof for P(ℕ) in [1]. A subfamily T ⊂ r.o.(ℝ) is shattering if for each w ∈ r.o.(ℝ), there are disjoint t, t′ ∈ T, both of which meet^{∗} w.

Lemma 3.2. If there is a tree T ⊂ r.o.(ℝ) which is shattering and has height at most κ, then there is a dense^{∗} tree of height at most κ

The main task in constructing our tree T is to ensure that ω_{1}-branches are nowhere dense. The key to ensuring this is to ensure that if λ is minimal such that some ω_{1}-branch is a subset of V [G_{λ}], then the branch generates a V [G_{λ}]-remote filter. To do this we use a weak enumeration or ♢-like principle on ω_{2}.

Following [6], we will call the set S described next, a self-indexing, totally reflecting stationary set. We will show how it can be usefully regarded as a weak ♢-like principle on ω_{2}. The “self-indexing” refers to property (1) and the totally reflecting refers to property (2). It is easily established that S is a stationary subset of the familiar structure ([ω_{2}]^{ω}, ⊂).

Definition 3.3. A family S = {S_{α} : α ∈ ω_{2}, cf(α) = ω} is a self-indexing, totally reflecting stationary set if the following are satisfied:

1. S_{α} is a countable cofinal subset of α,

2. for each λ with uncountable cofinality, {S_{α} : α ∈ λ} is stationary in [λ]^{ω}.

Here is why we call this a weak diamond principle.

Proposition 3.4. For each λ ∈ ω_{2} with uncountable cofinality and for each ξ in λ, the set {d ∈ λ : ξ ∈ S_{d}} is stationary in λ.

Proof. Let {Y_{β} : β ∈ ω_{1}} be any continous increasing chain of countable sets whose union is λ and ensure that ξ ∈ Y_{0}. Notice that {Y_{β} : β ∈ C} is a cub in the structure ([λ]^{ω}, ⊂). Define, for β ∈ ω_{1}, g(β) = sup(Y_{β}). Clearly, g is continuous and so there is a cub C ⊂ ω_{1} such that g restricted to C is strictly increasing. Now let D be any cub in λ such that every member of D has cofinality ω. Notice that C(D) = {β ∈ C : g(β) ∈ D} is cub in ω_{1} and therefore, {Y_{β} : β ∈ C(D)} is also cub in [λ]^{ω}. Finally then, since {S_{α} : α ∈ λ} is stationary in [λ]^{ω}, there is a β ∈ C(D) and an α such that Y_{β} = S_{α}. So, we claim that α = g(β) completing the proof. Since β ∈ C(D), we know that g(β) has cofinality ω and that Y_{β} is cofinal in g(β). Therefore S_{α} is cofinal in g(β), which means g(β) = α since S_{α} is, by definition, cofinal in α.

The existence of such a stationary set is not a consequence of ZFC but it is an interesting and useful principle nonetheless. In the current situation we are using a countable support iteration hence there is an easy alternative by using the generic filter from μ to μ + ω to choose, under some suitable coding, a countable subset of μ which would do the job here as well. However this would not work with a finite support iteration. In any case, we may assume that there is such a set in V [G].

For the remainder we work in V [G].

Theorem 3.5. There is, in V [G], a shattering tree^{∗} for r.o.(ℝ) of height ω_{1} .

Proof. Let {x_{ξ} : ξ ∈ ω_{2}} be an enumeration (repetitions allowed) of r.o.(ℝ) such that r.o.(ℝ) ∩ V [G_{α}] is contained in {x_{ξ} : ξ < α + ω_{1}} for each α . This is easily done since r.o.(ℝ) ∩ V [G_{α}] has cardinality ℵ_{1} for each α < ω_{2}.

Fix a self-indexing totally reflecting stationary set, S = {S_{α} : α ∈ ω_{2}, cf(α) = ω}, and we construct our tree^{∗} T ⊂ r.o.(ℝ) . Although it is not strictly needed, for technical convenience in the definition of T , let S_{λ} = ∅ for each λ < ω_{2} of uncountable cofinality. For each t ∈ r.o.(ℝ), let β_{t} denote the minimum ordinal such that t ∈ V [G_{β_{t}}] .

