Submitted to

Topology Proceedings

Q-SETS AND NORMALITY OF Ψ-SPACES

FERNANDO HERNÁNDEZ-HERNÁNDEZ AND MICHAEL HRUŠÁK

Abstract. We tie some loose ends in the relationship between the normality of Ψ-spaces and the existence of Q-sets.

1. Introduction

The starting point for our considerations is the following classical theorem giving definitive solution to the separable case of the Normal Moore Space Problem.

Theorem 1.1 ([Tal77]). The following are equivalent:

(a) There is a Q-set.

(b) There is an uncountable normal Ψ-space.

(c) There is a separable normal non-metrizable Moore space.

For the sake of completeness we include a sketch of the proof of the theorem later in the text. We would like to mention that the implication (a) ⇒ (c) was originally proven by Bing in [Bin51] when he showed that the bubble space over a Q-set is a normal space. On the other hand, Heath proved the implication (c) ⇒ (a) in [Hea64].

An almost disjoint family A is a family of infinite subsets of ω (or any other countable set) such that A ∩ B is finite for distinct A, B ∈ A. The Ψ-space Ψ (A) associated to A is ω ∪ A where the points of ω are isolated and the basic neighbourhoods of A ∈ A are of the form {A} ∪ A \ F, where F ⊆ ω is finite. The space Ψ (A) is a first countable, separable, locally compact Moore space.

2000 Mathematics Subject Classification. Primary 54A25, 03E35.

Key words and phrases. Ψ-space, Q-set, λ-set, almost disjoint family, normal, pseudonormal.

The authors gratefully acknowledge support from PAPIIT grant IN106705.

The second author’s research was also partially supported by GAČR grant 201-03-0933 and CONACYT grant 40057-F.

An almost disjoint family A is said to be ℝ-embeddable if there exists a continuous f : Ψ (A) → ℝ such that f is an injective function and f (A) is an irrational number while f (n) is a rational one for all A ∈ A and all n ∈ ω. For an ℝ-embeddable family A, we keep the notation X_{A} = {x_{A} : A ∈ A}, where f (A) = x_{A} and f is a witness that A is ℝ-embedded.

In a conversation with members of the Toronto Set Theory seminar the following question was raised:

Question 1. If A is an uncountable ℝ-embeddable family, is Ψ (A) normal if and only if X_{A} is a Q-set?

There seemed to be some confusion between the question and an analogous question for subfamilies of the Cantor tree (see Proposition 2.2). We will show that, consistently, neither of the implications is true. We first show that in a model of Fleissner and Miller there is an ℝ-embeddable family A with Ψ (A) normal but X_{A} not a Q-set. Then we modify their method to show that there is a model of ZFC where there is an ℝ-embeddable family A for which X_{A} a Q-set, yet Ψ (A) is not normal.

2. Preliminaries

Q-SETS AND NORMALITY OF Ψ-SPACES 3

Our terminology is mostly standard: A ⊆^{∗} B means that A is almost contained in B; that is, A \ B is finite, A =^{∗} B means A ⊆^{∗} B and B ⊆^{∗} A. For functions f, g ∈ ω^{ω} we write f ≤^{∗} g to mean that there is some m ∈ ω such that f (n) ≤ g (n) for all n ≥ m. The bounding number in ω^{ω}, 𝔟, is the least cardinal of an ≤^{∗}-unbounded family of functions. The dominating number in ω^{ω}, 𝔡, is the least cardinal of a ≤^{∗}-cofinal family of functions. A Q-set is an uncountable set X of reals such that every subset of X is F_{σ} in X. A λ-set is an uncountable set X of reals such that every countable subset of X is G_{δ} in X. ZFC suffices to construct a λ-set of size 𝔟. We say that a subset A of ℝ is concentrated on a set C ⊆ ℝ if A \ U is countable for every open set U containing C. The set 2^{ω} is equipped with the product topology, that is the topology with basic open sets of the form [s] = {x ∈ 2^{ω} : s ⊆ x}, where s ∈ 2^{<ω}. The topology of P (ω) is that obtained via the identification of each subset of ω with its characteristic function. Given a set E ⊆ 2^{ω}, let Ê = {x ↾ n : x ∈ E, n ∈ ω } ⊆ 2^{<ω}. See [Eng89] for undefined topological notions and [Kun83] for set theoretical notions.

The following proposition is standard and easy to prove.

Proposition 2.1. Ψ (A) is a normal space if and only if for every B ⊆ A there is a J ⊆ ω such that

B = {A ∈ A : A ⊆^{∗} J} and A \ B = {A ∈ A : A ∩ J =^{∗} ∅} .

