INTERSECTION PROPERTIES OF OPEN SETS, II

I. JUHÁSZ, ZS. NAGY, L. SOUKUP, AND Z. SZENTMIKLÓSSY

Abstract. A topological space is called P_{2} (P_{3}, P_{<ω}) if and only if it does not contain two (three, finitely many) uncountable open sets with empty intersection. We show that (i) there are 0-dimensional P_{<ω} spaces of size 2^{ω}, (ii) there are compact P_{<ω} spaces of size ω_{1}, (iii) the existence of a Ψ-like examples for (ii) is independent of ZFC, (iv) it is consistent that 2^{ω} is as large as you wish but every first countable (and so every compact) P_{2} space has cardinality ≤ ω_{1}.

1. Introduction

In this paper we continue the investigations started in [4]. There P_{2} spaces, i.e. spaces having no two uncountable disjoint open subsets, were considered. We solve some problems which were left open in that paper and strengthen some of its results. First we introduce some strengthenings of notion P_{2}.

Definition 1.1. A topological space X has property P_{n}, where n is a natural number, if it is Hausdorff and given open sets U_{0}, U_{1}, …, U_{n−1} with ∩{U_{i} : i < n} = ∅ we have |U_{i}| ≤ ω for some i < n. We say X is P_{<ω} if and only if it is P_{n} for each n < ω. The space X is called strongly P_{2} provided the intersection of two uncountable open sets is uncountable.

Clearly P_{m} spaces are P_{n} for n < m and strongly P_{2} spaces have property P_{<ω}.

In section 2 we construct P_{<ω}-spaces: a 0-dimensional one of size 2^{ω} and two locally compact, first countable examples of size ω_{1}. One of the ZFC constructions of locally compact P_{<ω} spaces is due to S. Shelah [9] and it is included here with his kind permission.

1991 Mathematics Subject Classification. 54A25,03E35.

Key words and phrases. almost disjoint families, P_{2} spaces, Luzin gap.

The third author was supported by DFG (grant Ko 490/7-1) and Magyar Tudományért Alapítvány.

The preparation of this paper was supported by the Hungarian National Foundation for Scientific Research grant no. l908.

2 I. JUHÁSZ, ZS. NAGY, L. SOUKUP, AND Z. SZENTMIKLÓSSY

In section 3 we will see why the construction of a compact P_{3} space can be expected to be much harder than that of a P_{2} one: a Ψ-like example (see the definition 1.2 below), which worked in the P_{2} case, can not be constructed for the P_{3} case in ZFC. On the other hand we show (corollary 3.10) that it is consistent that 2^{ω} is as large as you wish and there is a Ψ-like P_{<ω}-space of size 2^{ω}.

Definition 1.2. (1) Given an almost disjoint family A ⊂ [ω]^{ω} we define the topological space X[A] as follows: its underlying set is ω ∪A, the elements of ω are isolated and the neighborhood base of A ∈ A in X[A] is B_{A} = {A ∪ {A} \ n : n ∈ ω}. (2) A topological space is Ψlike if and only if it is homeomorphic to X[A] for some almost disjoint A ⊂ [ω]^{ω}.

It is straightforward that Ψ-like spaces are locally compact, first countable and 0-dimensional.

In section 4 we present the proof of a partition theorem below the continuum which was used in [4] to show the it is consistent that 2^{ω} is big but every first countable (and so every compact) P_{2} space has cardinality ≤ ω_{2}.

Finally, in section 5 we strengthen this result (and solve problem 10 of [4]) by showing that it is consistent that 2^{ω} is as large as you wish but every first countable (and so every compact) P_{2} space is of size ≤ ω_{1}.

We use standard topological notation and terminology throughout, cf [6].

We would like to thank for the referee for his helpful suggestions and comments.

2. ZFC constructions of P_{<ω}-spaces

Proposition 2.1. There are 0-dimensional P_{<ω} spaces of size 2^{ω}.

First proof of proposition 2.1. We show that the space X from [4, theorem 3] is in fact P_{<ω}. To start with, we recall its definition. Fix an independent family F ⊂ [ω]^{ω} of size 2^{ω}. The underlying set of X will be ω ∪ F, where the elements of ω are isolated. The neighborhood base of F ∈ F will be B_{F} = {V_{F} (G) : G ∈ [F \ {F }]^{<ω}}, where V_{F} (G) = {F } ∪ F \ ∪G. If ⋃ F = ω then X will be 0-dimensional and T_{2}.

Let U_{0}, …, U_{n−1} be uncountable open subsets of X. For each i < n we can find {F_{ν}^{i} : ν < ω_{1}} ⊂ F and {G_{ν}^{i} : ν < ω_{1}} ⊂ [F]^{<ω} such that F_{ν}^{i} ∉ G_{ν}^{i}, the F_{ν}^{i} are all distinct and V_{F_{ν}^{i}}(G_{ν}^{i}) ⊂ U_{i}.

Define the set mapping f : ω_{1} ⟶ [ω_{1}]^{<ω} by the stipulation

f(ν) = {μ : ∃j < n F_{μ}^{j} ∈ ⋃_{i<n} G_{ν}^{i}}.

Since the F_{μ}^{j} are different we have |f(ν)| ≤ n| ⋃_{i<n} G_{ν}^{i}|. By Hajnal’s theorem on set mappings ([3]) there is an uncountable f-free subset I of ω_{1}. Let ν_{0}, …, ν_{n−1} be different elements of I. Then {F_{ν_{j}}^{j} : j < n} ∩ ⋃_{i<n} G_{ν_{i}}^{i} = ∅, so

∅ ≠ ⋂_{j<n} F_{ν_{j}}^{j} \ ⋂( ⋃_{i<n} G_{ν_{i}}^{i}) = ⋂_{i<n} V_{F_{ν_{i}}^{i}}(G_{ν_{i}}^{i}) ∩ ω ⊂ ⋂_{i<n} U_{i},

which completes the proof.

Second proof of proposition 2.1. Consider the space D∪{p} where |D| = 2^{ω}, the elements of D are isolated, and the co-countable subsets of D forms the neighbourhood base of p.

Let us recall the following theorem [8, theorem 4.4.4].

Theorem . If X is a P -space (i.e. the intersection of countable many open sets is open) with w(X) ≤ 2^{ω} then X can be embedded into βω\ω.

