Supremum vs. Maximum: λ-sets.

Fernando Hernández-Hernández

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

Abstract

We show that, relative to the existence of an inaccessible cardinal, it is consistent that there is no λ-set of maximal size and that in the absence of inaccessible cardinals there is a λ-set of maximal size.

AMS Classification: 03E35, 54A35.

Key words: λ-sets, special subsets of reals, strongly inaccessible cardinal, Hechler forcing, isomorphism of names.

1 Introduction

K. Kuratowski introduced rarefied or λ-sets in 1933. A set X ⊆ ℝ is a λ-set if every countable subset of X is relatively G_{δ} in X. N. Lusin showed that λ-sets of cardinality ℵ_{1} always exist and F. Rothberger improved Lusin’s result by showing that ZFC suffices to show that there is a λ-set of cardinality 𝔟 (see Theorem 2.4).

The motivation for this paper comes from Problem 291 in [Nyi90] which asks about pseudonormal Ψ-spaces. 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. 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

1 The author gratefully acknowledges partial research support from PAPIIT grant IN106705.

E-mail: fernando@matmor.unam.mx

Preprint submitted to Elsevier Preprint

form {A} ∪ A \ F, where F ⊆ ω is finite. The space Ψ (A) is always a first countable, separable, locally compact Moore space. It is known that if Ψ (A) is a pseudonormal space, then A is a λ-set as subspace of P (ω) , and if there is a λ-set of cardinality κ then there is an almost disjoint family A of cardinality κ such that Ψ (A) is a pseudonormal space (see [HH05] Proposition 2.2). F.B. Jones used λ-sets to construct a pseudonormal Moore space which served as the inspiration for nonmetrizable normal Moore spaces based on Q-sets. For more on λ-sets consult [Mil84] and [Mil93].

Problem 291 in [Nyi90] has two parts; in the first one P. Nyikos asked: Is there a pseudonormal Ψ-space of cardinality 𝔡? The first part of Problem 291 is equivalent to asking whether there is a λ-set of size 𝔡. A.W. Miller [Mil93, Theorem 22] showed that in the Cohen model (in which 𝔟 = ℵ_{1} and 𝔡 = ℵ_{2}), any λ-set of reals has size ℵ_{1}. Thus in the Cohen model the answer is “no”. Of course, in a model of 𝔟 = 𝔡 the answer is “yes”, see Theorem 2.4.

Problem 291 has second part where Nyikos asked: More generally, what is the maximum cardinality of a pseudonormal Ψ-space? This note is meant to show that the second part of Problem 291 is in general ill-posed. The results in this paper were announced in [HH05]. We show that: (1) In the absence of inaccessible cardinals, there is a λ-set of maximal cardinality and (2) assuming the existence of a strongly inaccessible cardinal it is consistent that: 𝔠 is a limit cardinal and for every κ < 𝔠 there is a λ-set of size κ, yet there is no λ-set of size 𝔠. That is, the maximum cardinality of a λ-set may not be attained.

Our terminology is mostly standard: 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. The set ω^{ω} is equipped with the product topology, that is the topology with basic open sets of the form [s] = {f ∈ ω^{ω} : s ⊆ f}, where s ∈ ω^{<ω}. See [Kun83] for undefined set theoretical notions.

2 The results.

The first result we are presenting shows that regarding the cardinalities of λsets, sup = max if there are no inaccessible cardinals below the continuum. In other words, the maximum of the cardinalities of λ-sets is attained, this shows that the second part of Problem 291 makes sense in the absence of inaccessible cardinals.

Theorem 2.1 If κ is a singular cardinal and there is a λ-set of cardinality μ for every cardinal μ < κ, then there is a λ-set of cardinality κ.

Proof. Let cf (κ) = μ and consider an increasing sequence ⟨κ_{α} : α < μ⟩ which is cofinal in κ. Choose a λ-set A ⊆ ℝ of cardinality μ and for each α < μ choose a λ-set X_{α} ⊆ ℝ of cardinality κ_{α}. Enumerate A as {x_{α} : α < μ}. We claim that X = _{⋃α<μ} {x_{α}} × X_{α} is a λ-set of cardinality κ. That |X| = κ is clear. If Y ∈ [X]^{ℵ_{0}} then there is some A_{0} = {x_{α_{n}} : n ∈ ω} ⊆ A such that Y is contained in _{⋃n∈ω} {x_{α_{n}}} × X_{α_{n}} and moreover Y_{n} = Y ∩ ({x_{α_{n}}} × X_{α_{n}}) is at most countable for every n ∈ ω. Since {x_{α_{n}} : n ∈ ω} is a G_{δ} subset of A, there are open sets B_{m}, for m ∈ ω, such that _{⋃n∈ω} {x_{α_{n}}} × X_{α_{n}} = _{⋂m∈ω}B_{m}.

