ON SUBSPACES OF PSEUDORADIAL SPACES

ALAN DOW AND JINYUAN ZHOU

Abstract. A topological space X is pseudoradial if each of its non closed subsets A has a sequence (not necessarily with countable length) convergent to outside of A. We prove the following results concerning pseudoradial spaces and the spaces ω ∪ {p}, where p is an ultrafilter on ω:

(i) CH implies that, for every ultrafilter p on ω, ω ∪ {p} is a subspace of some regular pseudoradial space.

(ii) There is a model in which, for each P-point p, ω ∪ {p} cannot be embedded in a regular pseudoradial space while there is a point q such that ω ∪ {q} is a subspace of a zero-dimensional Hausdorff pseudoradial space.

1. Introduction and Definitions

In [1] the authors asked if ω ∪ {p} is a subspace of a pseudoradial space for p ∈ βω \ ω, where ω ∪ {p} takes the subspace topology in βω, i.e, p is the only nonisolated point and a neighbourhood of p is A∪{p} for A ∈ p . It is proved in [9] that under the assumption 𝔭 = 𝔠, each space with countable tightness is a subspace of some pseudoradial space. It is essentially proved that, under the assumption 𝔭 = 𝔠 , ω ∪{p} is a subspace of a Hausdorff pseudoradial space for p ∈ βω \ω. However the pseudoradial space constructed in [9] is not regular. In a communication with the second author, J. Vaughan asked if it is possible to make the pseudoradial space in [9] regular. In [7] P. Nyikos asked whether each topological space can be embedded into a pseudoradial space (Problem 6.22 ). Our discussion will be focused on the class of regular spaces. We first prove, in section 2 , that, under CH, ω ∪ {p} is a subspace of a regular pseudoradial space for each ultrafilter p on ω . However, the question of whether CH can be replaced by MA remains open. In section 3, we prove that if the ground model satisfies CH, then in the forcing extension obtained by adding ℵ_{2} many Cohen reals, ω ∪ {p} , for any P-point p, is not a subspace of any regular pseudoradial space. In section 4, we show that, for a special ultrafilter p constructed in [5], if there exists a special stationary set and 𝔠 = ω_{2}, then ω ∪ {p} can be embedded into a zero-dimensional Hausdorff pseudoradial space. We also prove that it is consistent that 𝔠 = ω_{2} while there is such a stationary set. We conclude that there is a forcing extension in which, for every P-point p, ω ∪ {p} can not be embedded in any regular pseudoradial space while there is an ultrafilter q on ω such that ω ∪ {q} is a subspace of a zero-dimensional Hausdorff pseudoradial space.

1980 Mathematics Subject Classification (1985 Revision). Primary 54E35.

Key words and phrases. Forcing, CH, ultrafilter, zero-dimensional space, pseudoradial.

Definition 1.1. A sequence {x_{α} : α < λ} of points in X is said to be convergent to x provided that for each neighbourhood U of x, there exists α < λ such that {x_{β} : β > α} ⊆ U. (We will call any transfinite sequence a sequence).

A space X is radial if for each subset A of X and x in Ā, there is a sequence in A convergent to x. The pseudoradial spaces have weaker properties. This is analogous to the relationship between sequential spaces and Frechét spaces.

Definition 1.2. A subset A of a space of X is said to be radially closed if no sequence in A is convergent to outside of A. A space X is said to be pseudoradial provided that any radially closed subset of X is closed in X.

In this paper, a topological space is always regular. The set theory notions are standard. For example, if A and B are two sets, we write A =^{∗} B when A and B are equal modulo finite.

2. Under CH

If p is a P_{𝔠} point on ω , let {A_{α} : ω ≤ α < 𝔠} be a strictly decreasing modulo finite base for p and let B be the Boolean Algebra generated by {A_{α} : α ∈ 𝔠}∪[ω]^{<ω}, then it is easy to see that the Stone space of B is just ω ∪ [ω, 𝔠 + 1] where [ω, 𝔠 + 1] has the usual ordinal topology and ω ∪ {𝔠} is homeomorphic to ω ∪ {p}. An immediate corollary is that , under CH, if p is a P -point, then ω ∪ {p} can be embedded into a zero-dimensional Hausdorff compact pseudoradial space.

However, we have the following result about non P -points.

Lemma 2.1 (p=𝔠). If p is not a P -point, then ω ∪ {p} can be embedded in a zero-dimensional pseudo-radial space.

Proof: Let p = {A_{α} : α < 𝔠} such that {A_{i} : i < ω} witnesses that p is not a P -point. First of all, we will construct a base {B_{α} : α < 𝔠} of p and an almost disjoint family {C_{α} : α < 𝔠} such that, for ξ, α ∈ 𝔠, the following are true:

(i)_{α} B_{α} ⊆ A_{α},

(ii)_{ξα} ξ ≤ α → C_{α} ⊆^{∗} B_{ξ},
(iii)_{ξα} α < ξ → C_{α} ∩ B_{ξ} =^{∗} ∅.