According to this theorem the space D ∪ {p} can be regarded as a subspace of βω \ ω. Consider the subspace Y = ω ∪ D of βω. If U is an uncountable open subset of βω with |U ∩ Y | > ω then |U ∩ D| > ω, so p ∈ \overline{U ∩ D}. But ω is dense in βω, so U ∩D ⊂ \overline{U ∩ ω}. Hence p ∈ \overline{U ∩ ω}, i.e., U ∩ ω ∈ p. (Remember that the elements of βω \ ω are just the ultrafilters on ω). But the intersection of finitely many elements of p is not empty, so Y is P_{<ω}.

Proposition 2.2. There are first countable, locally compact, scattered P_{<ω} spaces of size ω_{1}.

We present two different examples. The first is due to S. Shelah [9].

First proof of proposition 2.2. The underlying set of our space X will be ω_{1} ×ω. For x ∈ ω_{1} ×ω write x = ⟨π_{0}(x), π_{1}(x)⟩. For A ⊂ ω_{1} ×ω and i < 2 put π_{i}(A) = π′′_{i}A. If α < ω_{1} let Y_{α} = {α} × ω and X_{α} = α × ω.

Let P_{0} be the family of pairs p = ⟨A, f⟩ which satisfy (i)–(iv) below:

(i) A ∈ [ω_{1} × ω]^{<ω}, (ii) f is a function, f : A × A ⟶ 2. To formulate (iii) and (iv) write U(x) = U^{p}(x) = {y ∈ A : f(y, x) = 1} for x ∈ A. (iii) x ∈ U(x), U(x) \ {x} ⊂ X_{π_{0}(x)}, (iv) x ∈ U(y) implies U(x) ⊂ U(y).

4 I. JUHÁSZ, ZS. NAGY, L. SOUKUP, AND Z. SZENTMIKLÓSSY

For p = ⟨A, f⟩ and q = ⟨B, g⟩ from P_{0} let

p ≤_{0} q if and only if A ⊃ B,
f ⊃ g,
∀x, y, z_{0}, …, z_{l−1} ∈ B
if U^{q}(x) ∩ U^{q}(y) = ⋃_{j<l} U^{q}(z_{j})
then U^{p}(x) ∩ U^{p}(y) = ⋃_{j<l} U^{p}(z_{j}).

For p ∈ P_{0} let

W(p) = {⟨x, s⟩ ∈ A^{p} × [A^{p}]^{<ω} : s ⊂ X_{π_{0}(x)}}.

For w = ⟨x, s⟩ ∈ W(p) let b(w) = x and U^{p}(w) = U^{p}(x) \ ⋃_{y∈s} U^{p}(y). Let

D(p) = {⟨w_{0}, w_{1}⟩ ∈ W(p) × W(p) : π_{0}(b(w_{0})) < π_{0}(b(w_{1}))}.

Now let P be the family of triples ⟨A, f, d⟩, where ⟨A, f⟩ ∈ P_{0}, d is a function, d : D(⟨A, f⟩) ⟶ ω, such that

(1) if d(⟨w_{0}, w_{1}⟩) = d(⟨w′_{0}, w′_{1}⟩) then π_{0}(b(w_{1})) ≠ π_{0}(b(w′_{1})),

(2) if d(⟨w_{0}, w_{1}⟩) = d(⟨w′_{0}, w′_{1}⟩), α ∈ π_{0}(A) and α < π_{0}(b(w_{0})) ≤ π_{0}(b(w′_{1})) then

U^{p}(w_{0}) ∩ U^{p}(w′_{1}) ∩ Y_{α} ≠ ∅.

Write p = ⟨A_{p}, f_{p}, d_{p}⟩ for p ∈ P. If p = ⟨A, f, d⟩ and q = ⟨B, g, e⟩ are from P take p ≤ q if and only if ⟨A, f⟩ ≤_{0} ⟨B, g⟩ and d ⊃ e. Let P = ⟨P, ≤⟩.

For α < ω_{1} let E_{α} = {p ∈ P : α ∈ π_{0}(A^{p})} and E = {E_{α} : α ∈ ω_{1}}.

Lemma 2.3. (ZFC) There is an P-generic filter over E.

Proof. This follows from [10], or Velleman’s ω-morass (a weak form of Martin’s axiom provable in ZFC, see [11]) can be applied for P and E. We explain the second argument more explicitly. By [11, Theorem 3.4] it is enough to prove that E is weakly ω-indiscernible for P (see [11, Definition 1.5]).

For n < ω, α < ω_{1} and an order-preserving function f : n ⟶ α define σ_{f} : P_{n} ⟶ P_{α} as follows. For x = ⟨ν, m⟩ ∈ n × ω let σ_{f}(x) = ⟨f(ν), m⟩. For ⟨x, s⟩ ∈ (n × ω) × [n × ω]^{<ω} let σ_{f}(⟨x, s⟩) =

⟨σ_{f}(x), σ′′_{f}s⟩. For p = ⟨A, f, d⟩ ∈ P take σ_{f}(p) = ⟨σ′′_{f}A, f′, d′⟩ ∈ P, where f′(σ_{f}(x), σ_{f}(y)) = f(x, y) and d′(⟨σ_{f}(w), σ_{f}(w′)⟩) = d(⟨w, w′⟩).

We should check conditions [11, Definition 1.5](1)–(5). All but (5) are clear. So let s < n < ω, f = f(s, n) (i.e. dom(f) = n, f(i) = i for i < s and f(i) = n + i − s for i ≥ s) and p ∈ P_{n}. We need to find a common extension of p and q = σ_{f}(p).

Let B = A_{p} ∪ A_{q} and g : B × B ⟶ 2 be the function defined by the stipulation g^{−1}{1} = f_{p}^{−1}{1} ∪ f_{q}^{−1}{1}. Clearly r = ⟨B, g⟩ is a common extension of p^{−} = ⟨A_{p}, f_{p}⟩ and q^{−} = ⟨A_{q}, f_{q}⟩ in P_{0}. Now for each \overline{w}_{0} = ⟨w_{0}, w_{1}⟩ ∈ D(p) \ D(q), \overline{w}_{1} = ⟨w′_{0}, w′_{1}⟩ ∈ D(q) \ D(p) and α ∈ π_{0}(B) with d_{p}(\overline{w}_{0}) = d_{q}(\overline{w}_{1}) and α < π_{0}(b(w_{0})) ≤ π_{0}(b(w′_{1})) pick a different element z_{\{\overline{w}_{0},\overline{w}_{1},α\}} ∈ Y_{α} \ B. Extend r to r′ = ⟨B′, g′⟩ ∈ P_{0} by adding the points z_{\{\overline{w}_{0},\overline{w}_{1},α\}} to B in such a way that for any x ∈ B we have z_{\{\overline{w}_{0},\overline{w}_{1},α\}} ∈ U^{r′}(x) if and only if b(w_{0}) ∈ U^{r}(x) or b(w′_{1}) ∈ U^{q}(x). Finally, taking d = d_{p} ∪ d_{q} we choose d′ : D(r′) ⟶ ω such that d′ ⊃ d, ran(d′ \ d) ∩ ran(d) = ∅ and d′ \ d is 1–1. Now r^{∗} = ⟨B′, g′, d′⟩ is a common extension of p and q = σ_{f}(p) in P.