It is remarkably simple. Let T be any maximal subset of r.o.(ℝ) with the following three properties:

1. ℝ is a member of T,

2. for t ∈ T, β_{t} is greater than β_{s} for all s ∈ T with t ⊂^{∗} s ,

3. for t ∈ T, let γ_{t} = sup{β_{s} : s ∈ T and t ⊂^{∗} s}, then for each ξ ∈ S_{γ}, the closure of t is contained in one of {x_{ξ}, ℝ \ x_{ξ}}.

The main point to the proof, then, is the following Lemma.

Lemma 3.6. Each ω_{1}-branch of T nowhere dense^{∗} .

Proof. Suppose that {t_{α} : α ∈ ω_{1}} is an ω_{1}-branch of T. For each α ∈ ω_{1}, let β_{α} = β_{t_{α}} and recall that this forms a strictly increasing sequence in ω_{2}. Let λ denote the supremum, and for each limit α, let γ_{α} denote the supremum of {β_{ζ} : ζ < α} . Observe that {γ_{α} : lim(α) and α ∈ ω_{1}} is a closed and unbounded subset of λ.

By Lemma 2.3, it suffices to show that {t_{α} : α ∈ ω_{1}} is a V [G_{λ}]-remote filter. So, let U ⊂ ℝ be any dense open set which is a member of V [G_{λ}]. Clearly for each interval (a, b) ⊂ ℝ, there are disjoint open subsets of (a, b) each of which has both a and b as limit points. Therefore, there is a regular open set x ∈ V [G_{λ}] such that x ⊂ U and the boundary of x contains the boundary of U . By Proposition 3.1 and the assumption on our enumeration, there is a ξ < λ such that x = x_{ξ} . By Lemma 3.4, there is a limit α < ω_{1} such that ξ ∈ S_{γ_{α}} . By the property of T, it follows that the closure of t_{α} is either contained in x_{ξ} or is contained in ℝ \ x_{ξ}. Clearly then the closure of t_{α} does not hit the boundary of x_{ξ} nor the boundary of U .

Clearly it follows from Lemma 3.6, that the height of T is at most ω_{1}. Now, to finish the proof of Theorem 3.5, we must show that T is shattering. Let w ∈ r.o.(ℝ) and assume that there is no pair of incomparable members of T which meet^{∗} w . Therefore the set C = {s ∈ T : s∩w ≠^{∗} ∅} is a chain. Suppose there is some s ∈ C, such that w′ = w \ s is unbounded. Let s be minimal; hence w′ ⊂^{∗} s′ for each s′ ^{∗}⊃ s and w′ is disjoint^{∗} from each t ∈ T . Therefore, if we replace w by w′ we can assume that the chain C is countable. On the other hand, if w \ s is bounded for each s ∈ C, then w diagonalizes C. By Lemma 3.6, it again follows that C is countable. Also, in both cases, we can assume that w ⊂^{∗} s for each s ∈ C .

Let γ be the supremum of {β_{s} : s ∈ C} and again use the countable completeness of r.o.(ℝ) to deduce that there is an unbounded w′ ⊂ w such that w′ refines {x_{ξ}, ℝ\ x_{ξ}} for each ξ ∈ S_{γ}. Finally, choose t ⊂ w′ such that β_{t} is greater than γ. It follows easily that T ∪ {t} satisfies all the properties required of T (except possibly the maximality), and this, of course, contradicts the maximality of T . Thus we conclude that T is shattering and this completes the proof of the theorem.

4. The existence of self-indexing stationary sets

Recall that a □ sequence is a sequence {C_{α} : α ∈ ω_{2}} such that for each limit α, C_{α} is a closed and unbounded subset of α and if α has cofinality ω, then C_{α} is countable. Furthermore if α is a limit point in C_{λ}, then C_{α} is equal to C_{λ} ∩ α. It follows that for λ with uncountable cofinality, C_{λ} has order type ω_{1}.

Todorcevic has constructed his well-known and important ρ-functions ([11]) from □ and as he informed the author, a self-indexing totally reflecting set can be constructed quite directly from such a ρ function.