On the other hand, Y_{m} is a G_{δ} subset of {x_{α_{m}}} × X_{α_{m}} and {x_{α_{m}}} × X_{α_{m}} is a G_{δ} subset of X as well, thus Y_{m} is a G_{δ} subset of X. Therefore we can fix open subsets U_{m,n} of X such that

(1) U_{m,n+1} ⊆ U_{m,n} for all n ∈ ω,

(2) U_{m,n} ∩ ({x_{α_{k}}} × X_{α_{k}}) = ∅ for all k < m, and

(3) Y_{m} = _{⋂n∈ω}U_{m,n}.

Let W_{n} = (_{⋃m∈ω}U_{m,n}) ∩ B_{n}, for n ∈ ω. Then W_{n} is open subset of X and it can be easily verified that Y = _{⋂n∈ω}W_{n}.

Corollary 2.2 Let Λ = sup {|X| : X is a λ-set}. If Λ is not an inaccessible cardinal then there is a λ-set X with |X| = Λ.

Now we turn to address the case where inaccessible cardinals exist. Here we see that it is consistent that the maximum of the cardinalities of λ-sets is never attained. It will be easier to change the real numbers ℝ by ω^{ω} and show the result for subsets of ω^{ω}. Since every λ-set is zero-dimensional, there is no difference to consider λ-sets in the real line or in the irrationals.

Theorem 2.3 If there is a strongly inaccessible cardinal κ then there is a model of ZFC in which there is a λ-set of every cardinality below 𝔠, yet there is no λ-set of size 𝔠.

The main idea to prove the theorem is to make the continuum a limit cardinal and to add a λ-set of each cardinal below 𝔠. The natural forcing to add a λ-set of cardinality μ is the finite support iteration ℙ_{μ} of length μ of Hechler forcing 𝔻. This is due to the fact that ℙ_{μ} adds a well-ordered family of functions in ω^{ω} of size μ, the dominating (or Hechler) reals. Thus ℙ_{μ} forces 𝔟 ≥ μ. Then the existence of a λ-set of cardinality μ follows from the following theorem. For a proof of it see [Mil93].

Theorem 2.4 (Rothberger) There exists a λ-set of cardinality 𝔟.

A second key ingredient of the idea is the following simple lemma. We give a proof for the sake of completeness.

Lemma 2.5 (Folklore) If X is a λ-set and ℙ is a c.c.c. forcing notion then ℙ preserves the fact that X is a λ-set.

Proof. Fix X and ℙ as in the statement of the Lemma. It is well-known that the cardinality of X is preserved. If Ȧ is a ℙ-name for a countable subset of X, then there is a countable B ⊆ X in the ground model such that ℙ ⊩ Ȧ ⊆ B. Since X is assumed a λ-set in the ground model, there are open sets U_{n} ⊆ X, n ∈ ω, such that U_{n} = B. Then working in the extension V [G], where G is a ℙ-generic filter over V, we can fix an enumeration {b_{n} : n ∈ ω} of B \ A and thus write A = _{⋂n∈ω}U_{n} \ {b_{n}}. Therefore A is a G_{δ} subset of X, in V [G] .

Thus we can use iterations of Hechler forcing to add λ-sets of every cardinality below the new 𝔠 and we can preserve those λ-sets if we use a c.c.c. forcing notion which puts together (in an independent manner) all the iterations that we are going to use. The forcing notion ℙ that we will use is defined next. Before doing that, recall that conditions in Hechler forcing 𝔻 are of the form ⟨s, f⟩ where s ∈ ω^{<ω} and f ∈ ω^{ω}, and the ordering is ⟨s, f⟩ ≤ ⟨t, g⟩ if and only if

(1) s ⊇ t,

(2) f ≥ g and

(3) (∀n ∈ dom (s) \ dom (t)) (s(n) ≥ g (n)).