Without loss of generality, we can assume that {A_{n} : n ∈ ω} is strictly decreasing with empty intersection. We also assume that A_{0} = ω. It is easy to construct {B_{ξ}, C_{ξ} : ξ ≤ ω} with B_{n} = A_{n} for n ∈ ω. Suppose we have constructed {B_{ξ}, C_{ξ} : ξ < α} with α ≥ ω and such that (i)_{ξ}, (ii)_{ξη} and (iii)_{ξη} are true for each ξ, η < α. We have to define B_{α} and C_{α}. For ξ ∈ α and n ∈ ω, if ξ ≥ ω, then C_{ξ} ⊆^{∗} A_{n}. By the fact that 𝔭 = 𝔠, there is a D ⊆ ω such that D is a pseudo-intersection of {A_{n} : n ∈ ω} and, for ω ≤ ξ < α, C_{ξ} ⊆^{∗} D. Indeed, we can take, for ω ≤ ξ < α, an f_{ξ} ∈ ^{ω}ω such that for each n, C_{ξ} − A_{n} ⊆ f_{ξ}(n). Since 𝔭 = 𝔠, 𝔟 = 𝔠 , hence we can choose f which is an upper bound of {f_{ξ} : ω ≤ ξ < α} in (^{ω}ω, <^{∗}). We assume f is strictly increasing. Let D = ⋃_{n∈ω}A_{n} ∩ f(n). It is not difficult to see

that D works. By the choice of D, we know that D is not a member of p. Let B_{α} = A_{α} ∩ B_{ω} − D. Let C_{α} be any pseudo-intersection of {B_{ξ} : ξ ≤ α}. It is easy to check that B_{α} and C_{α} satisfies the requirements.

Next we will use the family {B_{α}, C_{α} : α ∈ 𝔠} to construct a topology τ on X = ω∪((𝔠+1)×{0}) by assigning a neighbourhood system for each point. First of all, ω is an open discrete dense subset of ⟨X, τ⟩. For α < 𝔠, {(C_{α} − n)∪{ ⟨α, 0⟩} : n ∈ ω} is a neighbourhood base for ⟨α, 0⟩. Let {(B_{α} − n)∪{⟨ξ, 0⟩ : α ≤ ξ ≤ 𝔠}: α ∈ 𝔠, n ∈ ω} be a neighbourhood base for ⟨𝔠, 0⟩. It is easy to see that ⟨X, τ⟩ is a zero-dimensional Hausdorff pseudoradial space and the subspace ω ∪ {⟨𝔠, 0⟩} is homeomorphic to ω ∪ {p}. □

Thus we proved

Theorem 2.2 (CH). For every point p in ω^{∗}, ω ∪ {p} can be embedded in a zerodimensional pseudoradial space. □

By Theorem 3.1 in next section, the above lemma is not always true in ZFC.

3. In Cohen Model

We assume CH is true in this section.

Theorem 3.1 (CH). If ℙ = F_{n}(ω_{2}, 2), then in V^{ℙ}, for every P -point p , the space ω ∪ {p} can not be embedded in any regular pseudoradial space.

Let us just remark that the Theorem remains valid if CH is dropped and κ many Cohen reals are added for any regular κ > 𝔠.

Proof: Suppose

q ⊩ “ṗ is an ultrafilter on ω and ω ∪ {ṗ} is embedded into a regular pseudoradial space ⟨X, τ̇⟩” .

We have to prove that q ⊮ “ ṗ is a P-point ”. We will work in V^{Fn(λ,2)}, where λ is M ∩ ω_{2} for some elementary submodel of H_{θ} for certain regular cardinal θ. We prove some preliminary facts. We will assume CH is true in the remainder of the section. Without loss of generality, we assume that X is in the ground model and we identify ω ∪ {ṗ} with its copy in X. We furthermore assume that ṗ has the form {{π} × A_{π} : π ∈ dom(ṗ)}, where π is a nice name for a subset of ω. Let İ be a nice ℙ-name such that

1 ⊩ “ İ = {I ∈ [ω]^{ω} : I converges in ⟨X, τ̇⟩}”

We also fix a sufficiently large regular cardinal θ and an elementary submodel M of H_{θ} such that the following are true:

(i) M^{ω} ⊆ M, (ii) {ℙ, X, ṗ, τ̇, İ} ⊂ M and (iii) M has cardinality ℵ_{1}.

Let ℙ_{M} = ℙ ∩ M, X_{M} = X ∩ M and λ = M ∩ ω_{2}. For each ℙ-name Ẏ ∈ M, if 1 ⊩_{ℙ} “ Ẏ ⊆ X”, we can define a ℙ_{M}-name Ẏ_{M} as follows. For each x ∈ X_{M}, let A_{x} be a maximal antichain in {q ∈ ℙ_{M} : q ⊩_{ℙ} “ x ∈ Ẏ ”}. ℙ_{M} is actually Fn(λ, 2), therefore ℙ_{M} is completely embedded in ℙ. Hence A_{x} is also a maximal antichain in {q ∈ ℙ : q ⊩_{ℙ} “ x ∈ Ẏ ”}. Since M^{ω} ⊆ M and ℙ is ccc, we can choose A_{x} from M. Let Ẏ_{M} be {{x} × A_{x}: x ∈ X_{M}}. If, on the other hand, Ẏ is a ℙ-name such that 1 ⊩_{ℙ} “ Ẏ ⊆ PX”, we define a ℙ_{M}-name Ẏ_{M} to be

