PROPERTIES OF IDEALS ON THE

GENERALIZED CANTOR SPACES

Jan Kraszewski

Abstract. We define a class of productive σ−ideals of subsets of the Cantor space 2^{ω} and observe that both σ−ideals of meagre sets and of null sets are in this class. From every productive σ−ideal J we produce a σ−ideal J_{κ} of subsets of the generalized Cantor space 2^{κ}. In particular, starting from meagre sets and null sets in 2^{ω} we obtain meagre sets and null sets in 2^{κ}, respectively. Then we investigate additivity, covering, uniformity and cofinality of J_{κ}. For example, we show that

non(J) = non(J_{ω_{1}}) = non(J_{ω_{2}}).

Our results generalizes those from [5].

0. Introduction

In this paper we shall discuss the properties of canonical σ−ideals of subsets of generalized Cantor spaces 2^{κ}, for example the σ−ideal of null sets and of meagre sets.

In the 80’s several people investigated relations between the σ−ideal N of null subsets of the classical Cantor space 2^{ω} and the σ − ideal N_{κ} of null subsets of the generalized Cantor space 2^{κ} for some uncountable cardinal number κ. One of the most important questions was what were the connections between cardinal coefficients (such as add, cov, non and cof) of N and these of N_{κ}. The answer was given independently by Cichoń ([1], unpublished) and Fremlin ([5]). Both authors obtained almost the same results, except for two of them: Theorem 3.4 for null sets (only Fremlin) and Theorem 3.10 for null sets (only Cichoń).

A natural question arose whether measure-theoretic tools were really necessary to get these results. In this paper we give a complete answer to it. In order to do this we extract the combinatorial principles that are considered by both authors and show that similar results to those which were obtained by them can be proved for a much wider class of ideals.

In the first section we formulate a notion of productivity. If we identify 2^{ω} with its square using canonical homeomorphism then we can say a bit imprecisely that an ideal J of subsets of 2^{ω} is productive if for every A ∈ J the cylinder A × 2^{ω} is in J. We observe that σ−ideals of meagre sets and of null sets are productive. Moreover, there exists the least non-trivial productive σ −ideal on 2^{ω} called S_{2}.

Typeset by AMS-TEX

Key words and phrases. Cantor space, σ−ideals, null sets, meagre sets, cardinal functions.

Then we describe a method of constructing a σ−ideal J_{κ} of subsets of 2^{κ} from a given productive σ−ideal J of subsets of 2^{ω}. This method is reasonable because starting from meagre sets (null sets) in 2^{ω} we obtain meagre sets (resp. null sets) in 2^{κ} and it is our main tool throughout this paper.

In the next section we completely describe the additivity and (with some additional weak assumption) cofinality of J_{κ}. The third section is devoted to the uniformity of J_{κ}. The most surprising result is that for every productive σ−ideal J we have

non(J) = non(J_{ω_{1}}) = non(J_{ω_{2}}).

(for meagre sets it was proved by Veličkovič and remained unpublished). Moreover, uniformities of J_{κ}’s form an increasing sequence. However, it is possible to find bounds for non(J_{κ}) which depend only on κ, not on J:

log(κ) ≤ non(J_{κ}) ≤ (log(κ))^{ω}.

We also prove Theorem 3.9, which is a powerful tool for showing several absolute inequalities concerning non’s as well as some consistency results.

In the fourth section we deal with the covering number of J_{κ}. These results are quite different from the previous ones as covering numbers of J_{κ}’s form a decreasing sequence. We prove that this sequence stabilizes for some cardinal number κ ≤ 𝔠. In the last section we consider several models of Set Theory. In particular, we compute all cardinal coefficients of J_{κ} assuming Generalized Continuum Hypothesis and investigate a function F (κ) = cov(M_{κ}) (where M stands for the σ−ideal of meagre sets) in some models of set theory obtained by the method of forcing.

The paper contains the main part of the author’s Ph.D. thesis. The author would like to thank Professor J. Cichoń for his help during the preparation of this paper and Professor D. H. Fremlin for his fruitful remarks.

1. Basic definitions and facts

In this paper we deal with the generalized Cantor space 2^{κ} interpreted as the set of all functions from an infinite cardinal number κ into the set {0, 1}. This spaces are endowed with the standard product topology. Moreover, we consider the standard product measure on 2^{κ}.

We use standard set-theoretical notation and terminology derived from [12]. Let us remind that the cardinality of the set of all real numbers is denoted by 𝔠. The cardinality of a set X is denoted by |X|. If κ is a cardinal number then [X]^{κ} ([X]^{≤κ}) denotes the family of all subsets of the set X of cardinality κ (not greater than κ, respectively). X^{<ω} denotes the set of all finite sequences of elements of the set X. If φ : X → Y is a function then rng(φ) denotes the range of φ. If A ⊆ Y then φ^{−1}[A] denotes the pre-image of A.

The σ−ideals of meagre subsets and of null subsets of 2^{ω} are denoted by M and N, respectively.

In addition, we introduce some extra notation in order to simplify further considerations. Let λ and κ be any infinite cardinal numbers. We put Inj(λ, κ) = {φ ∈ κ^{λ} : φ is an injection}.

Definition. For A ⊆ 2^{κ} and φ ∈ Inj(ω, κ) we put φ ∗ A = {x ∘ φ : x ∈ A}. For B ⊆ 2^{ω} and φ ∈ Inj(ω, κ) we put B_{φ} = {x ∈ 2^{κ} : x ∘ φ ∈ B}.

It is reasonable to add some intuitions to this definition. Namely, if T is a subset of κ then we identify the generalized Cantor space 2^{κ} and the space 2^{T} ×2^{κ\T} using the canonical homeomorphism π_{T} defined by π_{T} (x) = (x ↾ T, x ↾ (κ \ T)). Then, if we treat 2^{κ} as a product 2^{rng(φ)} × 2^{κ\rng(φ)} for some φ ∈ Inj(ω, κ) and canonically identify 2^{rng(φ)} with 2^{ω} using φ then for any set A ⊆ 2^{κ} we can interpret the set φ ∗ A as a projection of A into 2^{rng(φ)}. Similarly, for any set B ⊆ 2^{ω} we can treat the set B_{φ} as a cylinder B × 2^{κ\rng(φ)}.

Obviously, φ ∗ A ⊆ 2^{ω} and B_{φ} ⊆ 2^{κ}. Another simple observation is that for B ⊆ 2^{ω} and φ ∈ Inj(ω, κ) we have φ ∗ B_{φ} = B. Similarly, for A ⊆ 2^{κ} and φ ∈ Inj(ω, κ) we have A ⊆ (φ ∗ A)_{φ}.