Definition 2.6 Let κ be a strongly inaccessible cardinal and let {μ_{α} : α < κ} be a fix enumeration of all regular cardinals below κ. Then define ℙ by p ∈ ℙ if and only if p ∈ ∏_{α<κ} ℙ_{μ_{α}} and supp (p) is finite, where ℙ_{μ_{α}} is a finite support iteration of length μ_{α} of Hechler forcing 𝔻.

Proof of Theorem 2.3. We use the forcing ℙ just described above. It is clear that ℙ ⊩ 𝔠 = κ. That ℙ is a c.c.c. forcing notion follows from the facts that finite support iteration of forcing notions with precaliber^{2} ℵ_{1} has precaliber ℵ_{1} and that a finite support product of forcing notions with precaliber ℵ_{1} has precaliber ℵ_{1} (see [BJ95, Chap. 1]).

On the other hand, to show that there is a λ-set of cardinality μ for every cardinal μ < κ, fix an arbitrary ℙ-generic filter G over V. As ℙ is a product of forcing notions, G can be seen as the product of G_{α}’s where G_{α} is ℙ_{μ_{α}}-generic over V. Moreover, given any cardinal μ < κ, there is some α < κ such that μ < μ_{α} and

V [G] = V[_{∏α<κ}G_{α}] = V [G_{α}] [_{∏β∈κ\{α}}G_{β}].

2 Recall that a forcing notion ℙ has precaliber κ if for every subset A of size κ there is B ⊆ A of size κ such that B is centred.

Since ℙ_{α} forces 𝔟 = μ_{α}, by Theorem 2.4 it follows that there is a λ-set X of cardinality μ_{α} in V [G_{μ}] and since the next part of the extension is given by a c.c.c. extension, Lemma 2.5 implies that X remains a λ-set in V [G]. Then that there is a λ-set in V [G] of cardinality μ follows from the fact that subsets of λ-sets are λ-sets themselves.

It remains to show that there is no λ-set of cardinality 𝔠 in the generic extension V [G]. This will follow by the next lemma.

Lemma 2.7 Let ℙ be the forcing notion of Definition 2.6 and let Y = {ḟ_{ξ} : ξ < κ} be a set of ℙ-names for reals which are forced by ℙ to be pairwise distinct. If G is a ℙ-generic filter over V, then {f_{ξ} : ξ < κ} is not a λ-set in V [G] .

We will spend the next section to give a proof of this lemma in a detailed way. The idea for the proof is to take a countable elementary submodel M of some big enough H (ϑ) such that ℙ, Y ∈ M, then we shall show that whenever Ẇ is a ℙ-name for a G_{δ} subset of ω^{ω}, there are κ-many ξ’s such that ℙ ⊩ ḟ_{ξ} ∈ Ẇ. In order to do this we will need to introduce several notions among which a notion of similarity between names for functions in ω^{ω} will be crucial. There the fact that Hechler forcing is definable will also be very important.

3 Proof the main lemma.

In this concluding section we are going to use an isomorphism of names type argument to give a proof of the main lemma and hence to complete the proof of Theorem 2.3. The use of arguments involving isomorphic names has been around in the literature for quite a long time without been explicitly mention. See for example [Hec72]. At the beginning of Section 3 in [Bre02] a particular argument of isomorphic names for Cohen forcing is explained in some detail. The main result in [HH05] also uses this type of argument.

A nice ℙ-name ḟ for a function in ω^{ω} will be a ℙ-name of the form

⋃n∈ω {⟨p, ⟨n, i_{n}^{p}⟩⟩ : p ∈ A_{n} & i_{n}^{p} ∈ ω} ,

where:

(1) A_{n} is a maximal antichain in ℙ for every n ∈ ω, and

(2) ⟨p, ⟨n, i_{n}^{p}⟩⟩ ∈ ḟ ⇒ p ⊩ ḟ (n) = i_{n}^{p}.

It is easy (see [Kun83, 5.12]) to show that whenever ℙ ⊩ ḟ ∈ ω^{ω} there is a nice name ḟ′ such that ℙ ⊩ ḟ = ḟ′; that is, every function in the generic extension has a nice ℙ-name.