The set J in the conclusion of this proposition is called partitioner for B and A \ B. Notice that if B = {A ∈ A : A ⊆^{∗} J}, then

B = ⋃_{n∈ω} ⋂_{m∈ω} {A ∈ A : m ∈ A \ n ⇒ m ∈ J};

that is, B is an F_{σ}-set of A as subspace of P (ω). Therefore, if Ψ (A) is a normal space then A is a Q-set (as subspace of P (ω)).

For x ∈ 2^{ω}, put A_{x} = {x ↾ n : n ∈ ω} and for a subset X of 2^{ω} let A_{X} = {A_{x} : x ∈ X}. Then A_{X} is an almost disjoint family of subsets of 2^{<ω}. Call this A_{X} the almost disjoint family corresponding to X.

Proposition 2.2 (Folklore). Let X ⊆ 2^{ω} and A_{X} be the almost disjoint family corresponding to X. Then X is a Q-set if and only if Ψ (A_{X}) is a normal space.

Proof. Assume X is a Q-set, B ⊆ A, and let B = {x ∈ X : A_{x} ∈ B}. Since X is a Q-set there are closed subsets F_{n} and G_{n} of X such that B = ⋃_{n∈ω} F_{n} and X \B = ⋃_{n∈ω} G_{n}. Define J_{o} = F̂_{0}, K_{0} = Ĝ_{0} \F̂_{0} and J_{n} = F̂_{n} \⋃_{i<n} Ĝ_{i} as well as K_{n} = Ĝ_{n} \⋃_{i≤n} F̂_{n} for n > 0. Put J = ⋃_{n∈ω} J_{n} and observe that J ∩ K_{m} =^{∗} ∅ for every m ∈ ω. If A_{x} ∈ B, then there is some n ∈ ω such that x ∈ F_{n}. Moreover, since each G_{i} is closed in X and G_{i}∩B = ∅, for i < n, there is some k ∈ ω such that [x ↾ k] ∩ ⋃_{i<n} G_{i} = ∅. This implies that A_{x} ⊆^{∗} J_{n} ⊆ J. Similarly if x ∈ X \ B there are k, m ∈ ω such that x ∈ G_{m} and [x ↾ k] ∩ ⋃_{i≤n} F_{i} = ∅; this implies A_{x} ∩ J =^{∗} ∅. By Proposition 2.1 this suffices to show that Ψ (A) is normal. The other direction follows from the comments following Proposition 2.1.

Proof of Theorem 1.1. The equivalence between (a) and (b) follows from Proposition 2.2. For (b) implies (c) we only need to recall that Ψ (A) is always a separable Moore space which is non-metrizable if A is uncountable. To show that (c) implies (b) first observe that a separable normal Moore space which is not metrizable has an uncountable closed discrete subset. This is a consequence of Bing’s theorem that a Moore space X is metrizable if and only if X is collectionwise normal. So, let X be a separable non-metrizable normal Moore space, let D be a countable dense subset of X and let F be a closed discrete uncountable subset of X disjoint from D. Taking sequences of elements in D which converge to the points in F we obtain an uncountable almost disjoint family A. The normality of X implies that of Ψ (A).

Restricting ourselves to ℝ-embeddable almost disjoint families does not impose any restriction due to the following simple observation.

Proposition 2.3. Every almost disjoint family A such that Ψ (A) is normal is ℝ-embeddable.

Proof. If Ψ (A) is a normal space the we can take any injection g : A → ℝ \ ℚ and use Tietze extension theorem to find a continuous extension \overline{g} : Ψ (A) → ℝ of g. Now \overline{g} can be modified to obtain a function f witnessing that A is ℝ-embedded; simply let f ↾ A = \overline{g} ↾ A and, for n ∈ ω, f (n) ∈ ℚ\ \{f (0) , …, f (n − 1)\} such that |f (n) − \overline{g} (n)| < \frac{1}{n}. The continuity of f follows from that of \overline{g}.

3. Main results

In this section we describe how to construct models of ZFC where either of the possible implications in Question 1 fails. With a little bit of care one can actually construct a single model where both implications fail.

Even though the belief in a positive answer for Question 1 was not correct, it was partially supported by the next theorem.

Theorem 3.1 (Folklore). MA_{σ-centered} implies that for an ℝ-embed-dable almost disjoint family A, Ψ (A) is normal if and only if X_{A} is a Q-set.