From now on let J be any σ−ideal of subsets of 2^{ω} and κ be any infinite cardinal number. We define

κ(J) = {A ⊆ 2^{κ} : (∃φ ∈ Inj(ω, κ)) φ ∗ A ∈ J }.

If A ⊆ 2^{κ} then any φ ∈ Inj(ω, κ) such that φ ∗ A ∈ J we called a witness for A. If I ⊆ J then κ(I) ⊆ κ(J). Moreover, we have J ⊆ ω(J), because for every A ∈ J the identity on ω is a witness for A. The σ−ideal generated by the family κ(J) we denote by J_{κ}.

Definition. The ideal J is productive if ω(J) ⊆ J.

We formulate some equivalent versions of this property now.

Fact 1.1. For any σ − ideal J of subsets of 2^{ω} the following conditions are equivalent:

(a) J is productive,

(b) (∀A ⊆ 2^{ω})(∀φ ∈ Inj(ω, ω))(φ ∗ A ∈ J ⇒ A ∈ J),

(c) (∀A ⊆ 2^{ω})(∀φ ∈ Inj(ω, ω))(A ∈ J ⇒ A_{φ} ∈ J).

Proof. The equivalence of conditions (a) and (b) follows directly from the definition of the productivity of J. Suppose now that (b) holds and let A ∈ J and φ ∈ Inj(ω, ω). As φ ∗ A_{φ} = A ∈ J then according to (b) we have A_{φ} ∈ J and, consequently, (c) holds as well.

On the other hand, let us assume (c) and take A ⊆ 2^{ω} and φ ∈ Inj(ω, ω) such that φ ∗ A ∈ J. Then (c) gives us that (φ ∗ A)_{φ} ∈ J. But as we know A ⊆ (φ ∗ A)_{φ} so A ∈ J. Hence (b) is true. □

We can intuitively interpret the definition of productivity in such a way that justifies its name. Namely, thanks to Fact 1.1 and the previous description, we can say that J is productive if for every φ ∈ Inj(ω, ω) and every set A ⊆ 2^{rng(φ)} if A is in J then the cylinder A × 2^{ω\rng(φ)} is in J.

Directly from their definitions we deduce that the σ−ideals of meagre subsets and of null subsets of 2^{ω} are productive. Also the well-known σ−ideal generated by closed null subsets of 2^{ω} is productive.

Not every σ−ideal of subsets of 2^{ω} is productive. However, every σ−ideal has its “productive closure”.

Fact 1.2. For any σ−ideal J of subsets of 2^{ω} the σ−ideal J_{ω} is productive.

Proof. It is enough to prove that ω(J_{ω}) ⊆ J_{ω}. Let A ⊆ 2^{ω} be a member of ω(J_{ω}) and let φ ∈ Inj(ω, ω) be its witness. It means that φ ∗ A ∈ J_{ω} which implies φ ∗ A ⊆ ⋃_{i<ω} A_{i} for some family {A_{i} : i < ω} ⊆ ω(J). Thus

A ⊆ (φ ∗ A)_{φ} ⊆ ( ⋃_{i<ω} A_{i})_{φ} = ⋃_{i<ω}(A_{i})_{φ}.

Let φ_{i} ∈ Inj(ω, ω) be a witness for A_{i} for each i < ω. To finish the proof we need the following simple observation:

Claim. For every A ⊆ 2^{ω} and φ, ψ ∈ Inj(ω, ω) we have ψ ∗ (φ ∗ A) = (φ ∘ ψ) ∗ A.

Hence for each i < ω we have

(φ ∘ φ_{i}) ∗ (A_{i})_{φ} = φ_{i} ∗ (φ ∗ (A_{i})_{φ}) = φ_{i} ∗ A_{i} ∈ J ,

so (A_{i})_{φ} is a member of ω(J) and φ ∘ φ_{i} is its witness. Consequently, ⋃_{i<ω}(A_{i})_{φ} is a member of J_{ω} and so is A. □

The σ−ideal [2^{ω}]^{≤ω} of all countable subsets of 2^{ω} is not productive. The σ−ideal ([2^{ω}]^{≤ω})_{ω} is the least non-trivial productive σ−ideal of subsets of the Cantor space. It is denoted by S_{2} and was investigated in [3]. Earlier it appeared in [13], but only incidentally. It occurred that properties of this σ−ideal are strongly connected with some intensively studied combinatorial properties of subsets of natural numbers (the splitting and reaping numbers). For example, non(S_{2}) = ℵ_{0}-𝔰 (see [11] for the definition of ℵ_{0}-𝔰 and more discussion).

It is worth noting that the σ −ideal J_{ω} is not necessarily proper even if J is proper. For example, if J is a σ−ideal orthogonal to S_{2} (i.e. there exist sets A ∈ J and B ∈ S_{2} such that A ∪ B = 2^{ω}) then J_{ω} must be improper because due to Fact 1.2 it is productive and thus contains S_{2}.

The following fact holds:

Fact 1.3. If J is productive then κ(J) = J_{κ} for any infinite cardinal number κ.

Proof. We left easy calculations to the reader. □

Throughout this paper we will investigate σ−ideals J_{κ} for a certain productive σ −ideal J of subsets of 2^{ω}. First, we have to make sure that it will not be art for art sake, namely if we consider the σ−ideal M of meagre subsets of 2^{ω} or the σ−ideal N of null subsets of 2^{ω} then M_{κ} and N_{κ} will be the σ−ideals of meagre or null subsets of 2^{κ}, respectively.

Fact 1.4. M_{κ} is the σ −ideal of meagre subsets of 2^{κ} in the standard product topology. N_{κ} is the σ−ideal of null subsets of 2^{κ} in the standard product measure.

Proof. 1. The meagre case. In order to show that every element of M_{κ} is meagre it is enough to observe that for every nowhere dense subset E of 2^{ω} and every φ ∈ Inj(ω, κ) the set E_{φ} is nowhere dense in 2^{κ}.

To complete the proof it is enough to show that every closed nowhere dense subset D of 2^{κ} is in M_{κ}. Let us remind that if σ is a finite partial function from κ into the set {0, 1} then the set

[σ]_{κ} = {x ∈ 2^{κ} : σ ⊆ x}

is open and the family of all subsets of 2^{κ} of this type is a basis for the standard product topology.