Now, let ℙ_{α} be a finite support iteration of length α of Hechler forcing 𝔻. By induction on α ≥ 1 we define when p is a good condition in ℙ_{α}. Firstly, for α = 1, every condition p in 𝔻 is a good condition. Suppose we have already defined good conditions in ℙ_{α}. If p ∈ ℙ_{α} ∗ 𝔻̇, then p is a good condition if p ↾ α is a good condition in ℙ_{α} and

p ↾ α ⊩ p (α) = ⟨s, ḟ⟩,

where s ∈ ω^{<ω} and ḟ is a nice ℙ_{α}-name for a function in ω^{ω} such that conditions p appearing in the nice ℙ_{α}-name ḟ are all good conditions in ℙ_{α}. If α is a limit ordinal and we have defined good conditions in ℙ_{β} for all β < α, then p ∈ ℙ_{α} is a good condition if p ↾ β is a good condition in ℙ_{β}, where β = max (supp (p)) + 1.

Lemma 3.1 For every α < κ, ℙ_{α} has a dense subset of good conditions.

Proof. We can prove this lemma by induction on α. Since ℙ_{α} is a finite support iteration, the limit steps of the induction are easy. To show the successor steps, assume ℙ_{α} has a dense subset of good conditions and let p be an arbitrary condition in ℙ_{α} ∗ 𝔻̇. Then p ↾ α ⊩ p (α) = ⟨ṡ, ḟ⟩ ∈ 𝔻̇ and using the density of good conditions in ℙ_{α} there is a good condition p̃ ↾ α ≤ p ↾ α and a nice ℙ_{α}-name ḟ′ such that p̃ ↾ α ⊩ ṡ = s ∈ ω^{<ω}, p̃ ↾ α ⊩ ḟ = ḟ′ and every condition q ∈ ℙ_{α} which appears in ḟ′ is a good condition in ℙ_{α}. Put p̃ = (p̃ ↾ α) ∪ {⟨α, ⟨s, ḟ′⟩⟩}. Then p̃ is a good condition in ℙ_{α} ∗ 𝔻̇ and p̃ ≤ p.

Now we extend the definition of good conditions to conditions in the whole forcing ℙ in the natural way: A condition p ∈ ℙ is a good condition if p (α) ∈ ℙ_{μ_{α}} is a good condition for every α < κ. The next lemma follows directly from Lemma 3.1.

Lemma 3.2 ℙ has a dense subset of good conditions.

As ℙ is a c.c.c. forcing notion, every function from ω into ω in a generic extension by ℙ has a nice ℙ-name formed by countable antichains of good conditions. If ḟ is a ℙ-name for an element of ω^{ω}, we say that ḟ is a good ℙ-name for a function in ω^{ω} if ḟ is a nice name and ⟨p, ⟨n, k⟩⟩ ∈ ḟ implies that p is a good condition in ℙ.

Again by induction we define the transitive support of good conditions in ℙ_{α}. For p ∈ 𝔻 we define the transitive support of p to be tr supp (p) = {0}. Suppose we have defined the transitive support for good conditions in ℙ_{α}, then for a good condition p ∈ ℙ_{α+1}, the transitive support of p is tr supp (p) = tr supp (p ↾ α) if p ↾ α ⊩ p (α) = 1_{𝔻}, and

tr supp (p) = tr supp (p ↾ α) ∪ {α} ∪ tr supp(ḟ)

if p ↾ α ⊮ p (α) = 1_{𝔻} and p ↾ α ⊩ p (α) = ⟨s, ḟ⟩, where tr supp(ḟ) = ⋃ {tr supp (q) : q ∈ A_{n}} with ⟨A_{n} : n ∈ ω⟩ being the sequence of maximal antichains in ℙ_{α} used to define the good ℙ_{α}-name ḟ. If α is a limit ordinal and we have defined the transitive support for good conditions in ℙ_{β} for all β < α, then for a good condition p ∈ ℙ_{α} we define the transitive support of p by

tr supp (p) = tr supp (p ↾ β) ,

where β = max (supp (p)) + 1.

For technical reasons, we assume, without loss of generality, that the transitive support of every good condition contains 0.