{⟨Ẏ_{M}, 1⟩ : Ẏ ∈ M and 1 ⊩_{ℙ} “ Ẏ ∈ Ẏ”}

Fact 3.2. Suppose G is a ℙ-generic over V, and σ ∈ M such that 1 ⊩_{ℙ} “ σ ⊆ X”, then the following hold:

(i) if 1 ⊩_{ℙ} “ σ ⊆ ω”, then σ_{M} ∈ M, moreover, if σ is a nice name we can define σ_{M} to be σ;

(ii) V [G] ⊧ “val(σ_{M}, G) = val(σ_{M}, M ∩ G) = val(σ, G) ∩ M”; and

(iii) if 1 ⊩_{ℙ} “ ṗ is a P-point ”, then, 1 ⊩_{ℙ_{M}} “ ṗ_{M} is an P-point ”.

Proof of Fact 3.2: (i) and (ii) are straightforward. For (iii), suppose that 1 ⊩_{ℙ} “ ṗ is a P-point ”. It is easy to see that 1 ⊩_{ℙ_{M}} “ ṗ_{M} is an ultrafilter on ω”. We prove that 1 ⊩_{ℙ_{M}} “ ṗ_{M} is countably complete ”. Suppose ḟ is a ℙ_{M}-name and q ∈ ℙ_{M} such that q forces ḟ is a function from ω onto ṗ_{M}. For each n ∈ ω, we can choose an antichain B_{n} = {q_{i}^{n} : i ∈ ω} which is maximal in {r ∈ ℙ_{M} : r ≤ q} and an associated set {π_{i}^{n} : i ∈ ω} ⊆ dom(ṗ_{M}) such that q_{i}^{n} forces ḟ(n) = π_{i}^{n} for each i ∈ ω. By (i), dom(ṗ_{M}) ⊆ M. Therefore the ℙ_{M}- name ρ = {⟨op⟨ň, π_{i}^{n}⟩, q_{i}^{n}⟩ : n, i ∈ ω} is in M, where op⟨ň, π_{i}^{n}⟩ is a ℙ_{M}-name such that val(op⟨ň, π_{i}^{n}⟩, G) is ⟨n, val(π_{i}^{n}, G)⟩ (See [6]). It is obvious that q ⊩ “ ḟ = ρ”.

Since q ⊩_{ℙ} “ ṗ is countably complete ”, We can choose a nice ℙ-name Ẏ ∈ M such that q ⊩_{ℙ} “ Ẏ is a pseudo-intersection of ran(ρ)” and 1 ⊩_{ℙ} “ Ẏ ∈ ṗ”. By (i) , we can define Ẏ_{M} to be Ẏ. It is clear that q ⊩_{ℙ_{M}} “ Ẏ_{M} = Ẏ ”, hence q ⊩_{ℙ_{M}} “ ran(ḟ) = ran(ρ) > Ẏ_{M} ∈ ṗ_{M}”.

It is obvious that ℙ_{M} is actually Fn(λ, 2). Let ℙ_{1} be ℙ_{M} and ℙ_{2} be Fn(ω_{2} −λ, 2), then ℙ is isomorphic to ℙ_{1} ×ℙ_{2}. For a ℙ-generic filter G, we will also refer to G∩M as G_{M} or as G_{1} and G_{2} will be G ∩ ℙ_{2} while V_{1} will be V [G_{1}]. To further simplify the notation, we will use Y to denote val(Ẏ , G) for a ℙ-name Ẏ. If Ẏ and Ẏ are in M, we will use Y_{M} and Y_{M} to denote val(Ẏ_{M}, G_{M}) and val(Ẏ_{M}, G_{M}) respectively. Thus p, τ, p_{M}, τ_{M} and etc. are well-defined when it is clear from context which ℙ-generic filter G is being used.

Fact 3.3. The following are true:

(i) 1 ⊩_{ℙ} “ ṗ_{M} = ṗ ∩ V_{1}, İ_{M} = İ ∩ V_{1}”; (ii) if ρ ∈ M is a ℙ-name such that 1 ⊩_{ℙ} “ ρ ⊆ P(ω)”, then 1 ⊩_{ℙ} “ ρ_{M} =

ρ ∩ V_{1}”; and (iii) 1 ⊩_{ℙ_{M}} “ İ_{M} = {a ∈ [ω]^{ω} : a converges in ⟨X_{M}, τ̇_{M}⟩}”.