Let {[σ_{i}]_{κ} : i ∈ I} be a maximal family of pairwise disjoint open basis subsets of 2^{κ} which are disjoint with the set D as well. The set I is at most countable due to countable chain condition of the space 2^{κ}. Thus there exists an injection φ ∈ Inj(ω, κ) such that ⋃_{i<ω} dom(σ_{i}) ⊆ rng(φ). Then σ_{i}∘ φ is a partial function from ω into the set {0, 1} and φ ∗ [σ_{i}]_{κ} = [σ_{i}∘ φ]_{ω} for each i ∈ I. Then {[σ_{i}∘ φ]_{ω} : i ∈ I} is a maximal family of pairwise disjoint open basis subsets of 2^{ω} which are disjoint with the set φ ∗ D. Consequently, the set φ ∗ D is a nowhere dense subset of 2^{ω}, which ends the proof.

2. The null case Let μ_{κ} denotes standard product measure on 2^{κ}. To begin with, let us observe that for every basis open set B ⊆ 2^{ω} of the form mentioned above and every φ ∈ Inj(ω, κ) we have μ_{ω}(B) = μ_{κ}(B_{φ}). Thus straight from the definition of a product measure we can deduce that every set from N_{κ} is null.

Conversely, let A be a null subset of 2^{κ}. For every natural number n there exists a family {σ_{i}^{n} : i < ω} of finite partial functions from κ into the set {0, 1} such that

A ⊆ ⋃_{i<ω}[σ_{i}^{n}]_{κ} and ∑_{i<ω} \frac{1}{2^{|σ_{i}^{n}|}} < \frac{1}{n + 1}.

Let φ ∈ Inj(ω, κ) be such that ⋃_{i,n<ω} dom(σ_{i}^{n}) ⊆ rng(φ). Then for every n ∈ ω we have φ ∗ A ⊆ ⋃_{i<ω}[σ_{i}^{n} ∘ φ]_{ω}. But

∑_{i<ω} \frac{1}{2^{|σ_{i}^{n}∘φ|}} = ∑_{i<ω} \frac{1}{2^{|σ_{i}^{n}|}} < \frac{1}{n + 1}

Thus φ ∗ A is a null subset of 2^{ω}, so A ∈ N_{κ}. □

We shall use in our further considerations the following simple lemma.

Lemma 1.5. Let J be productive. If A ∈ J_{κ} and φ ∈ Inj(ω, κ) is its witness then every ψ ∈ Inj(ω, κ) such that rng(φ) ⊆ rng(ψ) is also a witness for A.

Proof. Let η = ψ^{−1} ∘ φ. Then η ∈ Inj(ω, ω) and η ∗ (ψ ∗ A) = φ ∗ A ∈ J. Thus ψ ∗ A ∈ ω (J) = J. □

Suppose that J is an ideal of subsets of X. We say that a family B ⊆ J is cofinal with J if for each A ∈ J there exists such B ∈ B that A ⊆ B. We also call such a family B a base of J.

For an ideal J of subsets of the set X we consider the following cardinal numbers

add(J) = min{|A| : A ⊆ J & ⋃ A ∉ J },
cov(J) = min{|A| : A ⊆ J & ⋃ A = X},
non(J) = min{|B| : B ⊆ X & B ∉ J },
cof(J) = min{|B| : B ⊆ J & B is cofinal with J }.

They are called the additivity, the covering, the uniformity and the cofinality of J, respectively. Note that if J is a proper ideal containing all points (i.e. ⋃ J = X) then the following relations hold:

add(J) ≤ cov(J), add(J) ≤ non(J), cov(J) ≤ cof(J), non(J) ≤ cof(J).

Moreover, add(J) is regular and add(J) ≤ min{cf(non(J)), cf(cof(J))}.

There is a cardinal function which plays a big role in our further considerations, namely the cofinality of the σ−ideal of all countable subsets of κ. We denote it by c_{κ} and remind some of its properties:

Fact 1.6.

(a) κ ≤ λ ⇒ c_{κ} ≤ c_{λ};

(b) cf(κ) > ω ⇒ c_{κ} = max{κ, sup_{ω≤ξ<κ} c_{ξ}};

(c) ω = cf(κ) < κ ⇒ c_{κ} > κ;

(d) ω_{1} ≤ κ < ω_{ω} ⇒ c_{κ} = κ;

(e) κ ≥ 𝔠 ⇒ c_{κ} = κ^{ω}.

Proof. The proof of this fact can be found in [5] (see Theorem 6.17 (c)). □

2. Additivity and cofinality of J_{κ}

From now on we assume that J is a proper and productive σ−ideal of subsets of 2^{ω} containing all points (i.e. ⋃ J = 2^{ω}) and that κ ≥ ω_{1}.

Cardinal coefficients add, cov, non and cof of a σ−ideal J have a strong influence on those of J_{κ}. However, the additivity of J_{κ} is absolute.

Theorem 2.1. add(J_{κ}) = ω_{1}

Proof. For every ordinal number α < κ we define I_{α} = {β < κ : α·ω ≤ β < α·ω+ω} and F_{α} = {x ∈ 2^{κ} : x↾I_{α} ≡ 0}. Obviously, F_{α} ∈ J_{κ} for each α < κ. Let T be any uncountable subset of κ. Then

⋃_{α∈T} F_{α} ∉ J_{κ}.

Actually, if there exists a witness φ for ⋃_{α∈T} F_{α} then φ ∗ F_{α} ∈ J for every α ∈ T. But there is α_{0} ∈ T such that rng(φ) ∩ I_{α_{0}} = ∅ and, consequently, φ ∗ F_{α_{0}} = 2^{ω} which is a contradiction. □

Next results give us a precise description of the cofinality of described ideals.

Theorem 2.2. cof(J_{κ}) ≤ max{cof(J), c_{κ}}.

Proof. Let {B_{ξ} : ξ < λ} ⊆ J be a base of J and let {φ_{α} : α < η} ⊆ Inj(ω, κ) be a family such that {rng(φ_{α}) : α < η} is a base of the σ−ideal of all countable subsets of κ. We define

B = {(B_{ξ})_{φ_{α}} : ξ < λ, α < η}.

It is enough to show that B is a base of J_{κ}.

Let A ∈ J_{κ} and let φ be its witness. Thanks to the Lemma 1.5 we can assume that φ = φ_{α_{0}} for some α_{0} < η. Let Z = φ_{α_{0}} ∗ A. We have Z ∈ J and it is easy to observe that if Z ⊆ B_{ξ_{0}} for some ξ_{0} < λ then A ⊆ (B_{ξ_{0}})_{φ_{α_{0}}} ∈ B which ends the proof. □

Theorem 2.3. cof(J_{κ}) ≥ c_{κ}.

Proof. We define the family {F_{α} : α < ω_{1}} in the same way as in the proof of Theorem 2.1. For every B ⊆ 2^{κ} we put

F_{B} = {α : F_{α} ⊆ B}.