The motivation to introduce the transitive support for conditions is to define isomorphic good ℙ-names for functions in ω^{ω}. Before doing that, we need to extend the notion of transitive support for conditions in the whole forcing ℙ. This is easy as we can do it coordinatewise: If p ∈ ℙ is a good condition then the transitive support of p is given by

tr supp (p) = ⋃ {{μ_{α}} × tr supp (p (α)) : α ∈ supp (p)} .

It is worth observing that tr supp (p) is a countable subset of κ × κ and it is meant to give us a template of the parts of our forcing used to form a good condition. The definition of the transitive support of a good ℙ-name for a function in ω^{ω} has implicitly been given and it might be thought as a template or guide in making that good ℙ-name accurately over the whole forcing ℙ. If ḟ is a good ℙ-name for a function in ω^{ω}, then the support of ḟ is defined by

supp(ḟ) = ⋃ {supp (p) : (∃n, k ∈ ω) (⟨p, ⟨n, k⟩⟩ ∈ ḟ)}.

The transitive support of ḟ is simply

tr supp(ḟ) = ⋃ {{μ_{α}} × tr supp (p (α)) : α ∈ supp(ḟ)} .

Again by c.c.c.-ness of ℙ supp(ḟ) is a countable subset of κ for every good ℙ-name for a function from ω into ω and tr supp(ḟ) is a countable subset of κ × κ.

The definition of an equivalence relation between good ℙ-names for functions will be given in terms of equivalence between templates. First we detail the notion of isomorphic templates.^{3}

Definition 3.3 (1) A simple template is a countable subset of κ which (for technical reasons) contains 0.

(2) Two simple templates A and B are isomorphic if there is an order preserving bijection ψ : A → B such that:

(2a) if β ∈ A and for some γ ∈ ω_{1}, the next point of A is β + γ, then

ψ (β + γ) = ψ (β) + γ, and (2b) if β ∈ A and the next point γ of A is such that β + ω_{1} ≤ γ, then

ψ (β) + ω_{1} ≤ ψ (γ).

(3) A template T is a subset of κ × κ of the form T = ⋃ {{α} × A_{α} : α ∈ S} for an at most countable S ⊆ κ and A_{α} being a simple template for every α ∈ S.

(4) Two templates T = ⋃ {{α} × A_{α} : α ∈ S} and T′ = ⋃ {{α} × A′_{α} : α ∈ S′} are isomorphic if there exists a bijection φ : S → S′ such that:

(4a) φ (α) = α for all α ∈ S ∩ S′, (4b) for all α ∈ S, the simple templates A_{α} and A_{φ(α)} are isomorphic witnessed

by a bijection ψ_{α} : A_{α} → A′_{φ(α)}, and (4c) in case φ (α) = α, then A_{α} = A_{φ(α)} and ψ_{α} is the identity function.

(5) An isomorphism Φ between templates T and T′ is a bijection Φ : T → T′ which induces bijections φ : S → S′ and ψ_{α} : A_{α} → A_{φ(α)} which witness that T and T′ are isomorphic templates. We denote Φ(α, β) = ⟨φ (α) , ψ_{α} (β)⟩ for ⟨α, β⟩ ∈ T.

It is easy to see that this notion of isomorphism between templates defines an equivalence relation on the family of templates and that every template is isomorphic to some template contained in ω × ω_{2} and, therefore, there are only ℵ_{2}-many distinct types of templates. Now we will use isomorphism of templates to translate information between names. Note that if ḟ is a good ℙ-name for a function in ω^{ω} then tr supp(ḟ) is a template. We need to make precise what we understand by translating information between conditions.

Let ψ : A → B by an order-preserving function between simple templates A and B. For conditions p ∈ ℙ_{β} with tr supp (p) ⊆ A, define a new condition \overline{ψ} (p) by induction on β ≥ 1. If p ∈ ℙ_{1}, then \overline{ψ} (p) (0) = p (0). If we have defined \overline{ψ} for all good conditions in ℙ_{β} and p ∈ ℙ_{β+1} is a good condition then:

(1) If β ∉ A, then \overline{ψ} (p) = \overline{ψ} (p ↾ β), or

3 Here the word “template” has no relationship with the same word used in the theory of iterations along a template.

(2) if β ∈ A and p ↾ β ⊩ p (β) = ⟨s, ḟ⟩, then \overline{ψ} (p) ↾ ψ (β) = \overline{ψ} (p ↾ β) and \overline{ψ} (p) ↾ ψ (β) ⊩ \overline{ψ} (p) (ψ (β)) = ⟨s, \overline{ψ}(ḟ)⟩, where

\overline{ψ}(ḟ) = _{⋃n∈ω} \{⟨\overline{ψ} (q) , ⟨n, k_{n}^{q}⟩⟩ : q ∈ A_{n} & k_{n}^{q} ∈ ω\}

if ḟ = _{⋃n∈ω} {⟨q, ⟨n, k_{n}^{q}⟩⟩ : q ∈ A_{n} & k_{n}^{q} ∈ ω}.