Proof of Fact 3.3: (i) is a corollary of (ii). To prove (ii) , let ρ satisfy the conditions of (ii) and let G be a ℙ-generic filter over V. In V [G], take arbitrary a ∈ val(ρ, G) ∩ V_{1}. Let π ∈ dom(ρ) and σ be a nice ℙ_{M}-name such that a = val(π, G) = val(σ, G). Take a q in G which forces that π = σ ∈ ρ. Since σ, ρ are in M, q ∩ M ⊩_{ℙ} “ σ ∈ ρ”. Let Ẏ ∈ M such that 1 ⊩_{ℙ} “ Ẏ ∈ ρ and (σ ∈ ρ → Ẏ = σ)”, then val(Ẏ_{M}, G) ∈ val(ρ_{M}, G). But q ⊩_{ℙ} “ π = σ = Ẏ ”. Hence a = val(σ, G) = val(Ẏ , G) = val(Ẏ_{M}, G) ∈ val(ρ_{M}, G) Thus we proved 1 ⊩_{ℙ} “ ρ_{M} ⊇ ρ ∩ V_{1}”. By the definition of ρ_{M} , it is obvious that

1 ⊩_{ℙ} “ ρ_{M} ⊆ ρ ∩ V_{1}”.

Therefore 1 ⊩_{ℙ} “ ρ_{M} = ρ ∩ V_{1}”. We are left to prove (iii)

Let H be a ℙ_{M}-generic filter over V, there exists a ℙ-generic filter G over V such that H = G ∩ ℙ_{M}. We have to prove that, in V_{1} = V [G_{M}], I_{M} = {a ∈ [ω]^{ω} : a converges in ⟨X_{M}, τ_{M}⟩}

“ ⊆ ”: Let a ∈ I_{M}. We have to prove that a converges in ⟨X_{M}, τ_{M}⟩. There exits Ẏ_{M} ∈ dom(İ_{M}), such that a = Y_{M}. Since a ⊆ ω, by (ii) of Fact 3.2, Y_{M} = Y, hence a ∈ M[G]. In V [G], there exists x ∈ X, such that a converges to x in ⟨X, τ⟩. By elementarity, we can assume x in X_{M}. Since a ⊆ ω ⊆ X_{M} and ⟨X_{M}, τ_{M}⟩ is a weaker topology than ⟨X_{M}, τ ↾ X_{M}⟩, a is obviously converges to x in ⟨X_{M}, τ_{M}⟩.

“ ⊇ ”: Let a ∈ V_{1} ∩ [ω]^{ω} and for some x ∈ X_{M}, a converges to x in ⟨X_{M}, τ_{M}⟩. We have to show that a is in I_{M}. By (i), we only have to prove that a is in I, i.e.

V [G] ⊧ “a converges in ⟨X, τ⟩”.

By the facts that M^{ω} ⊆ M and that ℙ is ccc , any nice ℙ_{M}-name for a subset of ω is in M. In particular, a is in M[G]. By the choice of θ and the elementarity, it is sufficient to prove that

M[G] ⊧ “a → x in ⟨X, τ⟩” .

Let U be in τ ∩ M[G] containing x. We have to show that a is almost contained in U. There exists Ẏ ∈ M such that U = val(Ẏ , G), we can assume 1 ⊩_{ℙ} “ Ẏ ∈ τ”, hence Ẏ_{M} ∈ dom(τ̇_{M}). Notice that U ∩ X_{M} = U ∩ M = Y_{M} is in τ_{M}. Therefore a is almost contained in U ∩ X_{M} therefore in U.

Now we are ready to finish the proof. Assume that q does force ṗ is a P-point. We can assume that q is the empty function , i.e. q = 1 ( otherwise we can work instead with ℙ_{q} = {r ∈ ℙ : r ≤ q} which is isomorphic to ℙ). Suppose θ, M, and I are obtained as in the previous discussions and G is a ℙ-generic filter over V. Let a ∈ V [G] be a Cohen real over V_{1} such that a is in p. Since, ⟨X, τ⟩ is regular, we can take two open neighbourhoods of p , say W and U such that cl(W) ⊆ U and a = U ∩ω. Let b = W ∩ω. Given any I in I_{M}, let x be in X_{M} such that I converges to x. By (i) of the Fact 3.3, I also converges to x in ⟨X, τ⟩. If b ∩ I is infinite , then x is in cl(W) ⊆ U. Therefore I is almost contained in a. This contradicts the fact that a is a Cohen real over V_{1}. Thus we proved that for each I in I_{M} , b ∩ I is finite. Let τ̇, İ, ȧ, U̇, Ẇ , ḃ and ṗ denote both the ℙ-names and ℙ_{2}-names for the corresponding objects in V [G]. Choose an r ∈ G_{2}, such that r forces the above fact over V_{1}.