We have already noticed that if B ∈ J_{κ} then F_{B} is countable. Let {B_{ξ} : ξ < λ} ⊆ J_{κ} be a base of J_{κ}. We define

B = {F_{B_{ξ}} : ξ < λ}.

Let T be any countable subset of κ. Then ⋃_{α∈T} F_{α} ⊆ B_{ξ_{0}} for some ξ_{0} < λ. Hence T ⊆ F_{B_{ξ_{0}}} ∈ B. □

To complete the characterization of the cofinality we have to introduce an additional property called the Weak Fubini Property (WFP).

Definition. The ideal J of subsets of 2^{ω} has WFP if for every φ ∈ Inj(ω, ω) and every A ⊆ 2^{ω} if A_{φ} is in J then so is A.

The name can be explained by formulating the definition in terms of products, in a way that has been used before to explain the meaning of productivity. Namely, J has WFP if for every φ ∈ Inj(ω, ω) and every A ⊆ 2^{rng(φ)} if the cylinder A × 2^{ω\rng(φ)} is in J then its projection into 2^{rng(φ)}, that is A, is also in J (after identifying 2^{rng(φ)} with 2^{ω} using φ). The σ−ideals mentioned previously, i.e. σ− ideals of meagre sets and of null sets of 2^{ω}, σ−ideal generated by closed null subsets of 2^{ω} and S_{2} obviously have WFP.

We will need the following technical lemma.

Lemma 2.4. If J is a productive ideal of subsets of 2^{ω} having WFP then for every φ ∈ Inj(ω, κ) and every A ⊆ 2^{ω} if A_{φ} ∈ J_{κ} then A ∈ J.

Proof. We fix φ ∈ Inj(ω, κ) and A ⊆ 2^{ω} such that A_{φ} ∈ J_{κ}. Let ψ be a witness for A_{φ}. We can assume that rng(φ) ⊆ rng(ψ). Then η = ψ^{−1} ∘ φ is a member of Inj(ω, ω). A routine calculation shows that A_{η} ⊆ ψ ∗ A_{φ} ∈ J so we can apply WFP to η and A and obtain the needed result. □

Theorem 2.5. If J has WFP then cof(J_{κ}) ≥ cof(J).

Proof. Notice that the identity function on ω (denoted by id_{ω}) is a member of Inj(ω, κ). For a set B ⊆ 2^{κ} we construct a set

H_{B} = {x ∈ 2^{ω} : {x}_{id_{ω}} ⊆ B}.

We observe that (H_{B})_{id_{ω}} = ⋃_{x∈H_{B}}{x}_{id_{ω}} ⊆ B.

Let {B_{ξ} : ξ < λ} ⊆ J_{κ} be a base of J_{κ}. We define a family

B = {H_{B_{ξ}} : ξ < λ}.

For every ξ < λ we know that (H_{B_{ξ}})_{id_{ω}} ∈ J_{κ}. Applying the Lemma 2.4 to id_{ω} and the set H_{B_{ξ}} leads us to the conclusion that H_{B_{ξ}} ∈ J.

Now fix any A ∈ J. Then the set A_{id_{ω}} is in J_{κ} so it is contained in B_{ξ_{0}} for some ξ_{0} < λ. But this imply that A ⊆ H_{B_{ξ_{0}}} . Thus B is a base for J. □

Corollary 2.6. If J has WFP then cof(J_{κ}) = max{cof(J), c_{κ}}. □

It is worth observing that Theorems 2.2 and 2.3 do not need an assumption that J has WFP. The natural question is if Theorem 2.5 (and therefore Corollary 2.6) are true without any extra assumptions. Question. Is it true that for any productive σ−ideal J of subsets of 2^{ω} we have cof(J_{κ}) = max{cof(J), c_{κ}}?

3. Uniformity of J_{κ}

The uniformity of J_{κ} differs from two coefficients considered in the previous section because no straightforward formula for it was found. Nevertheless, we can prove a series of useful inequalities. The first well-known fact shows that non(J_{κ}) form an increasing sequence.

Fact 3.1. If λ ≤ κ then non(J_{λ}) ≤ non(J_{κ}).

Proof. Let A be a subset of 2^{κ} which is not in J_{κ}. We define a subset of 2^{λ} in the following way

A′ = {x↾λ : x ∈ A}.

It is enough to notice that if φ ∈ Inj(ω, λ) would be a witness for A′ then it would be a witness for A as well (as a member of Inj(ω, κ)). Consequently, A′ is not in J_{λ}. □

Theorem 3.2. non(J_{κ}) ≤ max{non(J), c_{κ}}.

Proof. Let A ⊆ 2^{ω} does not belong to J and let {φ_{α} : α < η} ⊆ Inj(ω, κ) be a family such that {rng(φ_{α}) : α < η} is a base of the σ−ideal of all countable subsets of κ. For every α < η we define a set

A_{α} = {x ∈ 2^{κ} : x ∘ φ_{α} ∈ A & x↾(κ \ rng(φ_{α})) ≡ 0)}.

Let A^{∗} = ⋃_{α<η} A_{α}. We will prove that A^{∗} ∉ J_{κ}.

Suppose otherwise and let φ ∈ Inj(ω, κ) be a witness for A^{∗}. As in the proof of Theorem 2.2 we can assume that φ = φ_{α_{0}} for some α_{0} < η. For every x ∈ A we can easily construct a function x′ ∈ A_{α_{0}} such that x′ ∘ φ_{α_{0}} = x. But this means that A ⊆ A_{α_{0}} ∈ J which is a contradiction. □

It occurs that we can find also some bounds for non(J_{κ}) which depend only on κ. For any infinite cardinal number κ we define

log(κ) = min{λ : 2^{λ} ≥ κ}.

Theorem 3.3. non(J_{κ}) ≥ log(κ).

Proof. We show that every subset of 2^{κ} of cardinality strictly smaller than log(κ) is in J_{κ}. Let λ < log(κ) and A = {x_{α} : α < λ}. For every ξ < κ we define a function f_{ξ} ∈ 2^{λ} in the following way:

f_{ξ}(α) = x_{α}(ξ).

We have 2^{λ} < κ so we can find a countable infinite set of indices P ⊆ κ and a function f ∈ 2^{λ} such that for every ξ ∈ P we have f_{ξ} = f. Straight from the definition of f_{ξ} we obtain that for every α ∈ λ the function x_{α} ↾P is constant. Thus if φ is any member of Inj(ω, κ) such that rng(φ) = P then the set φ ∗ A has at most two elements and, consequently, is in J. □

Theorem 3.4. If 2^{λ} ≥ κ then non(J_{κ}) ≤ λ^{ω}.