And if β ∈ A is a limit ordinal, we have defined \overline{ψ} for all good conditions in ℙ_{γ}, for all γ < β, and p ∈ ℙ_{β} is a good condition with γ_{0} = max (supp (p)) ∈ A then

\overline{ψ} (p) = \overline{ψ} (p ↾ ψ (γ_{0})) ∪ \{⟨γ, 1𝔻̇⟩ : ψ (γ_{0}) ≤ γ < ψ (β)\} .

It is not hard to check by induction that for every good condition p ∈ ℙ_{β} such that tr supp (p) ⊆ A the following equality holds

ψ [tr supp (p)] = tr supp (\overline{ψ} (p)).

Thus if p ∈ ℙ_{β} is a good condition and ψ is an isomorphism between tr supp (p) and some simple template A, then A = tr supp (\overline{ψ} (p)) .

Definition 3.4 Let Φ : T → T be an isomorphism between templates T and T′ where Φ (α, β) = (φ (α) , ψ_{α} (β)) for all ⟨α, β⟩ ∈ T. Let p ∈ ℙ be a good condition with tr supp (p) contained in T. The translated condition Φ (p) is defined by Φ (p) (α) = \overline{ψ_{α}} (p (α)) for all α < κ such that ⟨α, 0⟩ ∈ T and Φ (p) (α) = 1_{ℙ_{μα}} otherwise. Where \overline{ψ_{α}} is defined before, for every α.

The idea behind the previous definition is simple: Φ must translate p into Φ (p) using the templates as guides for making that translation accurate changing only the corresponding coordinates of conditions in the iterations ℙ_{μ_{α}}’s used to form the good ℙ-condition p. The function Φ can be thought as an isomorphism between the suborders of ℙ induced by the respective templates T and T′. The isomorphism Φ cannot be a regular embedding though. To avoid too many indexes we will write ψ_{α} (p) instead of ψ_{α} (p (α)). Finally, the main definition of the section.

Definition 3.5 Let ḟ = _{⋃n∈ω} {⟨p, ⟨n, i⟩⟩ : p ∈ A_{n}} and ġ = _{⋃n∈ω}{⟨q, ⟨n, i⟩⟩ : q ∈ B_{n}} be good ℙ-names for functions from ω into ω, we will say that they are isomorphic (ḟ ≈ ġ) if:

(1) The templates tr supp(ḟ) and tr supp(ġ) are isomorphic witnessed by an isomorphism Φ : tr supp(ḟ) → tr supp(ġ) and

(2) if ⟨p, ⟨n, i⟩⟩ ∈ ḟ, and ⟨q, ⟨n, j⟩⟩ ∈ ġ and q = Φ (p), then i = j.

We are now ready to finish the proof of the main lemma.

Proof of Lemma 2.7. Suppose that {ḟ_{ξ} : ξ < κ} are good ℙ-names for κmany distinct reals. We will show that if G is a ℙ-generic filter over V, then X = {f_{ξ} : ξ < κ} is not a λ-set in V [G]. Since subsets of λ-sets are λ-sets, to reach our goal we can thin out the original set in order to find a subset which is not a λ-set.

Because κ is a regular cardinal and supp(ḟ_{ξ}) is a countable subset of κ, for every ξ < κ, the family all supp(ḟ_{ξ}) necessarily has cardinality κ; otherwise, ⋃ξ<κ supp(ḟ_{ξ}) would be a bounded set in κ and using that κ is strongly inaccessible there are not enough countable subsets of a bounded subset of κ, therefore there are κ-many ξ’s for which the supports of ḟ_{ξ} coincide. However, ℙ ↾ A cannot code κ distinct functions for any countable A ⊆ κ. Henceforth, we assume that supp(ḟ_{ξ}) are all distinct.