Proof. For the forward implication just recall that by Jones’ Lemma, the cardinality of A must be less than 𝔠 and by a result of J. Silver (under MA_{σ-centered}) every set of reals of size less than 𝔠 is a Q-set. The reverse implication follows directly from the next lemma.

Lemma 3.2. Suppose that A is an ℝ-embeddable family such that X_{A} is a Q-set. If |A| < 𝔟, then Ψ (A) is a normal space.

Proof. For convenience we shall use the bubble space to show that Ψ (A) is normal. It is an old result of Bing’s that the bubble space over a Q-set is a normal space. See [Tal84, p. 709].

Let φ : Ψ (A) → ℝ witness that A is ℝ-embedded, let X_{A} = {x_{A} : A ∈ A} and put q_{n} = φ (n), for every n ∈ ω. By hypothesis X_{A} is a Q-set. In order to establish that Ψ (A) is normal, let B ⊆ A. As the bubble space over X_{A} is normal, there are basic neighbourhoods B_{x}, for x ∈ X_{A}, such that U_{B} = ⋃ {B_{x_{B}} : B ∈ B} and W_{B} = ⋃ {B_{x_{A}} : A ∈ A \ B} are disjoint.

It is clear that, for A ∈ A, there is a function f_{A} : A → ω satisfying

(1) f_{A} is non-decreasing,

(2) f_{A} is finite-to-one and

(3) eventually ⟨q_{n}, \frac{1}{f_{A}(n)}⟩ ∈ B_{x_{A}}.

The idea is that the function f_{A} can be used to lift the points in A to the neighbourhood B_{x_{A}}. Define g_{A} : ω → ω by

g_{A} (n) = max {a ∈ A : f_{A} (a) ≤ n} .

Since |A| < 𝔟 there is an increasing function g : ω → ω such that g_{A} ≤^{∗} g for all A ∈ A. Let f : ω → ω be defined by

f (n) = max {k ∈ ω : g (k) ≤ n}

for every n ∈ ω. Then f ↾ A ≤^{∗} f_{A} for every A ∈ A and f is finite to one. This f can be used to do the lifting of the points q_{n} in an uniform way for every A ∈ A, i.e. for large n ∈ A, ⟨q_{n}, \frac{1}{f(n)}⟩ ∈ B_{x_{A}} for every A ∈ A.

Then the partitioner for B and A \ B will be

J = \{n ∈ ω : ⟨q_{n}, \frac{1}{f (n)}⟩ ∈ U_{B}\}.

It is clear that B ∈ B ⇒ B ⊆^{∗} J and A ∈ A \ B ⇒ A ∩ J =^{∗} ∅.

Now we show that there is a model in which implication “if Ψ (A) is an uncountable normal space then X_{A} is Q-set in ℝ” fails.

Theorem 3.3. It is consistent that there is an uncountable A which is ℝ-embeddable and Ψ (A) is normal but X_{A} is not a Q-set.

Proof. In [FM80], a model is described where there is a Q-set X which consists of the irrationals concentrated on some countable set F of the irrationals disjoint from X. Partition ℚ into two disjoint dense sets D_{0} and D_{1}. By Propositions 2.2 and 2.3 there is a an ℝ-embedded almost disjoint family A_{0} of subsets of D_{0} such that X_{A_{0}} = X. Consider also the almost disjoint family A_{1} obtained by taking convergent sequences of elements in D_{1} to the elements of F. Since F is countable, Ψ (A_{1}) is normal. Letting A = A_{0} ∪ A_{1}, Ψ (A) is normal as it is the topological sum of Ψ (A_{0}) and Ψ (A_{1}) both of which are normal. However, as X is concentrated on F, X_{A} = X ∪ F cannot be a Q-set.^{1}

Theorem 3.4. It is consistent that there is an almost disjoint family A that is ℝ-embeddable, X_{A} is a Q-set, yet Ψ (A) is not a normal space.

Proof. We will use a finite support iteration ⟨ℙ_{α}, ℚ̇_{α} : α < ω_{2}⟩ of c.c.c. forcings that will be described next. The idea is that ℙ_{0} generically adds an ℝ-embeddable family A and the rest of the iteration makes X_{A} a Q-set without making Ψ (A) a normal space.

For technical reasons, fix families W_{n}, n ∈ ω, of open intervals with irrational endpoints such that

• the length of each interval in W_{n} is at most \frac{1}{n} and

• W_{n+1} refines W_{n}. Then W = ⋃_{n∈ω} W_{n} is a base for the rationals with the property

that given two elements of W, either one is contained in the other or they are disjoint. In particular, for every q ∈ ℚ and n ∈ ω there is a unique W = W (q, n) ∈ W_{n} such that q ∈ W.