Now work in V_{1}, let κ be a sufficiently large regular cardinal, let N be a countable submodel of H_{κ} such that N contains {τ̇, İ, ȧ, U̇, Ẇ , ḃ, ṗ, λ}. By (iii) of Fact 3.2, p_{M} is a P-point. We can take A in p_{M} such that for each B ∈ N ∩ p_{M}, A ⊆^{∗} B. Since in V [G], A is in p and ⟨X, τ⟩ is pseudoradial, there exists I ∈ I such that I ⊂ A. Notice that A ∈ M[G], and , according to [8], M[G] is an elementary submodel of H_{θ}^{V [G]}. By elementarity, we can assume I ∈ M[G]. Hence I ∈ V_{1}. Therefore , by (i) of Fact 3.3. I ∈ I_{M}. By the choice of r, r ⊩_{ℙ_{2}} “ ḃ ∩ I is finite ”, let s ∈ ℙ_{2} and m ∈ ω, such that r ≥ s ⊩_{ℙ_{2}} “ ḃ ∩ I ⊆ m”. Claim: s ∩ N ⊩_{ℙ_{2}} “ ḃ ∩ I ⊆ m”. Otherwise, let k ∈ I − m, such that, there is t ≤ s ∩ N, t ⊩_{ℙ_{2}} “ k ∈ ḃ”. Since k and ḃ are in N, we can assume t ∈ N. But then t and s are compatible while they force contradicting statements. Therefore we can assume s ∈ N. Let Z = {n : ∃t ≤ s, t ⊩_{ℙ_{2}} “ n ∈ ḃ”} ∈ N. Since Z ⊇ b, Z ∈ p_{M}. Now I ⊆ A ⊆^{∗} Z, take k ∈ I ∩ Z − m and t ≤ s such that t ⊩_{ℙ_{2}} “ k ∈ ḃ”. This contradicts the fact that s ⊩_{ℙ_{2}} “ ḃ ∩ I ⊆ m”. □

Remark: In a similar way, we can prove that in V^{Fn(ω_{2},2)}, for every ultrafilter p on ω, if p is a P -point limit of a sequence of P -points, then the space ω ∪ {p} can not be embedded in any regular pseudoradial space.

4. A stationary set in [ω_{2}]^{ω} and a special ultrafilter

In this section, we show that there is an ultrafilter p such that ω ∪ {p} can be embedded into a zero-dimensional Hausdorff pseudoradial space provided there is a reflecting self-indexed stationary set in [ω_{2}]^{ω} and 𝔠 is ω_{2}.

{S_{α} : α < ω_{2}} is a reflecting self-indexed stationary set in [ω_{2}]^{ω} if for any λ < ω_{2} with cf(λ) = ω_{1}, {S_{ξ} : ξ < λ and sup S_{ξ} = ξ} is stationary in [λ]^{ω}.

The reflecting self-indexed stationary set is very similar to the notion of stationary coding set defined in [10].

Lemma 4.1. Let V ⊧ CH and ℙ = {p ⊆ ω_{2}×[ω_{2}]^{ω} : p is a function with countable domain such that p(α) ⊆ α for α ∈ dom α}. Let ℙ have the inverse inclusion as its partial order. Then, in V^{ℙ}, there is a reflecting self-indexed stationary set in [ω_{2}]^{ω}and CH is true.

It is easy to see that ℙ is countably closed and ω_{2}-cc. Therefore CH still holds in V [P] and cardinals are preserved. We leave the proof to the reader.

Lemma 4.2. Reflecting self-indexed stationary set is preserved by ccc forcing

Proof: It is well-known that after ccc forcing every cub subset of ω_{1} contains a ground model cub. The same proof gives that this also holds for cub subsets of [λ]^{ω} for each λ < ω_{2}. □

Remark: S. Todorcěvić informed us that the existence of a reflecting self-indexed stationary set follows from the existence of the ρ-function in [3]. Indeed, one can easily produce such a stationary set directly form a □-sequence (See [4]).

Next we show that for the ultrafilter p stated in the following Theorem, it is possible to embed ω ∪ {p} into a zero-dimensional pseudoradial space.

Definition 4.3 ([2]). Given a family F of subsets of some set S. A family I is called a refinement of F if for each F in F , there is an I in I such that I ⊆ F. If I is an almost disjoint family we say I is an almost disjoint refinement of F.

Theorem 4.4 ([5]). There is an ultrafilter p on ω which has an almost disjoint refinement I and a base B such that

(i) for each subset I′ ∈ [I]^{<𝔠} there is a member of p which is almost disjoint with each member of I′,

(ii) for each B ∈ B and I ∈ I either B ∩ I is finite or B almost contains I,

and (iii) each member of I is almost disjoint with all but finitely many members of

B.

Theorem 4.5. Suppose p is the ultrafilter stated in the Theorem 4.4. If 𝔠 = ω_{2} and there is a reflecting self-indexed stationary set in [ω_{2}]^{ω}, then the space ω ∪ {p} can be embedded in a zero-dimensional pseudoradial space.

Corollary 4.6. It is consistent that, for every P-point p, ω ∪ {p} cannot be embedded into any regular pseudoradial space while there is an ultrafilter p on ω such that ω ∪ {p} is a subspace of a zero-dimensional pseudoradial space.

Proof: Let V be the model obtained through Lemma 4.1. Let Q = Fn (ω_{2}, 2). By Theorem 3.1, in V^{Q}, for every P-point p, ω ∪ {p} can not be embedded into any regular pseudoradial space.

On the other hand, by Lemma 4.2, in V^{Q}, there is a reflecting self-indexed stationary set on [ω_{2}]^{ω} and 𝔠 is ω_{2}. Therefore, by Theorem 4.5, there is an ultrafilter p on ω such that ω ∪ {p} is a subspace of a zero-dimensional pseudoradial space. □

We will need the following Lemma in the proof of Theorem 4.5.