As κ is a regular cardinal, we can also apply the Δ-system lemma to the family of supports of those good ℙ-names ḟ_{ξ}’s. Thus we assume that {supp(ḟ_{ξ}) : ξ < κ} forms Δ-system with root R. Moreover, due to R is at most countable, there must be κ-many ξ’s such that the ℙ-names ḟ_{ξ} are exactly the same name over R. Since there are only ℵ_{2}-many possible types for those ḟ_{ξ} good ℙ-names we can assume further that any two ḟ_{ξ} and ḟ_{η} are isomorphic names for ξ, η < κ.

Let M be a countable elementary submodel of H (ϑ) for some large enough regular cardinal ϑ such that ℙ and {ḟ_{ξ} : ξ < κ} are elements of M. Let Ẇ be a ℙ-name for a G_{δ}-subset of ω^{ω} such that ℙ ⊩ X ∩ Ẇ ⊇ {ḟ_{ξ} : ξ ∈ M ∩ κ}. Fix ℙ-names Ẇ_{m} for open subsets of ω^{ω}, m ∈ ω, such that ℙ ⊩ Ẇ = _{⋂m∈ω}Ẇ_{m}. Consider now another countable elementary submodel N of H (ϑ) containing everything that is relevant for the proof and such that M ∈ N and the ℙnames Ẇ, Ẇ_{n} are all elements of N. Inside N, we can fix an enumeration {ξ_{n} : n ∈ ω} of M ∩ κ.

Working inside N, for every m ∈ ω, we can find a ℙ-name ġ_{m} ∈ N for a function from ω into ω such that

ℙ ⊩ [ḟ_{ξ_{n}} ↾ ġ_{m} (n)] ⊆ Ẇ_{m},

for every n ∈ ω.

Fix ℙ-names U̇, U̇_{m} ∈ N, for m ∈ ω, such that ℙ ⊩ U̇_{m} = _{⋃n∈ω}[ḟ_{ξ_{n}} ↾ ġ_{m} (n)] and ℙ ⊩ U̇ = _{⋂m∈ω}U̇_{m}. Then U is forced to be a G_{δ} subset of W. To finish the proof we prove the following claim:

Claim 1 If ξ ∈ κ \ N and supp(ḟ_{ξ}) ∩ N = ∅, then ℙ ⊩ ḟ_{ξ} ∈ U̇.

In order to establish the claim let ξ be any such ordinal outside N and fix m ∈ ω. It suffices to show that ḟ_{ξ} is forced by a dense set of conditions to be in U̇_{m}. So, fix a condition r ∈ ℙ. Without loss of generality, assume r is a good ℙ-condition.

Firstly, recall that ḟ_{ξ} is a good ℙ-name for a function in ω^{ω}, say

ḟ_{ξ} = _{⋃i∈ω} {⟨p, ⟨i, j_{i}^{p}⟩⟩ : p ∈ A_{i} & j_{i}^{p} ∈ ω} .

We are assuming that {supp(ḟ_{ξ}) : ξ < κ} forms a Δ-system, by elementarity we can assume that its root R ∈ M, hence R ⊆ M. As supp (r) is finite there is an n ∈ ω such that

supp (r) ∩ (supp(ḟ_{ξ_{n}}) \ R) = ∅.

Suppose that the good ℙ-name ḟ_{ξ_{n}} is_{⋃i∈ω} {⟨q, ⟨i, j_{i}^{q}⟩⟩ : q ∈ A′_{i} & j_{i}^{q} ∈ ω}. Since we are assuming that ḟ_{ξ} and ḟ_{ξ_{n}} are isomorphic names, there is a bijection Φ : tr supp(ḟ_{ξ}) → tr supp(ḟ_{ξ_{n}}) witnessing that ḟ_{ξ} ≈ ḟ_{ξ_{n}}. By definition, Φ induces bijective functions ψ_{α} from the column α of tr supp(ḟ_{ξ}) onto the column φ (α) of tr supp(ḟ_{ξ_{n}}). For α ∈ supp(ḟ_{ξ}) ∩ supp (r), the function ψ_{α} can be extended to include tr supp (r (α)) in its domain preserving the ordering between ordinals.^{4} Denote that extension of ψ_{α} again by ψ_{α}. Define a new condition r_{0} ∈ ℙ by

r_{0} (α′) = \{ \begin{matrix}r (α′) & if α′ ∈ N \ (supp(ḟ_{ξ_{n}}) \ R) , \\ \overline{ψ_{α}} (r) & if α′ = φ (α) & α ∈ supp(ḟ_{ξ}) ∩ supp (r) , \\ 1_{ℙ_{μ_{α′}}} & otherwise, \end{matrix}

for all α′ < κ. Then r_{0} ≤ r ↾ N. Now extend r_{0} to some condition r_{1} such that supp (r_{1}) ⊆ N and