1From the referee’s report: “The example of a normal ℝ-embedded Ψ(A)

such that X_{A} is not a Q-set is a little unsatisfying, in that it is homeomorphic to the topological sum of two Ψ(A)’s, each over a Q-set (one countable). I wonder if every example must be like that. Surely there would at least have to be an uncountable subset of X_{A} which is a Q-set.

Define ℙ_{0} by p ∈ ℙ_{0} if and only if p = (a, k) ∈ Fn (ω_{1} × ω, ℚ)×ω. For p = (a_{p}, k_{p}) and q = (a_{q}, k_{q}) declare p ≤ q if and only if

(1) a_{p} ⊇ a_{q} and k_{p} ≥ k_{q}, and

(2) if a_{q} (α, ) and a_{p} (α, ) are the obvious functions and n_{q}^{α} is the largest element in the domain of a_{q} (α, ) then

a_{p} (α, n) ∈ W (a_{q}(α, n_{q}^{α}), k_{q})

for all n ∈ dom (a_{p} (α, )) \ dom (a_{q} (α, )). Moreover, this holds for all α ∈ ω_{1} such that there is some n ∈ ω for which (α, n) ∈ dom (a_{q}).^{2}

Then, if G_{0} is a generic filter in ℙ_{0}, G_{0} codes a family of sequences a_{α} : ω → ℚ given by a_{α} (n) = a_{p} (α, n) for some p ∈ G_{0}. Thus ℙ_{0} actually adds ω_{1} Cohen reals —the limits of the Cauchy sequences ⟨a_{α} (n) : n ∈ ω⟩. It is clear that letting A_{α} = {a_{α} (n) : n ∈ ω} we have an ℝ-embedded almost disjoint family A = {A_{α} : α < ω_{1}} in V [G_{0}].

Let B = {B_{n} : n ∈ ω} be a base for the topology of ℝ. For Y ⊆ X_{A} let ℙ (Y) be the set of all r such that

(1) r is finite subset of ω × (B ∪ Y) ,

(2) for B ∈ B and x ∈ Y the set r satisfies ⟨n, B⟩ ∈ r and

⟨n, x⟩ ∈ r implies x ∉ B. The ordering on ℙ (Y) is just r ≤ r′ if and only if r ⊇ r′. This forcing is defined in [FM80].

The forcing ℙ (Y) makes Y an F_{σ} subset of X_{A}. Indeed, if G is a generic filter in ℙ (Y) over a model V, then set for each n ∈ ω,

U_{n} = {x ∈ X_{A} : (∃r ∈ G) (∃B ∈ B) (⟨n, B⟩ ∈ r ∧ x ∈ B)} ,

and set K = ⋃ {X \ U_{n} : n ∈ ω}. Then each U_{n} is open relative to X_{A} and K is and F_{σ} relative to X_{A}. To see that K = Y proceed as follows. For all x ∈ Y and r ∈ ℙ (Y), there are r′ ⊇ r and n ∈ ω such that ⟨n, x⟩ ∈ r′. Then x ∈ X \ U_{n} ⊆ K. On the other hand, if x ∈ X_{A} \ Y, r ∈ ℙ (Y) and n ∈ ω there are r′ ⊇ r and B ∈ B such that ⟨n, B⟩ ∈ r′ and x ∈ B. Then x ∈ U_{n} for all n ∈ ω.

Assume the ground model V is a model of CH. Using the usual bookkeeping techniques we can take ℚ_{α} to be such that ℙ_{α} ⊩ ℚ̇_{α} = ℙ(Ẏ_{α}) where Ẏ_{α} is a ℙ_{α}-name for a subset of X_{A} and doing this in such a way that every subset of X_{A} eventually appears as some Y_{α}.

2The point of choosing the families W_{n} is to make the order of ℙ_{0} transitive.

Thus ℙ_{ω_{2}} will force A to be an ℝ-embedded almost disjoint family and X_{A} to be a Q-set. Fix a generic G ⊆ ℙ_{ω_{2}} over the ground model V.

To show that Ψ (A) is not normal in V [G] it suffices to find a C ⊆ A such that C and A \ C cannot be separated. Consider simply C = {A_{α} : α ∈ ω_{1} is a limit ordinal}. Suppose J ⊆ ℚ is such that

C = {A ∈ A : A ⊆^{∗} J} and A \ C = {A ∈ A : A ∩ J =^{∗} ∅} .