Lemma 4.7. Suppose R is a subset of space X, x ∈ X, and U is a local subbase at x, i.e, the intersections of finitely many members of U constitute a neighbourhood base at x. If κ is any cardinal number such that the following are true,

(i) for all U′ ∈ [U]^{<κ}, R ∩ ⋂ U′ is not empty ; (ii) for all y ∈ R, {U ∈ U : y ∉ U} has cardinality at most κ,

Then there is a sequence {y_{α} : α < κ} in R convergent to x.

Proof: For each y ∈ R, let {U_{α}^{y} : α < κ} enumerate, with possible repetition, the set {U ∈ U : y ∉ U}. It is straightforward to define, by the induction, a sequence {y_{α} : α < κ} such that for all α < κ

y_{α} ∈ R ∩ ⋂{U_{η}^{y_{ξ}}: η < α, ξ < α} (∗)

We show that {y_{α}}_{α<κ} is convergent to x by showing that each member of U contains a tail of the sequence. Suppose U ∈ U. We have to show that there is an α < κ, such that , for all β ∈ κ, β > α implies y_{β} ∈ U. If {y_{α} : α < κ} is contained in U. We are done. Otherwise, choose any ξ such that y_{ξ} ∉ U. Take η < κ such that U = U_{η}^{y_{ξ}}. Let α = max{η, ξ}, then, for any β > α, by (∗), y_{β} ∈ U. We are done. □

The rest of the section is to prove Theorem 4.5

Proof of Theorem 4.5 : We first fix a reflecting self-indexed stationary set {S_{α}: α ∈ ω_{2}} in [ω_{2}]^{ω} . Let p = {A_{α}: α < ω_{2}} be the ultrafilter in Theorem 4.4. Let {I_{A}: A ∈ p} be the almost disjoint refinement of p such that I_{A} ⊆ A and B is the base with the properties in Theorem 4.4. By induction we can define an almost disjoint family {a_{α}: α < ω_{2}} and a base {F_{α}: α < ω_{2}} of p such that

(i) (∀α)(a_{α} ⊆ F_{α} ⊆ A_{α}),

(ii) (∀α , β)(α < β) implies a_{α} ∩ F_{β} is finite.

(iii) (∀α) there are only finitely many β such that a_{α} ∩ F_{β} is infinite and if

a_{α} ∩ F_{β} is infinite, then a_{α} ⊆ F_{β}, (iv) (∀λ) if cf(λ) is uncountable, then {F_{α}: α < λ} is a filter base. We construct {a_{α}, F_{α}: α < ω_{1}η} by induction on η. At stage η, the only non

trivial case is when η is a successor ordinal. Suppose η = ξ + 1. We define a_{ω_{1}ξ+α} and F_{ω_{1}ξ+α} for α ∈ ω_{1} by induction on ω_{1}. Let f: ω_{1} → [ω_{1}ξ + ω_{1}]^{<ω} be an onto mapping such that f(α) ⊆ ω_{1}ξ + α for each α ∈ ω_{1}. At stage α, take a B in B such that B is almost disjoint with a_{ζ} for each ζ < ω_{1}ξ + α. Now simply define F_{ω_{1}ξ+α} be A_{ω_{1}ξ+α} ∩ B ∩ ⋂{F_{ζ}: ζ ∈ f(α)} and let a_{ω_{1}ξ+α} be I_{F_{ω_{1}ξ+α}}.

Let H = {b ⊆ ω: (∀α ∈ ω_{2})(a_{α} ⊆^{∗} b)}. Let A be the subalgebra of P (ω) generated by [ω]^{<ω} ∪ {a_{α}, F_{α}: α ∈ ω_{2}} ∪ H

We will define a subspace X of the Stone space St(A) such that X contains a copy of ω ∪ {p} and X is pseudoradial. X will be of the form ω ∪ {x_{α} : α ∈ Γ} , where Γ is a subset of ω_{2} and each x_{α} is an ultrafilter of A . We specify the set Γ and x_{α} for α ∈ Γ as follows.

For convenience, we define a subset x_{α}^{+} of F for each α ∈ ω_{2}. For successor ordinal α + 1, x_{α+1}^{+} is the finite of F_{β}’s which contains a_{α}. For limit α with cf(α) = ω, x_{α}^{+} is {F_{α}: α ∈ S_{α}} , where S_{α} is from the reflecting self-indexed stationary set. For limit λ with cf(λ) = ω_{1}, x_{λ}^{+} is {F_{α}: α < λ}.

First of all, Γ contains all the successor ordinals and the ordinals with uncountable cofinality.

For each α, since no member of A splits a_{α}, the cofinite subsets of a_{α} generate an ultrafilter on A. Let x_{α+1} be this ultrafilter. The point x_{ω_{2}} will simply be the ultrafilter p ∩ A. Before we handle other ordinals, we prove the following.

Fact 4.8. If λ < ω_{2} has cofinality ω_{1}, then x_{λ}^{+} ∪{ω −F_{β} : F_{β} ∉ x_{λ}^{+}}∪H is actually {F_{α}, ω \ F_{β}: α < λ and β ≥ λ} ∪ H and generates an ultrafilter x_{λ} on A. If α < ω_{2} has cofinality ω and x_{α}^{+}∪{ω−F_{β} : F_{β} ∉ x_{α}^{+}}∪H has the finite intersection property, then it generates an ultrafilter x_{α} on A.