Let G be an E-generic filter over P. The set A = ⋃{A_{p} : p ∈ G} is uncountable because π_{0}(A) = ω_{1} by the genericity of G. Take U(x) = ⋃{U^{p}(x) : x ∈ A^{p}} and

B_{x} = {U(x) \ ⋃_{j<l} U(z_{j}) : l < ω, z_{0}, … z_{l−1} ∈ X_{<π_{0}(x)} ∩ A}.

Let X = ⟨A, τ⟩, where B_{x} is the neighborhood base of x in X.

It is straightforward that X is scattered, 0-dimensional and locally compact.

Finally we show that X is strongly P_{2}. Let

W = {⟨x, s⟩ ∈ A × [A]^{<ω} : s ⊂ X_{π_{0}(x)}}.

For w = ⟨x, s⟩ ∈ W put U_{w} = U(x) \ ⋃_{x′∈s} U(x′).

Assume now that V_{0} and V_{1} are uncountable open subsets of X. We can find a sequence S = {⟨w_{0}^{α}, w_{1}^{α}⟩ : α < ω_{1}} ⊂ [W]^{2} such that

(i) U_{w_{i}^{α}} ⊂ V_{i} for i < 2 and α < ω_{1}, (ii) α + 1 = π_{0}(b(w_{0}^{α})) < π_{0}(b(w_{1}^{α})). Let d = ⋃{d_{p} : p ∈ G}. Then S ⊂ dom(d), so there is an uncountable I ⊂ ω_{1} such that d is constant on {⟨w_{0}^{α}, w_{1}^{α}⟩ : α ∈ I}. Then, by the definition of P, we have U_{w_{α}^{0}} ∩ U_{w_{β}^{1}} ∩ Y_{α} ≠ ∅ for α < β ∈ I. So the intersection V_{0} and V_{1} is uncountable.

Second proof of proposition 2.2. The construction will be divided into two parts. First we introduce the notion of α-good spaces and we show that ω_{1}-good spaces are P_{<ω}. Then in the second part we construct an ω_{1}-good space in ZF C.

Definition 2.4. Let X = ⟨ν × ω, τ⟩ be a locally compact scattered topological space of height ν ≤ ω_{1} such that the α^{th} level of X is just

6 I. JUHÁSZ, ZS. NAGY, L. SOUKUP, AND Z. SZENTMIKLÓSSY

X_{α} = {α} × ω. Write X^{(n)} = ν × {n}. We say that X is ν-good if and only if

(a) X^{(n)} is a closed subspace of X and the natural bijection between X^{(n)} and ν is a homeomorphism,

(b) for each limit ordinal β < ν and each α < β if x ∈ X_{β} and U is an open neighborhood of x then the set

K_{x,U,α} = {n ∈ ω : U ∩ X^{(n)} ∩ ( ⋃_{α≤γ<β} X_{γ}) = ∅}

is finite.

Lemma 2.5. An ω_{1}-good space X is strongly P_{2}.

Proof of lemma 2.5. Write X_{<α} for ⋃_{ν<α} X_{ν}. Let U and V be uncountable open subsets of X. Let

I = {n ∈ ω : {α : ⟨α, n⟩ ∈ U} is stationary}.

By (a) we have | ⋃{X^{(n)} : n ∈ I} \ U| ≤ ω. Since the ideal NS_{ω_{1}} is σ-complete, the set A = {α < ω_{1} : ∃n ∈ ω \ I ⟨α, n⟩ ∈ U} is not stationary. Hence there is α < ω_{1} such that U ∩ X_{α} ⊂ {α} × I. But X_{α} is dense in X \ X_{<α}, so U ∩ X_{α} is dense in U \ X_{<α}. It follows that I is infinite.

We show (U ∩ V) \ X_{<α} ≠ ∅ for each α < ω_{1}.

Indeed pick y ∈ V ∩X_{β} where α < β is limit. By (b) the set K_{y,V,α} is finite, so we can choose n ∈ I \K_{y,V,α}. Then V ∩X^{(n)}∩(X_{<β} \X_{<α}) ≠ ∅. But |X^{(n)} \ U| ≤ ω, so V ∩ U ∩ (X_{<β} \ X_{<α}) ≠ ∅ provided that α is large enough, which completes the proof of the lemma.

So it is enough to construct an ω_{1}-good space.

Lemma 2.6. If ν < ω_{1} and X = ⟨ν × ω, τ⟩ is a ν-good space then there is a ν + 1-good space Y such that X is a subspace of Y.

Proof of lemma 2.6. First recall that X is collectionwise normal because it is countable and regular. During the construction we have to distinguish two cases.

Case 1. ν = μ + 1.

Let {x_{n,i} : n, i < ω} be a 1–1 enumeration of {μ} × ω. The family {x_{n,i} : n, i < ω} is closed and discrete, so applying the collectionwise normality of X we can find a closed discrete family {U_{n,i} : n, i < ω} of clopen subsets of X with x_{n,i} ∈ U_{n,i}. Take W_{n,i} = {⟨ν, n⟩} ∪ ⋃{U_{n,j} : i ≤ j < ω}. Let the neighborhood base of ⟨ν, n⟩ in Y be W_{n} = {W_{n,i} : i < ω}.

Case 2. ν is limit.