By genericity, J is dense and co-dense subset of the rationals. The conclusion will then follow from the next lemma.

Lemma 3.5. In V [G], the set {α ∈ ω_{1} : A_{α} ⊆^{∗} J} is at most countable for every dense and co-dense J ⊆ ℚ.

To prove the lemma we first show that ℙ_{ω_{2}} is “semi-Cohen” (see [BJZ97]).

Lemma 3.6. If p ∈ ℙ_{ω_{2}} and M is a countable elementary submodel of some big enough H (θ) such that ℙ_{ω_{2}} ∈ M, then there is a \overline{p} ∈ ℙ_{ω_{2}} ∩ M such that whenever r ∈ ℙ_{ω_{2}} ∩ M is such that r ≤ \overline{p}, r is compatible with p.

Proof. For every p ∈ ℙ_{ω_{2}} there is some β < ω_{2} such that p ∈ ℙ_{β}, thus the proof can be done by induction on β ≤ ω_{2}. For β = 0 and for β limit it is easy to do and we only show the details for successor steps.

Suppose the lemma holds for ℙ_{β} and p ∈ ℙ_{β+1}. There is a p′ ∈ ℙ_{β} such that

p′ ⊩ p (β) = {⟨n_{i}, B_{i}⟩ : i < n_{0}} ∪ {⟨n_{i}, ẋ_{γ_{i}}⟩ : i < n_{2}} ,

for some n_{0}, n_{1}, n_{2} ∈ ω with n_{1} ≤ n_{2}, and such that {γ_{i} : i < n_{1}} ⊆ M while {γ_{i} : n_{1} ≤ i < n_{2}} ∩ M = ∅.

By induction hypothesis there exists a \overline{p′} ∈ ℙ_{β} ∩M satisfying the conclusion for p′ instead of p. Find {γ′_{i} : n_{1} ≤ i < n_{2}} ⊆ M disjoint from \{γ_{i} : i < n_{2}\} and define \overline{p} ∈ M as follows:

\overline{p} (0) = \overline{p′} (0)∪\{⟨⟨γ′_{i}, m, q_{j}⟩ , n⟩ : ⟨⟨γ_{i}, m, q_{j}⟩ , n⟩ ∈ p′ (0) ∧ i < n_{2}\}

\overline{p} (α) = \overline{p′} (α) for 0 < α < β, and

\overline{p} (β) = (p (β) ∩ M) ∪ \{⟨n_{i}, ẋ_{γ}′_{i}⟩ : i ∈ n_{2}\}

Then \overline{p} satisfies the requirements. Indeed, if r ∈ ℙ_{β+1} ∩ M and r ≤ \overline{p} then r ↾ β is compatible with p ↾ β, thus p and r can only be incompatible if r forces ẋ_{γ_{i}} inside some B used in p (β), for some i ∈ n_{2}. However, r (and p) can only decide a finite initial part of Ȧ_{γ_{i}} of which ẋ_{γ_{i}} is the limit. Then there must exist q_{m} ∈ ℚ and k ∈ ω such that the interval W (q_{m}, k) is contained in B, r ⊩ q_{m} ∈ Ȧ_{γ_{i}} and any mutual extension of p and r forces the new points of Ȧ_{γ_{i}} to be in W (q_{m}, k). Since the same initial part of Ȧ_{γ_{i}} that p uses is also used by \overline{p} to determine another point ẋ_{γ}′_{i}, r would also force q_{m} inside Ȧ_{γ}′_{i} and promise that the new points for Ȧ_{γ}′_{i} are in W (q_{m}, k). In other words, r would force ẋ_{γ}′_{i} inside B as well. This is a contradiction as \overline{p} ⊩ ẋ_{γ}′_{i} ∉ B.

Now we are ready to give the proof of the main lemma.

Proof of Lemma 3.5. Fix J ⊆ ℚ as in the statement of the lemma and let J̇ be a ℙ_{ω_{2}}-name for J. Let M ≺ H (θ) be countable, with θ being a large enough regular cardinal, J̇ ∈ M and M containing everything that is relevant for the proof of the lemma. We now show that ℙ_{ω_{2}} ⊩ Ȧ_{δ} ⊈^{∗} J̇ for δ ∉ M.