Here is a simple direct construction from the square sequence. For each δ in ω_{2}, let <_{δ} denote a well-ordering of δ in order type at most ω_{1}. Define S_{μ} for μ < ω_{2} with countable cofinality inductively as follows. If the limits of C_{μ}, denoted C′_{μ}, are cofinal in μ, then set S_{μ} = ⋃{S_{δ} : δ ∈ C′_{μ}}. Otherwise, for each δ ∈ C′_{μ}, let ξ(δ) denote the <_{δ}’th minimal member which is not in S_{γ} for any γ ∈ C′_{μ}, and set

S_{μ} = C_{μ} ∪ ⋃{S_{δ} ∪ {ξ(δ)} : δ ∈ C′_{μ}} .

For λ with uncountable cofinality, it is easy to see that {S_{δ} : δ ∈ C′_{λ}} is a continuous increasing sequence. It is also easy to see that every member of λ is in the union. This means that this set is actually a cub in [λ]^{ω}.

Proposition 4.1. Martin’s Maximum implies that there is no self-indexing totally reflecting stationary set.

Proof. Assume that S_{μ} is a countable subset of μ for each μ ∈ S_{0}^{2} where S_{0}^{2} denotes the set of all countable cofinality limits in ω_{2}. For each γ ∈ ω_{2}, let T_{γ} = {μ : γ ∈ T_{γ}}. Notice that {T_{γ} : γ ∈ ω_{2}} is a point-countable family. There are at most countably many of the T_{γ} which contain the intersection of some cub with S_{0}^{2} since the intersection of ω_{1} many cub’s in ω_{2} is again a cub. Fix any γ such that T_{γ} does not contain such a cub, i.e. S_{0}^{2} \ T_{γ} is stationary. Martin’s Maximum implies that every stationary subset of S_{0}^{2} contains a closed copy of ω_{1}, hence there is some λ with uncountable cofinality such that S_{0}^{2} \ T_{γ} contains a cub C in λ. It follows then that γ is not in S_{μ} for all μ ∈ C which implies that {S_{δ} : δ ∈ λ} is not stationary in [λ]^{ω}.

It is not known if ZFC implies the weaker principle obtained by replacing “all λ with cofinality ω_{1}” by “a stationary set of λ with cofinality ω_{1}”. This principle is discussed in [6] and such a set is called a self-indexing reflecting stationary set.

References

1. B. Balcar, J. Pelant, and P. Simon, The space of ultrafilters on N covered by nowhere dense sets, Fundamenta Mathematica (1980), 11–24.

2. J. Baumgartner, Iterated forcing, Surveys in set theory (A. R. D. Mathias, ed.), Cambridge Univ. Press, 1983, pp. 1–59.

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

4. A. Dow, Tree Π-bases for βN − N in various models, Top. and its Appl. 33 (1989), 3–19.

5. , Set theory in topology, Recent Progress in General Topology (M. Hušek and J. van Mill, eds.), Elsevier Science Publishers B.V., 1992, pp. 169–197.

6. A. Dow and J. Zhou, On subspaces of pseudoradial spaces, Proc. AMS (to appear), 1997.

7. Martin Goldstern, Tools for your forcing construction:set theory of the reals (ramat gan, 1991), Israel Math. Conf. Proc., 6, Bar-Ilan Univ., Ramat Gan, 1993, pp. 305–360.

8. S. Shelah, Proper forcing, Springer Lecture Notes, 1982.

9. , On cardinal invariants of the continuum, Axiomatic Set Theory, Contemporary Mathematics, AMS, 1984.

10. Otmar Spinas and Saharon Shelah, The distributivity numbers of √(ω)/fin and its squares,

Number 494 of Shelah Archives.

11. S. Todorcevic, Partitioning pairs of countable ordinals, Acta Mathematica 159 (1987), 261– 294.

12. Eric K. van Douwen, Transfer of information about βn \ n via open remainder maps, Illinois J. Math. 34 (1990), 769–792.

13. Jan van Mill and Scott W. Williams, A compact F -space not co-absolute with βn−n, Topology Appl. 15 (1983), 59–64.