Let {γ_{n} : n < ω} be a strictly increasing, cofinal sequence in ν. By induction choose ordinals δ_{n} < ν and distinct points {x_{n,i} : n ≤ i < ω} ⊂ X such that

(i) γ_{n} < δ_{n}, (ii) x_{n,i} ∈ (δ_{n} \ γ_{n}) × {n}. Put E_{n} = (ν \δ_{n})×{n}. Then F = {E_{n} : n ∈ ω}∪{x_{n,i} : n ≤ i ∈ ω}

is a closed, discrete family of closed sets because for each α < ν the set {F ∈ F : F ∩ X_{<α} ≠ ∅} is finite. Applying the collectionwise normality of X we can find a closed, discrete family {U_{n} : n ∈ ω}∪{V_{n,i} : n ≤ i < ω} of clopen subsets of X such that E_{n} ⊂ U_{n} and x_{n,i} ∈ V_{n,i}. For each x = ⟨α, m⟩ ∈ X fix a countable neighborhood base B_{x} of x containing only clopen subsets of X_{≤α}. The neighborhood base of ⟨ν, n⟩ in Y will be generated by the sets

W_{n,j,y,B} = ( ⋃_{j≤i<ω} V_{n,i}) ∪ (U_{n} \ B)

where n ≤ j < ω, y ∈ U_{n} and B ∈ B_{y}. It is easy to see that Y satisfies the requirements

Now define, by induction on ν, ν-good spaces X_{ν} for ν ≤ ω_{1} which extend each other. For successor ν we can apply lemma 2.6. For limit ν we can simply take the union.

3. Ψ-like spaces

All examples of locally compact P_{2} spaces in [4, theorem 3,4 and 8] are Ψ-like. In this section we will see why the examples of first countable, locally compact P_{<ω} spaces constructed in the previous section in ZFC are not Ψ-like.

Theorem 3.1. If MA_{ω_{1}}(σ-linked) holds then there is no Ψ-like P_{3}- space.

Theorem 3.1 follows from the following combinatorial result.

Theorem 3.2. (MA_{κ}(σ-linked)) If {F_{α} : α < κ} ⊂ [ω]^{ω} is an almost disjoint family then ω and κ have partitions (a_{0}, a_{1}, a_{2}) and (I_{0}, I_{1}, I_{2}), respectively, with |a_{i}| = ω and |I_{i}| = κ such that

∀i ∈ 3 ∀α ∈ I_{i} |F_{α} ∩ a_{i}| < ω.  (∗)

Remark . As the referee pointed out, MA(σ-centered) does not imply this statement because the strong Luzin property (see definition 3.8 below) of an almost disjoint family is preserved by any σ-centered forcing. As it was recalled in [4] partitioning ω into two pieces is also not enough: a Luzin gap , i.e, an almost disjoint family {B_{β} : β < ω_{1}} ⊂ [ω]^{ω} such that for each B ∈ [ω]^{ω} either the set {β < ω_{1} : |B_{β} ∩B| < ω} or the set {β < ω_{1} : |B_{β} \ B| < ω} is at most countable, can be constructed in ZFC.

Proof of theorem 3.2. For I ⊂ κ write F_{I} = ⋃_{α∈I} F_{α}. Define the poset P = ⟨P, ≤⟩ as follows. Its underlying set P consists of 7-tuples ⟨m, A_{i}, I_{i} : i ∈ 3⟩ satisfying (i)–(iii) below:

(i) m ∈ ω, A_{i} ∈ [ω]^{<ω}, I_{i} ∈ [κ]^{<ω}, (ii) (A_{0}, A_{1}, A_{2}) is a partition of m, (iii) F_{I_{i}} ∩ F_{I_{j}} ⊂ m for i ≠ j.

Write p = ⟨m^{p}, A_{i}^{p}, I_{i}^{p} : i ∈ 3⟩ and I^{p} = ⋃_{i∈3} I_{i}^{p} for p ∈ P. Given p, q ∈ P we set p ≤ q if and only if

(a) m^{q} ≤ m^{p},

(b) A_{i}^{q} = A_{i}^{p} ∩ m^{q} for i ∈ 3,

(c) I_{i}^{q} ⊂ I_{i}^{p} for i ∈ 3,

(d) F_{I_{i}^{q}} ∩ (A_{i}^{p} \ A_{i}^{q}) = ∅ for i ∈ 3. Obviously ≤ is a partial order on P.

For m ∈ ω and i ∈ 3 put

D_{m,i} = {p ∈ P : A_{i}^{p} \ m ≠ ∅}

and

D_{m} = {p ∈ P : m^{p} ≥ m}.

Lemma 3.3. ∀m ∈ ω ∀i ∈ 3 ∀q ∃p ∈ D_{m,i} p ≤ q ∧ I^{p} = I^{q}.

Proof. Let q ∈ P with A_{i}^{q} ⊂ m. We can assume that m ≥ m^{q}. Pick m^{∗} ∈ (ω \ F_{I_{i}^{q}}) \ m. Choose a function f : [m^{q}, m^{∗}] ⟶ 3 such that f(m^{∗}) = i and k ∉ F_{I_{f(k)}^{q}} for k ∈ [m^{q}, m^{∗}). Such a function exists by

(iii). Let A_{j} = A_{j}^{q} ∪ f^{−1}{j} for j < 3 and p = ⟨m^{∗} + 1, A_{j}, F_{j}^{q} : j ∈ 3⟩. Obviously p ∈ P and m^{∗} witnesses p ∈ D_{m,i}. Moreover, p ≤ q. Indeed, conditions (a)–(c) are trivial and (d) holds because A_{j}^{p} \ A_{j}^{q} = f^{−1}{j}.

Lemma 3.4. ∀m ∈ ω ∀q ∃p ∈ D_{m} p ≤ q ∧ I^{p} = I^{q}.

Proof. Straightforward by lemma 3.3.

Lemma 3.5. If α ∉ I^{q} then for each i ∈ 3 there is a p ≤ q with α ∈ I_{i}^{p}.

Proof. We can assume that i = 0. Fix m ∈ ω with F_{α} ∩ F_{I^{q}} ⊂ m. Using lemma 3.4 pick r ∈ D_{m} such that r ≤ q and I^{q} = I^{r}. Put p = ⟨m^{r}, A_{j}^{r} : j ∈ 3, I_{0}^{r} ∪ {α}, I_{1}^{r}, I_{2}^{r}⟩. Then p ∈ P because (iii) holds by F_{α} ∩ F_{I^{p}\{α}} = F_{α} ∩ F_{I^{q}} ⊂ m ⊆ m^{r} = m^{p}. Since p ≤ q is trivial, the lemma is proved.

For α ∈ κ put

E_{α,i} = {p ∈ P : I_{i}^{p} ∩ [ωα, ωα + ω) ≠ ∅}

and

E_{α} = {p ∈ P : α ∈ I^{p}}.

Lemma 3.6. Both E_{α,i} and E_{α} are dense in P.

Proof. Straightforward by lemma 3.5.

Lemma 3.7. P is σ-linked.

Proof. By MA_{ω_{1}}(σ-linked) we have κ < 2^{ω} and so we can choose a countable dense set D in the product space 3^{κ}. For each p = ⟨m, A, I_{i} : i ∈ 3⟩ ∈ P pick d_{p} ∈ D such that d_{p}(α) = i whenever i ∈ 3 and α ∈ I_{i}.