Proof. The proof is based on an idea from [5]. According to the Hewitt-Marczewski-Pondiczery Theorem (see [4]) there exists a dense subset D of 2^{κ} of cardinality λ. As the space 2^{κ} is compact so every sequence s ∈ D^{ω} has a cluster point s^{∗}. Let

D^{∗} = {s^{∗} : s ∈ D^{ω}}.

We will show that D^{∗} ∉ J_{κ}.

Let φ be any member of Inj(ω, κ). We fix x ∈ 2^{κ} and for each natural number n we choose a function x_{n} ∈ D such that

(∀i < n) x_{n}(φ(i)) = x(φ(i)).

Let s^{∗} be a cluster point of the sequence s = {x_{n} : n < ω} ∈ D^{ω} which is in D^{∗}. Then

s^{∗} ↾rng(φ) = x↾rng(φ).

But it means that φ ∗ D^{∗} = 2^{ω} so none φ ∈ Inj(ω, κ) can be a witness for D^{∗}. □

Corollary 3.5. log(κ) ≤ non(J_{κ}) ≤ (log(κ))^{ω}. □

Corollary 3.6. non(J_{2^{κ}}) ≤ κ^{ω}. □

A consequence of these results is that the sequence non(J_{κ}) cannot be constant. However, it is bounded by 𝔠 for long.

Corollary 3.7. (a) non(J_{2^{𝔠}}) ≤ 𝔠,

(b) There exists κ = (2^{𝔠})^{+} such that non(J_{κ}) > non(J),

(c) If 2^{𝔠^{+}} > 2^{𝔠} then for each cardinal number ξ such that (2^{𝔠})^{+} ≤ ξ ≤ 2^{𝔠^{+}} we have non(J_{ξ}) = 𝔠^{+}.

Proof. Applying Corollary 3.6 for κ = 𝔠 we obtain (a). Moreover, according to Theorem 3.3 for κ = (2^{𝔠})^{+} we have non(J_{κ}) ≥ log(κ) > 𝔠. But non(J) ≤ 𝔠 which ends the proof of (b). Finally, Corollary 3.6 for κ = 𝔠^{+} together with the previous observation give us the proof of (c). □

The following theorem was proved in a weaker version by Fremlin in [6] for the σ−ideal of null sets.

Theorem 3.8. If cf(κ) > ω then non(J_{κ}) ≤ max{cf(κ), sup{non(J_{λ}) : λ < κ}}.

Proof. Let {λ_{α} : α < cf(κ)} be a cofinal sequence in κ. For each α < cf(κ) let A_{α} ⊆ 2^{λ_{α}} be a set which is not in J_{λ_{α}} such that |A_{α}| = non(J_{λ_{α}}). For each f ∈ A_{α} we fix f^{∗} ∈ 2^{κ} such that f^{∗} ↾ λ_{α} = f. Let A_{α}^{∗} = {f^{∗} : f ∈ A_{α}} and let A^{∗} = ⋃_{α<cf(κ)} A_{α}^{∗}. Obviously,

|A|^{∗} ≤ ∑_{α<cf(κ)} |A_{α}| ≤ max{cf(κ), sup{non(J_{λ}) : λ < κ}}.

To complete the proof we have to show that A^{∗} is not in J_{κ}. Suppose otherwise and let φ ∈ Inj(ω, κ) be a witness for A^{∗}. Uncountability of cf(κ) implies that rng(φ) ⊆ λ_{α} for some α < cf(κ). Thus we can treat φ as a member of Inj(ω, λ_{α}), too. So

φ ∗ A_{α} = φ ∗ A_{α}^{∗} ⊆ φ ∗ A^{∗} ∈ J .

But φ ∗ A_{α} ∉ J, which is a contradiction. □

The next theorem is a powerful tool which allows us to show several absolute inequalities concerning non’s as well as some consistency results.

If f and g are functions with the same domain D then we define

E(f, g) = {x ∈ D : f(x) = g(x)}.

Let us observe that the cardinal function cov(J) can be considered not only for ideals but also for arbitrary families of sets.

Theorem 3.9. Suppose that there exists a family A ⊆ λ^{κ} such that |A| = η and

cov({E(f, g) : f, g ∈ A & f ≠ g}) > ω.

Then non(J_{η}) ≤ max{κ, non(J_{λ})}.

Proof. Let A = {f_{α} : α < η}. To begin with, we show that for every I ∈ [η]^{ω} there exists ξ_{I} < κ such that for every different α, β ∈ I we have f_{α}(ξ_{I}) ≠ f_{β}(ξ_{I}). In order to do this fix I ∈ [η]^{ω}. From the assumption we know that

⋂_{{α,β}∈[I]^{2}} E^{c}(f_{α}, f_{β}) ≠ ∅.

Let ξ_{I} be any member of this intersection. It is easy to check that ξ_{I} fulfils the needed condition.

Now, let A be a subset of 2^{λ} which is not in J_{λ}. For every x ∈ A and ξ < κ we define a function x_{ξ} ∈ 2^{η} in the following way

x_{ξ}(α) = x(f_{α}(ξ)).

Let A^{∗} = {x_{ξ} : x ∈ A & ξ < κ}. We will show that A^{∗} ∉ J_{η}.

Suppose otherwise and let φ ∈ Inj(ω, η) be a witness for A^{∗}. In particular, the set φ ∗ {x_{ξ_{rng(φ)}} : x ∈ A} is in J. We define a function ψ in the following way

ψ(n) = f_{φ(n)}(ξ_{rng(φ)}).

Straight from the definition of ξ_{rng(φ)} we obtain that ψ ∈ Inj(ω, κ). Simple calculations show that ψ ∗ A = φ ∗ {x_{ξ_{rng(φ)}} : x ∈ A}, which is a contradiction. □

In order to apply this theorem efficiently, we look for a relatively big family A ⊆ κ^{κ} such that every pair of functions from this family agree on the set from a certain σ−ideal of subsets of κ. The easiest such σ−ideal is [κ]^{<κ}, on condition cf(κ) > ω.

We say that two functions f, g ∈ λ^{κ} are eventually different if there exists α < κ such that E(f, g) ⊆ α (in particular, E(f, g) ∈ [κ]^{<κ}).

Theorem 3.10. If cf(κ) > ω then non(J_{κ^{+}}) ≤ max{κ, non(J_{κ})}.

Proof. Using the standard diagonal argument we can show that there exists a family A ⊆ κ^{κ} of pairwise eventually different functions of size κ^{+}. Thanks to uncountable cofinality of κ the assumption of Theorem 3.9 is fulfilled and we are done. □

As a special case of this theorem we obtain a quite surprising result. It was proved by Veličkovič in a special case of meagre sets but the proof remained unpublished

Corollary 3.11. For any productive σ−ideal J we have non(J) = non(J_{ω_{1}} ) = non(J_{ω_{2}} ).