(1) r_{1} decides the value of ġ_{m} (n); that is, r_{1} ⊩ ġ_{m} (n) = k,

(2) r_{1} decides ḟ_{ξ_{n}} ↾ k; that is, r_{1} ⊩ f_{ξ_{n}} ↾ k = t for some t : k → ω,

(3) r_{1} is a good condition.

Once again, for each φ (α) ∈ supp(ḟ_{ξ_{n}}) ∩ supp (r_{1}), the function ψ_{α} can be extended in such a way that tr supp (r_{1} (φ (α))) is included in its range and the ordering between ordinals is still preserved. One more time, we denote

4 It is here that clauses (2a) and (2b) of Definition 3.3 play their role in the definition of isomorphic names. Note that the extension of ψ_{α} may not be an isomorphism of simple templates though.

that extension again by ψ_{α}. Define another new condition r_{2} ∈ ℙ by

r_{2} (α) = \{ \begin{matrix}r_{1} (α) & if α ∈ N, \\ \overline{ψ_{α}^{−1}} (r_{1}) & if φ (α) ∈ supp(ḟ_{ξ_{n}}) ∩ supp (r_{1}) , \\ r (α) & otherwise, \end{matrix}

for all α < κ. Then the condition r_{2} has the following properties:

(1) r_{2} ≤ r_{1} and r_{2} ≤ r,

(2) r_{2} ⊩ ḟ_{ξ} ↾ k = ḟ_{ξ_{α}} ↾ k, and

(3) r_{2} ⊩ ḟ_{ξ} ∈ U_{m}.

This completes the proof of Lemma 2.7 and hence it completes the proof of Theorem 2.3 as well.

At first glance, the fact that κ is a strongly inaccessible cardinal does not seem to be important and one could think that the method might work as well for a singular cardinal κ. Of course, we know that it cannot be true because of Theorem 2.1. We can even give a simple ℙ-name for a λ-set of maximal cardinality. Suppose that μ = cf (κ) and consider a strictly increasing sequence ⟨κ_{α} : α < μ⟩ cofinal in κ and such that κ_{α} > μ for all α < μ. Let ḟ_{α,β} be a ℙ-name for the β-th generic real added by ℙ_{α} (β < α) and let ḣ_{α,β,γ,δ} be such that ℙ ⊩ ḣ_{α,β,γ,δ} = max{ḟ_{α,γ}, ḟ_{β,δ}}, for all α, γ < μ, κ_{α} < β < κ_{α+1} and δ < κ_{α+1}. Then supp(ḣ_{α,β,γ,δ}) = {γ, β} and by genericity we are considering 𝔠 = (κ)^{V} distinct functions. However, there is no Δ-system of κ-many distinct supports.

Acknowledgement. I am very grateful to Michael Hrušák for suggesting the research here contained, sharing his insight and clever ideas not only about what is written here but always.

References

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

[Bre02] Jörg Brendle, Mad families and iteration theory, Logic and algebra, Contemp. Math., vol. 302, Amer. Math. Soc., Providence, RI, 2002, pp. 1– 31.

[Hec72] Stephen H. Hechler, Short complete nested sequences in βN\N and small maximal almost-disjoint families, General Topology and Appl. 2 (1972),

139–149.

[HH05] Fernando Hernández-Hernández and Michael Hrušák, Q-sets and normality of Ψ-spaces, Topology Proceedings 29 (2005), no. 1, 155–165.

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

[Mil84] Arnold W. Miller, Special subsets of the real line, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 201–233.

[Mil93] , Special sets of reals, Set theory of the reals (Ramat Gan, 1991), Israel Math. Conf. Proc., vol. 6, Bar-Ilan Univ., Ramat Gan, 1993, pp. 415– 431.

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