Let p_{0} and p_{1} be elements of P, p_{j} = ⟨m^{j}, A_{i}^{j}, I_{i}^{j} : i ∈ 3⟩. We show that if

(1) m^{0} = m^{1},

(2) A_{i}^{0} = A_{i}^{1} for i ∈ 3,

(3) d_{p_{0}} = d_{p_{1}}, then p_{0} and p_{1} are compatible. Clearly this statement yields that P is σ-linked.

Let I^{j} = ⋃_{i∈3} I_{i}^{j} for j ∈ 2 and I = I^{0} ∩ I^{1}. Fix first m′ ≥ m with

F_{I^{0}\I} ∩ F_{I^{1}\I} ⊂ m′.  (†)

If k ∈ [m, m′) and j ∈ 2 then, by (iii), there is at most one i_{j}(k) ∈ 3 with k ∈ F_{I_{i_{j} (k)}^{j}}. Choose g(k) ∈ 3 \ {i_{0}(k), i_{1}(k)}. Then k ∉ F_{I_{g(k)}^{0}∪I_{g(k)}^{1}}.

Define the partition (A′_{0}, A′_{1}, A′_{2}) of m′ by the equations A′_{i} = A_{i} ∪ g^{−1}{i} for i < 3.

Put p = ⟨m′, A′_{i}, I_{i}^{0} ∪ I_{i}^{1} : i ∈ 3⟩.

Check first p ∈ P. Conditions (i) and (ii) are trivial. Since I_{i}^{0} ∩ I = I_{i}^{1} ∩ I by (3) we have

F_{I_{i}^{0}∪I_{i}^{1}}∩F_{I_{j}^{0}} = (F_{I_{i}^{0}∩I}∩F_{I_{j}^{0}})∪(F_{I_{j}^{1}\I}∩F_{I_{j}^{1}})∪(F_{I_{i}^{0}\I}∩F_{I_{j}^{1}\I}) ⊂ m∪m∪m′ = m′

by (iii) for p_{0} and p_{1} and by (†). Thus (iii) holds for p. Finally we show that p is a common extension of p_{0} and p_{1}. Conditions (a), (b) and (c) obviously hold. But A_{i}^{p}\A_{i}^{p_{0}} = A′_{i}\A_{i} = g^{−1}{i} and g^{−1}{i}∩F_{I_{i}^{0}∪I_{i}^{0}} = ∅ by the choice of g, so (d) is also satisfied. The lemma is proved.

We are now ready to conclude the proof of the theorem. Let

D = {D_{m}, D_{m,i} : m ∈ ω, i ∈ 3} ∪ {E_{α}, E_{α,i} : α ∈ κ, i ∈ 3}.

By lemma 3.3-3.6, D is a family of dense subsets of P. Since P is σ-linked, MA_{κ}(σ-linked) implies that there is a D-generic filter G over

10 I. JUHÁSZ, ZS. NAGY, L. SOUKUP, AND Z. SZENTMIKLÓSSY

P. Put a_{i} = ⋃{A_{i}^{p} : p ∈ G} and I_{i} = ⋃{I_{i}^{p} : p ∈ G}. Then (ii) and (b) imply a_{i} ∩a_{j} = ∅ for i ≠ j. Since D_{m} ∩G ≠ ∅, we have a_{0} ∪a_{1} ∪a_{2} ⊃ m. Moreover D_{m,i} ∩ G ≠ ∅ implies a_{i} ⊄ m. Thus (a_{0}, a_{1}, a_{2}) is a partition of ω into infinite pieces. Similar arguments show that (I_{0}, I_{1}, I_{2}) is a partition of κ into subsets of size κ.

Finally let i ∈ 3 and α ∈ I_{i}. Pick q ∈ G with α ∈ I_{i}^{q}. Then

F_{α} ∩ a_{i} ⊂ F_{I_{i}^{q}} ∩ a_{i} = ⋃_{p∈G,p≤q}(F_{I_{i}^{q}} ∩ A_{i}^{p}) ⊂ A^{q} ⊂ m^{q} by (d).

By the previous theorem the existence of a Ψ-like P_{3} space of size ω_{1} is not provable in ZFC. On the other hand, we will see that it is also consistent that 2^{ω} is arbitrarily large and there is a Ψ-like P_{<ω}-space of size 2^{ω}.

We start with a definition.

Definition 3.8. Let A ⊂ [ω]^{ω} be an almost disjoint family. We say that A is a strong Luzin gap if and only if given finitely many uncountable subsets A_{0}, …, A_{n−1} of A we have that ⋂_{i<n} (⋃ A_{i}) is infinite.

It is easy to see that A is a strong Luzin gap if and only if the corresponding Ψ-like space X[A] (see definition 1.2) is P_{<ω}.

In [4, theorem 6] a Luzin gap was obtained by a c.c.c forcing due to Hechler [5].

We show that this almost disjoint family is in fact strongly Luzin. First we recall some notations and definitions from [4]. Let κ > ω be a fixed regular cardinal. Write D = [κ]^{<ω} × κ. Let

P = {p : p is a function, dom(p) ∈ D, ran(p) ⊂ 2}

If p, p′ ∈ P, dom(p) = A × n, dom(p′) = A′ × n′ put p ≤ p′ if and only if p ⊃ p′ and for each k ∈ n \ n′ we have |{α : p(α, k) = 1}| = 1.

If G is P -generic then let A_{α} = {k : ∃p ∈ G p(α, k) = 1}. Take A[G] = {A_{α} : α < κ}.

Lemma 3.9. A[G] is a strong Luzin gap in V^{P}.

Proof. Assume on the contrary, that n and m are natural numbers, p^{∥}— “{ν̇_{i} : i < n} are injective functions from ω_{1} to κ and

( ⋂_{i<n} ⋃_{α<ω_{1}} A_{ν̇_{i}(α)}) ⊂ m.”  (∗)

For each α < ω_{1} pick p_{α} ≤_{P} p and ν_{α,i} ∈ ω_{1} for i < n such that

p_{α}^{∥}—“ν̇_{i}(α) = ν̂_{α,i} for each i < n.”