Proof. According to Fact 3.1 it is enough to show that non(J) ≥ non(J_{ω_{1}}) ≥ non(J_{ω_{2}}). Applying Theorem 3.2 (and Fact 1.6) we obtain

non(J_{ω_{1}}) ≤ max{c_{ω_{1}}, non(J)} = max{ω_{1}, non(J)} = non(J).

Furthermore, we get from Theorem 3.10 that non(J_{ω_{2}}) ≤ max{ω_{1}, non(J_{ω_{1}})} = non(J_{ω_{1}}) which ends the proof. □

This result cannot be improved. For instance, assuming Generalized Continuum Hypothesis we obtain

ω_{1} = non(J) = non(J_{ω_{1}}) = non(J_{ω_{2}}) < non(J_{ω_{3}}) = ω_{2}.

Next two corollaries are other examples of applying Theorem 3.10 and give us slightly stronger results concerning non(J_{κ}) in some special cases.

12 JAN KRASZEWSKI

Corollary 3.12. If κ < ω_{ω} then non(J_{κ^{+}}) ≤ max{κ, non(J)}. Proof. It is enough to iterate Theorem 3.10 finitely many times and use Corollary 3.11. □ Corollary 3.13. If non(J) = ω_{n} for some natural number n then non(J) = non(J_{ω_{n+1}}). Proof. We obtain from Corollary 3.12 that non(J_{ω_{n+1}}) ≤ max{ω_{n}, non(J)} = ω_{n}. Hence, according to Fact 3.1 the proof is finished. □

In [9] we can find another example of a big family of pairwise disjoint functions which allows us to formulate the following theorem. Theorem 3.14. non(J_{2^{κ^{+}}} ) ≤ max{κ^{+}, non(J_{2^{κ}})}. Proof. For every X ⊆ κ^{+} we define a function f_{X} : κ^{+} → [κ^{+}]^{≤κ} (and |[κ^{+}]^{≤κ}| = 2^{κ}) in the following way:

f_{X}(α) = X ∩ α.

Let A = {f_{X} : X ⊆ κ^{+}}. Thus |A| = 2^{κ^{+}}. Obviously, for different X, Y ⊆ κ^{+} we have E(f_{X}, f_{Y}) ⊆ α < κ^{+} for some α and thanks to the regularity of κ^{+} we can apply Theorem 3.9, which ends the proof. □

As before, we can extract from this theorem some interesting corollaries. Corollary 3.15. If ξ is a cardinal number such that 𝔠 ≤ ξ ≤ 2^{ω_{1}} then non(J_{ξ}) = non(J_{𝔠} ). Proof. It is enough to apply Theorem 3.14 for κ = ω. □ Corollary 3.16. If κ < ω_{ω} then non(J_{2^{κ}}) ≤ max{κ, non(J_{𝔠} )}. Proof. It is enough to iterate Theorem 3.14 finitely many times and use Corollary 3.15. □

In general, constructing big families of pairwise eventually different functions demands sophisticated combinatorial assumptions. For example, we have the following fact. Fact 3.17. ([10], Theorem 3. 1) Assume that 𝔠 < ω_{ω_{1}} and 𝔠 < 2^{ω_{1}} . Then there exists a family A ⊆ ω_{1}^{ω_{1}} of pairwise eventually different functions of size 2^{ω_{1}} . □

Corollary 3.18. Under the assumptions of Fact 3.17, if ξ ≤ 2^{ω_{1}} is an infinite cardinal number then non(J_{ξ}) = non(J). Proof. Straightforward from Theorem 3.9 and Fact 3.17. □ Question. Is it consistent with ZFC that non(J) < non(J_{𝔠})?

4. Covering of J_{κ}

As in a case of the uniformity, no straightforward formula for the covering of J_{κ} was found. However, the situation is different because cov(J_{κ}) form a decreasing sequence.

Fact 4.1. If λ ≤ κ then cov(J_{λ}) ≥ cov(J_{κ}).

Proof. Let {A_{ξ} : ξ < η} ⊆ J_{λ} be a covering family for 2^{λ} (i.e. ⋃_{ξ<η} A_{ξ} = 2^{λ}). We notice that if A ⊆ J_{λ} with a witness φ ∈ Inj(ω, λ) then a set

K_{A} = {x ∈ 2^{κ} : x↾λ ∈ A}

is in J_{κ} with the same witness, treated as a member of Inj(ω, κ).

To finish the proof we observe that a family {K_{A_{ξ}} : ξ < η} ⊆ J_{κ} is a covering family for 2^{κ}. □

This fact implies that results concerning coverings of J_{κ}’s are of totally different type than theorems proved so far.

Theorem 4.2. If cov(J_{κ}) ≤ λ ≤ κ then cov(J_{κ}) = cov(J_{λ}).

Proof. According to Fact 4.1 it is enough to show that cov(J_{κ}) ≥ cov(J_{λ}). Let {A_{ξ} : ξ < η} ⊆ J_{κ} be a covering family for 2^{κ} and for each ξ < η let φ_{ξ} ∈ Inj(ω, κ) be a witness for A_{ξ}. We know that η ≤ λ so there exists a set B ⊆ κ such that

⋃_{ξ<η} rng(φ_{ξ}) ⊆ B

and we can find a bijection Φ from λ onto B. Thus for each ξ < η the set Φ ∗ A_{ξ} = {x ∘ Φ : x ∈ A_{ξ}} ⊆ 2^{λ} is in J_{λ} (because the function Φ^{−1} ∘ φ_{ξ} ∈ Inj(ω, λ) is its witness).

Fix x ∈ 2^{λ}. Let x′ be any function from 2^{κ} such that x′ ↾B = x ∘ Φ^{−1} ↾B. Then x′ ∈ A_{ξ_{0}} for some ξ_{0} < η and x = x′ ∘Φ ∈ Φ∗A_{ξ_{0}}. Hence the family {Φ∗A_{ξ} : ξ < η} is a covering family for 2^{λ}. □

Let us consider a function from infinite cardinal numbers into themselves given by a formula F (κ) = cov(J_{κ}). According to Fact 4.1 this function is decreasing. The following corollaries show that its diagram intersects nonempty with the diagonal and the function stabilizes from this point.

Corollary 4.3. If κ ≥ 𝔠 then cov(J_{κ}) = cov(J_{𝔠}).