Proof of Fact 4.8: If λ < ω_{2} has cofinality ω_{1}, then {F_{α}: α < λ} is a filter base and it is easy to see that {F_{α}, ω \ F_{β}: α < λ and β ≥ λ} ∪ H has finite intersection property. To prove that it generates an ultrafilter, it is sufficient to show that for each β ∈ ω_{2}, ω \ a_{β} contains some member of the family. Indeed, if β < λ, then F_{β+1} is almost disjoint with a_{β} hence F_{β+1} ⊆^{∗} ω \ a_{β}. If β ≥ λ, then ω \ F_{β} is almost contained in ω \ a_{β} . The proof for the case when α has countable cofinality is virtually the same.

We resume the proof of the Theorem. For λ with cofinality ω_{1}, let λ be in Γ and x_{λ} be the ultrafilter defined Fact in 4.8. For an ordinal α with cf(α) = ω, α ∈ Γ if x_{α}^{+} ∪ {ω − F_{β} : F_{β} ∉ x_{α}^{+}} ∪ H generate a filter, we let x_{α} be the ultrafilter defined in Fact 4.8

By above discussion Γ is well defined and, for each α ∈ Γ, x_{α} is a point in the Stone space St(A). The following claim is trivial.

Claim 4.1. ω ∪ {x_{ω_{2}}} is homomorphic to ω ∪ {p}

For simplicity we will abuse the notation of Stone duality. If F ⊂ ω is a member of A, we also let F denote the set of points of X which as an ultrafilter on A contain F as a member. Of course, F is clopen, hence, to say x_{α} ∈ F is equivalent to saying F ∈ x_{α}.

Let X_{0} be all x_{α} with α a successor and X_{1} be all x_{α} with cf(α) = ω. Recall that x_{α+1} corresponds to a_{α}. Let X_{2} be all x_{α} such that α has uncountable cofinality.

Fact 4.9. The sets X_{1} ∪ X_{2} and X_{2} are closed in X. X_{2} is homomorphic to a subspace of ω_{2}, namely {α ∈ ω_{2} : cf(α) is uncountable } and so X_{2} is radial.

Proof of Fact 4.9 For the first statement of the claim, the only case that requires proof is that a point x_{α} of X_{1} is not in the closure of X_{2}. If α is countable, then ω − F_{α+1} is disjoint from X_{2}. On the other hand, if α is uncountable, then we can find γ < β < α, so that F_{β} ∈ x_{α}^{+} and F_{γ} ∉ x_{α}^{+}. Note that F_{β} \F_{γ} is not a member of x_{λ} for any λ with uncountable cofinality. For the second statement, it is sufficient to see that, for any λ < ω_{2} with uncountably cofinality, the typical neighhourhood of x_{λ} in the subspace X_{2} is (F_{α} \F_{λ})∩X_{2} = {x_{ξ} ∈ X_{2} : α < ξ ≤ λ} for some α < λ. Similarly, a typical neighbourhood of x_{ω_{2}} is {x_{ξ} ∈ X_{2} : α < ξ ≤ ω_{2}}.

Claim 4.2 will finish the proof of Theorem 4.5.

Claim 4.2. X is a zero-dimensional pseudoradial space.

Proof of Claim 4.2: Since X is a subspace of a St(A), X is zero-dimensional. Suppose X is not pseudoradial. Let R be a radially closed subset of X which is not closed. Let x_{α} ∈ cl R \ R and α is the minimal such index. Obviously x_{α} ∈ X_{1} ∪ X_{2} because other points have countable neighbourhood bases. We will prove x_{α} ∈ R to produce a contradiction. We will need the following fact.

Fact 4.10. x_{α} ∈ cl(R \ ω)

Proof of Fact 4.10: Suppose x_{α} ∉ cl(R \ ω). Then x_{α} ∈ cl(R ∩ ω) \ cl(R \ ω). Let U be a clopen neighbourhood of x_{α} such that U ∩(R \ω) = ∅. Hence b = R ∩U is a subset of ω. Since R is radially closed and U is closed , b is also radially closed. For any β ∈ ω_{2}, if a_{β} ∩ b is infinite, then x_{β+1} would be in b contradicting that b ⊆ ω. Hence, b is almost disjoint with each a_{β}, i.e, ω \ b is in H. Thus ω \ b are in the ultrafilter x_{α}, which contradicts that x_{α} is in the closure of b.

Since R \ ω is also radially closed in X, we can now assume R ∩ ω = ∅. By Fact 4.9, X_{2} is radial and closed so we can assume that R ∩ X_{2} = ∅ by again restricting to a neighbourhood of x_{α}. For the rest of the proof we assume that R ⊆ X_{0} ∪ X_{1}. Recall that x_{α} ∈ X_{1} ∪ X_{2}.