Without loss of generality the ν_{α,i} are pairwise different. Let dom(p_{α}) = D_{α} × k_{α}. We can assume that k_{α} are independent of α, k_{α} = k ≥ m, {D_{α} : α < ω_{1}} forms a Δ-system with kernel D, |D_{α}| = |D_{β}|, {ν_{α,i} : i < n} ⊂ D_{α} \ D and denoting by σ_{α,β} the unique order preserving bijection between D_{α} and D_{β} we have that σ_{α,β} gives an isomorphism between p_{α} and p_{β} and σ_{α,β}(ν_{α,i}) = ν_{β,i} for i < n.

Let α_{0} < … α_{n−1} < ω_{1}. Define the condition q ∈ P as follows:

1. dom(q) = (⋃_{i<n} D_{α_{i}}) × k + 1,

2. q ⊃ ⋃_{i<n} p_{α_{i}},

3. q(ν_{α_{i},i}, k) = 1 for i < n,

4. q(ξ, k) = 0 provided ξ ∈ ⋃_{i,n} D_{α_{i}} \ {ν_{α_{0},0}, …, ν_{α_{n−1},n−1}}.

Then q is a condition which extends p_{α_{i}} for i < n, but q^{∥}—k ∈ ⋂_{i<n} A_{ν_{α_{i},i}} which contradicts (∗) because k ≥ m. This concludes the proof of the lemma.

This lemma yields the following corollary.

Corollary 3.10. If ZF is consistent then so is ZFC + 2^{ω} = κ is as large as you wish + there is a Ψ-like P_{<ω} space of size 2^{ω}.

4. A positive partition theorem below 2^{ω}

In this section we present the proof of a theorem of Szentmiklóssy which was announced and applied in [4].

Given A, B ⊂ κ we denote by [A; B] the set {{α, β} : α ∈ A, β ∈ B, α < β}.

Definition 4.1. Given cardinals κ, λ and μ the partition relation κ ⟶ ((λ; λ))_{μ}^{2} holds if and only if ∀f : [κ]^{2} ⟶ μ ∃A, B ∈ [κ]^{λ} ∃ξ < μ

( tp(A) = tp(B) = λ ∧ sup A ≤ sup B ∧ f′′[A; B] = {ξ}).

Theorem 4.2. If ZF is consistent then so is ZFC + 2^{ω} = κ ≥ ω_{3} + ω_{3} ⟶ ((ω_{1}; ω_{1}))_{ω_{1}}^{2}.

The proof is based on the following lemma. To simplify its formulation we write P = Fn(ω_{3}, ω_{1}; ω_{1}). For f, g, h ∈ P we write f = g ∪^{.} h to means that f = g ∪ h and g ∩ h = ∅.

Lemma 4.3. Assume GCH. If f : [ω_{3}]^{2} ⟶ P then there is some C ∈ [ω_{3}]^{ω_{2}} and there are elements {p, p(ξ), q(η), r(ξ, η) : ξ, η ∈ C, ξ < η} ⊂ P with the following properties:

(i) ∀ξ < η ∈ C f(ξ, η) = p ∪^{.} p(ξ) ∪^{.} q(η) ∪^{.} r(ξ, η),

(ii) ∀ξ ∈ C {f(ξ, η) : η ∈ C \ ξ + 1} forms a Δ-system with kernel

p ∪^{.} p(ξ),

(iii) ∀η ∈ C {f(ξ, η) : ξ ∈ C ∩ η} forms a Δ-system with kernel

p ∪^{.} q(η),

(iv) the p(ξ) have disjoint supports,

(v) the q(η) have disjoint supports.

Proof of lemma. Fix a large enough regular cardinal τ and let N be a countable elementary submodel of H(τ) = ⟨H(τ), ∈, <⟩ with f ∈ N, where H(τ) is the family of the sets whose transitive closure has cardinality < τ.

Let G = {g ∈ N : g is a function from [ω_{3}]^{<ω} to ω_{1} } and H = {h ∈

N : h maps [ω_{3}]^{2} to ω_{1}}.

Sublemma 4.3.1. There is D ⊂ ω_{3} of order type ω_{2} + 1 which is end-homogeneous for all g ∈ G and homogeneous for all h ∈ H.

Proof of sublemma. By 2^{ω_{1}} = ω_{2} we can choose an increasing sequence ⟨N_{α} : α ≤ ω_{2}⟩ of elementary submodels of H(τ) such that N ∈ N_{0}, |N_{α}| = ω_{2} and [N_{α}]^{ω_{1}} ⊂ N_{α}.

Since G, H ∈ N_{0}, they have enumerations \overline{G} = ⟨g_{n} : n ∈ ω⟩ and \overline{H} = ⟨h_{n} : n ∈ ω⟩ in it.

Pick an arbitrary x ∈ ω_{3} \ sup(N_{ω_{2}} ∩ ω_{3}).

By transfinite recursion on α, define an increasing sequence {x_{α} : α < ω_{2}} ⊂ ω_{3} with x_{α} ∈ N_{α} such that

∀n∀a ∈ [{x_{β} : β < α}]^{<ω}g_{n}(a, x_{α}) = g_{n}(a, x).  (∗)

This can be done by [N_{α}]^{ω_{1}} ⊂ N_{α} ≺ H(τ) and x ∈ ω_{3} \ N_{α}.

Color the elements of {x_{α} : α < ω_{2}} with the H-colors of the pair {x_{α}, x}:

F (x_{α}) = ⟨h_{n}(x_{α}, x) : n < ω⟩ .

The range of F has cardinality ≤ ω_{1}^{ω} = ω_{1} by CH, so there is an F - homogeneous D′ ⊂ {x_{α} : α < ω_{2}} of size ω_{2}, and it is easily seen that D = D′ ∪ {x} satisfies the requirement of the sublemma.

To simplify our notation we will assume that D is just ω_{2} + 1.

Sublemma 4.3.2. ∀ξ < ω_{2} ∃δ(ξ) < ω_{2} {f(ξ, η) : η ∈ ω_{2} + 1 \ δ(ξ)} forms a Δ-system.

Proof of sublemma. For ξ < η < η′ ≤ ω_{2} let d(ξ, η, η′) = dom(f(ξ, η))∩ dom(f(ξ, η′)). Then d(ξ, η, η′) is one of the ω_{1} many countable subsets of dom(f(ξ, η)), so, by the end-homogeneity, d(ξ, η, η′) is independent of η′ for ξ < η < η′ ≤ ω_{2}. Denote this common value by h(ξ, η). Unfortunately we can’t apply the end-homogeneity for h(ξ, η), because its range may have large cardinality. But we can argue in the following way. We know that η < η′ implies h(ξ, η) ⊂ h(ξ, η′), because for any η′ < η′′ we have dom(f(ξ, η)) ∩ dom(f(ξ, η′′)) = h(ξ, η) = dom(f(ξ, η)) ∩ dom(f(ξ, η′)), so h(ξ, η) ⊂ dom(f(ξ, η′)) ∩ dom(f(ξ, η′′)) = h(ξ, η′).