Proof. (cf [5], Theorem 6.17 (d)(iii)) Notice that cov(J) ≤ 𝔠 as the family of all singletons is a covering family for 2^{ω}. Therefore, we have cov(J_{κ}) ≤ cov(J) ≤ 𝔠 ≤ κ and it is enough to apply Theorem 4.2. □

Corollary 4.4. If cov(J_{𝔠}) = λ then cov(J_{λ}) = λ.

Proof. We know from Fact 4.1 that cov(J_{κ}) ≤ cov(J) ≤ 𝔠. Applying Theorem 4.2 to κ = 𝔠 we obtain that cov(J_{λ}) = cov(J_{𝔠}) = λ. □

Corollary 4.5. If cov(J_{λ}) > λ for some cardinal number λ then (∀κ ≥ λ) cov(J_{κ}) > λ.

Proof. It is enough to show that if there exists a cardinal number κ such that κ ≥ λ and cov(J_{κ}) ≤ λ then cov(J_{λ}) ≤ λ. But this is true according to Theorem 4.2. □

If the covering of J_{κ} is relatively small we have a bound for it depending only on κ.

Theorem 4.6. If cov(J_{κ}) < add(J) then cov(J_{κ}) ≤ c_{κ}.

Proof. Let {A_{ξ} : ξ < η} ⊆ J_{κ} be a covering family for 2^{κ} and for each ξ < η let ψ_{ξ} ∈ Inj(ω, κ) be a witness for A_{ξ}. Let {φ_{α} : α < c_{κ}} ⊆ Inj(ω, κ) be a family such that {rng(φ_{α}) : α < c_{κ}} is a base of the σ−ideal of all countable subsets of κ. We have to show that if η < add(J) then we can construct a covering family for 2^{κ} of cardinality at most c_{κ}.

For each α < c_{κ} we define a set

T_{α} = {ξ < η : rng(ψ_{ξ}) ⊆ rng(φ_{α})}.

According to the Lemma 1.5 the function φ_{α} is a common witness for all A_{ξ} such that ξ ∈ T_{α}. But the set T_{α} has cardinality strictly smaller than add(J), which means that

φ_{α} ∗ ( ⋃_{ξ∈T_{α}} A_{ξ}) = ⋃_{ξ∈T_{α}}(φ_{α} ∗ A_{ξ}) ∈ J .

Let C_{α} = ⋃_{ξ∈T_{α}} A_{ξ}. We showed above that C_{α} ∈ J_{κ}. Moreover, we know that ⋃_{α<c_{κ}} C_{α} = ⋃_{ξ<η} A_{ξ} (because ⋃_{α<c_{κ}} T_{α} = η) so the family {C_{α} : α < c_{κ}} is a covering family for 2^{κ}. □

Corollary 4.7. If cov(J_{ω_{1}} ) < add(J) then (∀κ ≥ ω_{1}) cov(J_{κ}) = ω_{1}.

Proof. We deduce from Fact 1.6 that c_{ω_{1}} = ω_{1} and then use Fact 4.1 and Theorem 4.6. □

If we add certain assumptions about J and 𝔠, we will be able to show that the range of a function F (x) = cov(J_{x}) has at most two elements. If the function F is not constant then we can precisely show the place where it changes its value.

Corollary 4.8. If add(J) = cov(J) then for every natural number n

cov(J_{ω_{n}}) < cov(J) ⇒ cov(J_{ω_{n}}) ≤ ω_{n}.

Proof. As in the previous proof using the fact that for every natural number n we have c_{ω_{n}} = ω_{n}. □

Corollary 4.9. If add(J) = cov(J) and 𝔠 = ω_{n} where n is a natural number then there exists at most one natural number m such that cov(J_{ω_{m}}) > cov(J_{ω_{m+1}}). If such m really exists then cov(J_{ω_{m+1}}) = cov(J_{𝔠}) = ω_{m+1}.

Proof. Let n be a natural number. We know from Corollary 4.4 that if cov(J_{𝔠}) = ω_{m+1} for some m ∈ ω then cov(J_{κ}) = ω_{m+1} for all κ such that ω_{m+1} ≤ κ ≤ 𝔠.

On the other hand if cov(J_{ω_{i}}) > cov(J_{ω_{i+1}}) for some i < m then cov(J) > cov(J_{ω_{i+1}}) and we know from Corollary 4.8 that

cov(J_{ω_{i+1}}) ≤ ω_{i+1} < ω_{m+1} = cov(J_{ω_{m+1}})

which contradicts Fact 4.1. □

5. Several models

In this section we investigate the cardinal coefficients of classical σ −ideals in some models of the Set Theory. Let us assume, as usual, that J is a proper and productive σ−ideal of subsets of 2^{ω} containing all points.

Theorem 5.1. Let κ be an uncountable cardinal number. If GCH holds then

(a) add(J_{κ}) = cov(J_{κ}) = ω_{1}.

(b) (1) If cf(κ) = ω then non(J_{κ}) = κ^{+}; (2) if κ = λ^{+} and cf(λ) > ω then non(J_{κ}) = λ;

(3) otherwise, non(J_{κ}) = κ.

(c) If κ ≥ ω_{2} then (1) If cf(κ) = ω then cof(J_{κ}) = κ^{+};

(2) If cf(κ) > ω then cof(J_{κ}) = κ.

Proof. (cf [5], Theorem 6.18)

(a) Obvious from Fact 4.1.

(b) If κ = λ^{+} then log(κ) = λ and according to the Corollary 3.5 we have λ ≤ non(J_{κ}) ≤ λ^{ω} ≤ λ^{+}. Thus if cf(λ) > ω then λ^{ω} = λ, so non(J_{κ}) = λ. If cf(λ) = ω then non(J_{κ}) must be λ^{+} because it has an uncountable cofinality. If κ is a limit cardinal number then log(κ) = κ so κ ≤ non(J_{κ}) ≤ κ^{ω} ≤ κ^{+} by Corollary 3.5. As before, cf(κ) > ω implies that non(J_{κ}) = κ and cf(κ) = ω implies that non(J_{κ}) = κ^{+}.

(c) As cof(J) ≤ 2^{𝔠} = ω_{2} so we deduce from Theorems 2.2 and 2.3 and Fact 1.6 that for κ ≥ ω_{2} we have

cof(J_{κ}) = c_{κ} = κ^{ω}.

As in the previous part of this proof, cf(κ) > ω implies cof(J_{κ}) = κ and cf(κ) = ω implies cof(J_{κ}) = κ^{+} (because cf(cof(J_{κ})) is uncountable). □

Remark. We are not able to compute precisely cof(J_{ω_{1}}) in the previous theorem without any extra assumptions. But for example, if cof(J) = 𝔠 = ω_{1} (for instance, J has a base consisting of Borel sets) then cof(J_{ω_{1}}) = ω_{1}. Moreover, if J has WFP then cof(J_{ω_{1}}) = cof(J).