We claim that α ≠ ω_{2} . Indeed fix any M ≺ H(ω_{3}) such that |M| = ω_{1} and M contains all relevant sets such as R and X, and so that the cofinality of δ = M ∩ ω_{2} is uncountable. We show that x_{δ} is in the closure of R which will contradicts that R is assumed to be disjoint from X_{2} and that α is the minimal index of a limit point which is not in R. A typical neighbourhood of x_{δ} has the form H ∩⋂{F_{ξ} : ξ ∈ f}\⋃{F_{γ} : γ ∈ g} where f ∈ [δ]^{<ω}, g ∈ [ω_{2} \δ]^{<ω} and H ∈ H. However, ⋂{F_{ξ} : ξ ∈ f} is a neighbourhood of x_{ω_{2}} and is a member of M. Choose any β ∈ M such that x_{β} ∈ R ∩ ⋂{F_{ξ} : ξ ∈ f}. Clearly x_{β} ∉ F_{γ} for all γ > δ, hence x_{α} is in the above neighbourhood of x_{δ}.

Next we apply Lemma 4.7 to prove that there exists a sequence in R convergent to x_{α} and thus finish the proof.

Case 1. cf(α) = ω

To apply Lemma 4.7, let κ be ω, U be x_{α}^{+} ∪ {ω \ F_{β}: x_{β} ∉ x_{α}^{+}} ∪ H, and x be x_{α}. U is a local subbase of x. We show x, R and U satisfies the conditions in Lemma 4.7. Since x ∈ cl(R), the condition (i) is trivial. To verify the condition (ii), let x_{ξ} ∈ R. Then cf(ξ) ≤ ω. It is easy to see that

{U ∈ U: x_{ξ} ∉ U} ⊆ x_{α}^{+} ∪ {ω \ F: F ∈ x_{ξ}^{+}}.

But both x_{α}^{+} and x_{β}^{+} are countable. Therefore {U ∈ U: x_{ξ} ∉ U} is countable.

Case 2. cf(α) = ω_{1}.

We let κ be ω_{1}, and U be x_{α}^{+} ∪ {ω \ F_{β}: x_{β} ∉ x_{α}^{+}} ∪ H, and x be x_{α}. Again U is a local subbase at x. The second condition of Lemma 4.7 is verified as in the Case 1. We are left to show that for any U′ ∈ [U]^{ω}, R ∩ ⋂ U′ ≠ ∅. Since for each H in H, R ⊆ X \ ω ⊆ H, we can assume U′ = {F_{α_{i}}, ω \ F_{β_{i}} : i ∈ ω} . We take a countable elementary submodel M of H(ω_{3}), such that {X, R, U′, α} ⊆ M and M ∩ α = S_{γ} and γ = sup(M ∩ α), where S_{γ} is from the the reflecting self-indexed stationary set. Since S_{γ} = M ∩ α, x_{γ}^{+} is a filter base, x_{γ}^{+} ∪ {ω \ F_{ξ} : F_{ξ} ∉ x_{γ}^{+}} ∪ H has finite intersection property. Therefore it generates an ultrafilter on A. Therefore γ ∈ Γ, i.e, x_{γ} is a point of our X. Next we show that x_{γ} ∈ R ∩ ⋂ U′.

For i ∈ ω, since, U′ ∈ M, α_{i} ∈ M ∩ α = S_{γ}. Thus, F_{α_{i}} ∈ x_{γ}^{+} while F_{β_{i}} ∉ x_{γ}^{+} because β_{i} ≥ α > γ. This proves that x_{γ} ∈ ⋂ U′.

To prove x_{γ} ∈ R, it is sufficient to prove that x_{γ} ∈ cl(R) because α is the minimal index such that x_{α} ∈ cl(R) \ R. Take a neighbourhood W of x_{γ}, we have to show R ∩ W ≠ ∅. We can assume W = ⋂ F \ ⋃ G, where F and G are finite subsets of x_{γ}^{+} and B \ x_{γ}^{+} respectively. Obviously F and G ∩ M are members of M. If F_{ξ} ∈ G ∩ M, then by the definition of x_{γ}^{+} and the fact S_{γ} = M ∩ α, ξ ≥ α. Therefore, (R ∩ ⋂ F) \ ⋃(G ∩ M) is a neighbourhood of x_{α} and it is in M. Take a x_{ξ} ∈ M ∩R∩⋂ F \⋃(G∩M). Since x_{ξ}^{+} is a countable member of M, x_{ξ}^{+} ⊆ M, hence G\M is disjoint with x_{ξ}^{+}. Therefore x_{ξ} ∉ ⋃(G\M) and x_{ξ} ∈ R∩⋂ F \⋃ G = R∩W. We are done. □

We finish with the following question.

Question 4.11. Does MA or 𝔭 = 𝔠 implies that for each ultrafilter p on ω, ω ∪ {p} is a subspace of a regular pseudo-radial space?

By Lemma 2.1, the answer is yes for the non P-point p.

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Department of Mathematics, York University, 4700 Keele Street, North York, Ontario, Canada M3J 1P3

E-mail: Alan Dow: Alan.Dow@mathstat.yorku.ca

E-mail: Jinyuan Zhou: Jinyuan.Zhou@mathstat.yorku.ca