But h(ξ, η) is a countable set, so

∀ξ < ω_{2} ∃δ(ξ) < ω_{2} ∀η, η′ ≥ δ(ξ) h(ξ, η) = h(ξ, η′).

But this means that {dom(f(ξ, η)) : η > δ(ξ)} forms a Δ-system with kernel h(ξ, δ(ξ)). But f(ξ, η′)⌈h(ξ, δ(ξ)) is independent of η′ by the end homogeneity, so the functions f(ξ, η′) are pairwise compatible for δ(ξ) ≤ η′ ≤ ω_{2}.

Applying in four steps this sublemma, a Δ-system argument, transfinite recursion and a Δ-system argument again, we can find a set D ∈ [ω_{2} + 1]^{ω_{2}} with ω_{2} ∈ D such that

(1) ∀ξ ∈ D {f(ξ, η) : η ∈ D \ ξ + 1} forms a Δ-system with kernel p^{Δ}(ξ). We write f(ξ, η) = p^{Δ}(ξ) ∪^{.} r^{+}(ξ, η),

(2) {p^{Δ}(ξ) : ξ ∈ D} forms a Δ-system with kernel p. We write

p^{Δ}(ξ) = p ∪^{.} p(ξ). So f(ξ, η) = p ∪^{.} p(ξ) ∪^{.} r^{+}(ξ, η).

(3) ∀η ∈ D ∀ξ, ξ′, η′ ∈ D ∩ η if ξ′ < η′ then

( dom p(η) ∪ dom r^{+}(ξ, η)) ∩ dom f(ξ′, η′) = ∅,

(4) {f(ξ, ω_{2}) : ξ ∈ D} forms a Δ-system. By the end-homogeneity of D, it follows that {f(ξ, η) : ξ ∈ D ∩ η} also forms a Δ-system with some kernel q^{Δ}(η) for all η ∈ D. Write q^{Δ}(η) = p ∪ q(η).

Consider the increasing enumeration {δ_{ν} : ν < ω_{2}} of D and let C = {δ_{ν} : ν is limit}.

Claim 1. If ξ′ < η′ < ξ < η are from D, then dom f(ξ′, η′)∩dom f(ξ, η) = dom(p)

Indeed, f(ξ, η) = p ∪^{.} p(ξ) ∪^{.} r^{+}(ξ, η) and both dom(p(ξ)) and dom r^{+}(ξ, η) are disjoint from dom(f(ξ′, η′)) by (3).

Claim 2. If ξ′ < η′ < η are from C, then dom f(ξ′, η′) ∩ dom q(η) = ∅

Indeed, pick ξ ∈ D ∩ (η′, η), observe q(η) ⊂ f(ξ, η), and apply claim 1.

Claim 3. If η′ < η are from C, then dom(q(η′)) ∩ dom(q(η)) = ∅.

Let ξ′ ∈ D ∩ ν′. Then q(η′) ⊂ f(ξ′, η′) and apply claim 2.

Claim 4. If ξ < η are from C, then dom(p(ξ)) ∩ dom(q(η)) = ∅.

Indeed, pick η^{∗} < ξ^{∗} ∈ D ∩ (ξ, η). Then dom(p(ξ)) ∩ dom(q(η)) ⊂ dom f(ξ, η^{∗}) ∩ dom(f(ξ^{∗}, η) \ p)) = ∅ by claim 1.

So if you take r(ξ, η) = f(ξ, η) \ (p(ξ) ∪ q(η) ∪ p) for ξ < η ∈ C, then the set C and the conditions {p, p(ξ), q(η), r(ξ, η) : ξ, η ∈ C, ξ < η} satisfy (i)–(v). The lemma is proved.

Proof of theorem 4.2. We start with a model of ZFC + GCH and fix an arbitrarily large regular cardinal κ ≥ ω_{3}. Let C_{ω_{1}} = Fn(ω_{1}, 2, ω) and consider the poset Q = ∏_{ω_{1},ω}^{κ} C_{ω_{1}} = ⟨Fn(κ, C_{ω_{1}}ω_{1}), ≤_{ω}⟩ where f ≤_{ω} g if and only if dom(f) ⊃ dom(g), f(α) ≤_{C_{ω_{1}}} g(α) for each α ∈ dom(g) and |{α ∈ dom(g) : f(α) ≠ g(α)}| < ω (see [2]). We will show that the model V^{Q} satisfies our requirements.

It is known that V^{Q} ⊧ 2^{ω} = κ, Q is proper and the cardinals in V and in V^{Q} are the same (see [2]). Obviously the next lemma concludes the proof.

Lemma 4.4. V^{Q} ⊧ “∀g : [ω_{3}]^{2} ⟶ ω_{1} ∃A, B ⊂ ω_{3} tp(A) = tp(B) =

ω_{2}, sup A = sup B and ∃α ∈ ω_{1} ∀S ∈ [A]^{ω} ∃B_{S} ∈ [B]^{ω_{2}} g′′[S, B_{S}] = {α}”.

Proof of lemma. Assume ġ is a name of a function from [ω_{3}]^{2} to ω_{1}.

For each ξ < η < ω_{3} pick a condition s(ξ, η) ∈ Q and an ordinal α(ξ, η) ∈ ω_{1} with s(ξ, η)^{∥}— “ġ(ξ, η) = α(ξ, η)”. We can assume that dom(s(ξ, η)) ⊂ ω_{3}.

Fix an enumeration of C_{ω_{1}} in V, {s_{ν} : ν < ω_{1}}, and define a bijection F between ∏_{ω_{1},ω}^{ω_{3}} C_{ω_{1}} and Fn(ω_{3}, ω_{1}; ω_{1}) as follows:

F (s) = h if and only if dom(h) = s

and s(ν) = c_{h(ν)} for all ν ∈ dom(h).