Suppose not, there is some δ ∉ M and there is p ∈ ℙ_{ω_{2}} such that, for some n ∈ ω, p ⊩ Ȧ_{δ} \ J̇ ⊆ \{q_{0}, …, q_{n}\}. Let \overline{p} ∈ M ∩ ℙ_{ω_{2}} be as in the conclusion of Lemma 3.6. Since \overline{p} ⊩ “ℚ \ J̇ is dense in ℝ”, \overline{p} can be extended inside M to some r ∈ ℙ_{ω_{2}} ∩ M such that, for some k ∈ ω and large enough m ∈ ω,

• r ⊩ q_{k} ∉ J̇,

• q_{k} is a fresh new rational, i.e. q_{k} ∉ {q_{0}, …, q_{n}} and q_{k} is not in the range of p (0) (δ, ), and

• q_{k} ∈ W (q_{n}, m).

Then r is compatible with p, so we can find a common extension r′ such that r′ ⊩ q_{k} ∈ Ȧ_{δ} which contradicts that p ⊩ q_{k} ∉ Ȧ_{δ}.

We conclude with some remarks on Problem 291 in [Nyi90] which has already been implicitly solved. First, let us recall that a space is pseudonormal if every pair of disjoint closed sets can be separated by disjoint open sets, provided at least one of them is countable. P. Nyikos asks: Is there a pseudonormal Ψ-space of cardinality 𝔡?

10 FERNANDO HERNÁNDEZ-HERNÁNDEZ AND MICHAEL HRUŠÁK

This is equivalent to asking whether there is a λ-set of size 𝔡. For if Ψ (A) is a pseudonormal space, then A is a λ-set as subspace of P (ω) , and if X ⊆ 2^{ω} is a λ-set then Ψ (A_{X}) is a pseudonormal space (the proof of this is analogous to the proof of Proposition 2.2). A.W. Miller [Mil93, Theorem 22] showed that in the Cohen model (in which 𝔟 = ℵ_{1} and 𝔡 = ℵ_{2}), any λ-set has size ℵ_{1}.

Problem 291 also has a second part where Nyikos asks: More generally, what is the maximum cardinality of a pseudonormal Ψspace? To this we have two comments.

First, it is not true that such maximum can always be attained. The maximum exists in the absence of inaccessible cardinals, however, assuming the existence of a strongly inaccessible cardinal it is possible to construct a model of ZFC where 𝔠 is a limit cardinal, for every κ < 𝔠 there is a λ-set of size κ, yet there is no λ-set of size 𝔠. The proof of this will appear elsewhere.

The other comment is that there does not seem to be any reasonable combinatorial upper bound. This is largely due to the fact that λ-sets are preserved by c.c.c. forcing extensions. For example, it is consistent that there is a λ-set of size 𝔠 = ℵ_{2}, yet all cardinal invariants in the Cichoń diagram are equal to ℵ_{1}.

References

[Bin51] R. H. Bing. Metrization of topological spaces. Canadian J. Math., 3:175–186, 1951.

[BJZ97] Bohuslav Balcar, Thomas Jech, and Jindřich Zapletal. Semi-Cohen Boolean algebras. Ann. Pure Appl. Logic, 87(3):187–208, 1997.

[Eng89] Ryszard Engelking. General topology, volume 6 of Sigma Series in Pure Mathematics. Heldermann Verlag, Berlin, second edition, 1989. Translated from Polish by the author.

[FM80] William G. Fleissner and Arnold W. Miller. On Q sets. Proc. Amer. Math. Soc., 78(2):280–284, 1980.

[Hea64] R. W. Heath. Screenability, pointwise paracompactness, and metrization of Moore spaces. Canad. J. Math., 16:763–770, 1964.

[Kun83] Kenneth Kunen. Set theory, volume 102 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam, 1983. An introduction to independence proofs, Reprint of the 1980 original.

[Mil93] Arnold W. Miller. Special sets of reals. In Set theory of the reals (Ramat Gan, 1991), volume 6 of Israel Math. Conf. Proc., pages 415–431. Bar-Ilan Univ., Ramat Gan, 1993.

[Nyi90] Peter Nyikos. On first countable, countably compact spaces. III. The problem of obtaining separable noncompact examples. In Open problems in topology, pages 127–161. North-Holland, Amsterdam, 1990.

[Tal77] F. D. Tall. Set-theoretic consistency results and topological theorems concerning the normal Moore space conjecture and related problems. Dissertationes Math. (Rozprawy Mat.), 148:53, 1977.

[Tal84] Franklin D. Tall. Normality versus collectionwise normality. In Handbook of set-theoretic topology, pages 685–732. North-Holland, Amsterdam, 1984.

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

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

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

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