Consider now the function f : [ω_{3}]^{2} ⟶ P defined by the formula f(ξ, η) = F (s(ξ, η)) × {⟨ω_{3}, α(ξ, η)⟩}. Formally, the range of f is P × (ω_{3}×ω_{1}) = Fn(ω_{3}+1, ω_{1}; ω_{1}), but this poset is isomorphic to P. Applying lemma 4.3 we can get a C ∈ [ω_{3}]^{ω_{2}} and elements {p, p(ξ), q(η), r(ξ, η) : ξ, η ∈ C, ξ < η} ⊂ Q and α ∈ ω_{1} satisfying (i)–(v) above and α(ξ, η) = α for each ξ < η ∈ C. Let Ȧ and Ḃ be Q-names of subsets C such that for all ξ ∈ C we have [[ξ ∈ Ȧ]] = F^{−1}(p ∪ p(ξ)) and [[η ∈ Ḃ]] = F^{−1}(p ∪ q(η)). It is clear that F^{−1}(p)^{∥}—|Ȧ| = |Ḃ| = ω_{2}.

Let S be a countable subset of A in V^{Q}. Since Q is proper, there is a countable T in V with S ⊂ T.

Let B^{∗} = {η ∈ B : g′′[T ∩ A, {η}] = {α}} and Ḃ^{∗} be a name for this set.

It is enough to show that F^{−1}(p)^{∥}—|B^{∗}| = ω_{2}. Assume on the contrary that r ≤ F^{−1}(p), ρ ∈ C and r^{∥}— “B^{∗} ⊂ B∩ρ”, i.e., B^{∗} is bounded in B.

Let E = dom(p) ∪ dom(r) ∪ ⋃{dom(p(ξ)) : ξ ∈ T }. Pick σ ∈ C \ ρ such that dom(q(σ) ∪ dom(r(ξ, σ) ∩ E = ∅ for each ξ ∈ T. Since r^{∥}—|S| = ω, there is ξ ∈ T such that r and p ∪ p(ξ) is compatible.

Let r^{∗} = r∧(p∪p(ξ)∧(q(σ)∪r(ξ, σ). Then r^{∗} forces a contradiction.

This completes the proof of the theorem.

5. Models without large first countable P_{2}-spaces

Hajnal and Juhász [4, Problem 10] asked if it is consistent to assume that 2^{ω} ≥ ω_{2} and every first countable (or compact) space with property P_{2} has cardinality ≤ ω_{1}. We give an affirmative answer here. We will argue in the following way. First we quote the definition of principle C(κ) from [7], then we show that C(ω_{2}) implies that every first countable P_{2} space has cardinality ≤ ω_{1}, finally we will cite a theorem from [7] saying that C(ω_{2}) is consistent with any cardinal arithmetic.

Definition 5.1. (See [7].) Let κ be an infinite cardinal. We say that principle C(κ) holds if and only if for each family {A(ξ, n), B(ξ, n) : ξ ∈ κ, n ∈ ω} ⊂ [ω]^{ω} either (i) or (ii) below holds:

(i) ∃C ∈ [κ]^{κ} ∀n, m ∈ ω ∀ξ ≠ ζ ∈ C A(ξ, n) ∩ B(ζ, m) ≠ ∅,

(ii) ∃D, E ∈ [κ]^{κ} ∃n, m ∈ ω ∀ξ ∈ D ∀ζ ∈ E A(ξ, n) ∩ B(ζ, m) = ∅.

Theorem 5.2. If C(κ) holds then each first countable, separable Hausdorff space X of size κ contains two disjoint open sets U and V of cardinality κ.

Proof. Let S be a countable dense subset of X. For each x ∈ X fix a neighborhood base {U(x, n) : n ∈ ω} of x in X. Take A(x, n) = B(x, n) = U(x, n) ∩ S and apply C(κ). Since X is T_{2}, there is no C ∈ [X]^{κ} satisfying 5.1(i). So there are D, E ∈ [X]^{κ} and n, m ∈ ω such that U(x, n) ∩ U(y, m) ∩ S = ∅ whenever x ∈ D and y ∈ E. But S is dense in X, therefore U = ⋃{U(x, n) : x ∈ D} and V = ⋃{U(y, m) : y ∈ E} are disjoint and of cardinality κ.

It was proved in [7] that starting from a model of CH, after adding λ-many Cohen reals by the poset P = Fn(λ, 2, ω), we have that C(ω_{2}) holds in V^{P}. Since P_{2} spaces are separable as it was observed in the proof of [4, theorem 1], theorem 5.2 yields the following corollary.

Corollary 5.3. If ZF is consistent then so is ZFC + “2^{ω} is as large as you wish” + “every first countable P_{2} space has cardinality ≤ ω_{1}.”

References

[1] J. E. Baumgartner, S. Shelah, Remarks on superatomic Boolean algebras, Ann. Pure Appl. Logic, 33 (l987), no. 2, 1-9-129.

[2] S. Fuchino, S. Shelah, L. Soukup, Beating with sticks and clubs, 1994, preprint.

[3] A. Hajnal, Proof of a conjecture of S. Ruziewicz, Fund. Math. 50 (1961), 123-128.

[4] A. Hajnal, I. Juhász, Intersection properties of open sets, Topology and its Application, 19 (l985), 201–209.

[5] S. H. Hechler, Short complete nested sequences in βN − N and small maximal almost disjoint families, Gen. Top. Appl. 2 (1972) 139-149.

[6] I. Juhász, Cardinal Functions - Ten Years Later, Math. Center Tracts 123, Amsterdam, 1980.

[7] I. Juhász, S. Soukup, Z. Szentmiklóssy, Combinatorial principles from adding Cohen reals, to appear in Proc. of Logic Colloquium 95.

[8] J. Van Mill An introduction to βN, in Handbook of Set-Theoretic Topology, Ed. K. Kunen J.E Vaughan, Elsevier Publ. B. V. 1984.

[9] S. Shelah, private communication, 1993.

[10] S. Shelah, Models with second order properties III. Omitting types in λ^{+} for L(Q), Proc. of a Berlin Workshop July 1977, Archive f. Math. Logik 12 (l980) 1-11.

[11] D. Velleman ω-morasses and a weak form of Martin’s axiom provable in ZFC, Trans. Amer. Math. Soc. 285 (1984), no 2. 617–627

Mathematical Institute of the Hungarian Academy of Sciences, Budapest, Reáltanoda u, 13-15, 1053, Hungary

E-mail address: juhasz@math-inst.hu

Mathematical Institute of the Hungarian Academy of Sciences, Budapest, Reáltanoda u, 13-15, 1053, Hungary

E-mail address: zsiga@math-inst.hu

Mathematical Institute of the Hungarian Academy of Sciences, Budapest, Reáltanoda u, 13-15, 1053, Hungary

E-mail address: soukup@math-inst.hu

Department of Analysis, Eötvös University, Budapest, Múzeum krt, 6–8, 1088, Hungary

E-mail address: szentmiklossy@math-inst.hu