Another interesting result can be obtained by applying Martin’s Axiom (MA).

Theorem 5.2. If MA holds then for each κ ≤ 2^{𝔠} we have non(J_{κ}) = non(J) = 𝔠.

Proof. Straight from definitions we have 𝔰 ≤ ℵ_{0}-𝔰, where 𝔰 is the classical splitting number. We know that under MA we have 𝔰 = 𝔠 (see for example [16]). Therefore non(S_{2}) = ℵ_{0}-𝔰 = 𝔠. As it is easy to observe, if I ⊆ J then non(I) ≤ non(J). Thus non(J) = 𝔠. So it is enough to apply Fact 3.1 and the Corollary 3.7. □

Remark. As one can see, we can obtain the same result replacing MA by a weaker combinatorial property i.e. ℵ_{0}-𝔰 = 𝔠. It is worth observing that assuming another property (which is also implied by MA), namely (∀ω ≤ κ < 𝔠)(2^{κ} = 𝔠) we get another, weaker result: non(J_{𝔠^{+}}) = 𝔠. The proof is straightforward from Corollary 3.5.

The above results are true for every productive σ −ideal J. Now we focus on the σ−ideal of meagre sets.

The set A is called a (κ, λ) − Luzin set for a σ−ideal J if A is of cardinality κ and for every set B ∈ J we have |A ∩ B| < λ. For more information about Luzin sets see [2]. In particular, the following fact holds.

Fact 5.3. Let κ be an uncountable cardinal number. If there exists a (κ, ω_{1}) − Luzin set for a σ−ideal J then cov(J) ≥ κ and non(J) = ω_{1}. □

In our further considerations we will use the standard method of forcing (see [8] for more details) to obtain required models.

Theorem 5.4. Let λ be a cardinal number of uncountable cofinality and assume that GCH holds in the ground model V. Let V[G] be a generic extension obtained by adding λ Cohen reals to the ground model. Then in V[G]

(a) (∀κ ≥ ω)(cov(M_{κ}) = λ = 𝔠);

(b) (∀ω ≤ κ ≤ λ)(non(M_{κ}) = ω_{1});

(c) for κ ≥ λ, non(M_{κ}) is computed as in the Theorem 5.1.

Proof. It is well-known (see for example [14]) that V[G] ⊧ 𝔠 = λ and that added Cohen reals form a (λ, ω_{1})–Luzin set for M. Moreover, for each κ ≤ λ there exists also a (λ, ω_{1})–Luzin set for M_{κ}. It can be easily obtained by glueing the Cohen reals into functions from 2^{κ}. Then straight from Fact 5.3 and Corollary 4.3 we get (a) and (b). To prove (c) it is enough to observe that V[G] ⊧ 2^{κ} = κ^{+} for such κ’s. Consequently, we can compute non(M_{κ}) exactly as in the proof of Theorem 5.1(b) (using Corollary 3.5). □

The situations of adding κ Cohen reals for κ’s of countable cofinality is a bit more complicated.

Theorem 5.5. Assume that GCH holds in the ground model and let V[G] be a generic extension obtained by adding ω_{ω} Cohen reals to the ground model. Then in V[G]

(a) cov(M) = ω_{ω+1} = 𝔠 and (∀κ ≥ ω_{1})(cov(M_{κ}) = ω_{ω});

(b) (∀ω ≤ κ ≤ ω_{ω})(non(M_{κ}) = ω_{1});

(c) for κ ≥ ω_{ω+2}, non(M_{κ}) is computed as in the Theorem 5.1.

Proof. We know that V[G] ⊧ 𝔠 = ω_{ω+1}. The same argument as in the previous proof shows that for each infinite κ ≤ ω_{ω}, there exist an (ω_{ω}, ω_{1})–Luzin set for M_{κ}. This proves (b) and shows that for every such κ, cov(M_{κ}) ≥ ω_{ω}. Miller showed in [15] that cf(cov(M)) is uncountable, so cov(M) = ω_{ω+1}. In the same paper he proved that cov(M_{ω+1}) = ω_{ω} which together with Corollaries 4.3 and 4.4 end the proof of (a).

Finally, (c) is true as V[G] ⊧ (∀κ ≥ ω_{ω+1})(2^{κ} = κ^{+}) so we can apply Corollary 3.5 to compute non(M_{κ}) for κ ≥ ω_{ω+2}. □

Question. What is the value of non(M_{ω_{ω+1}}) in the model described in Theorem 5.5?

Remark. Let us notice that Theorems 5.4 and 5.5 remain true if we replace meagre sets by null sets and Cohen reals by random reals. In the case of the first theorem the proof is analogous. In the case of the latter one we have to use additionally Theorem 3 from [15].

We introduce a well-known combinatorial principle called “stick” and denoted by ( |•) and a cardinal number |• connected with it. For more details see [7].

Definition. ( |•): There exists a family T ⊆ [ω_{1}]^{ω} of cardinality ω_{1} such that for any X ∈ [ω_{1}]^{ω_{1}} there exists T ∈ T with T ⊆ X. |• = min{|T | : T ⊆ [ω_{1}]^{ω} & (∀X ∈ [ω_{1}]^{ω_{1}})(∃T ∈ T )T ⊆ X}.

Let us observe that ( |•) ⟺ |• = ω_{1}. The following simple fact holds.

Fact 5.6. Let J be any productive and proper σ−ideal of subsets of 2^{ω} containing all points. Then

cov(J_{ω_{1}} ) ≤ |• .

Proof. Let T be any family of countable infinite subsets of ω_{1} such that for each X ∈ [ω_{1}]^{ω_{1}} there exists T ∈ T with T ⊆ X. Let 0_{T} and 1_{T} denote functions constantly equal to 0 and 1, respectively, on its domain, which is the set T. Let [0_{T}] = {x ∈ 2^{ω} : x↾T ≡ 0} and [1_{T}] = {x ∈ 2^{ω} : x↾T ≡ 1}. Then for every T ∈ T both sets [0_{T}] and [1_{T}] are in J and any φ ∈ Inj(ω, ω_{1}) such that rng(φ) = T is their witness. Thus the family

F = {[0_{T}], [1_{T}] : T ∈ T } ⊆ J

is a covering family. To prove this it is enough to observe that for any function x ∈ 2^{ω} either its pre-image of 0 or its pre-image of 1 is uncountable. Hence we can find a set T ∈ T contained in one of these pre-images and, consequently, either x ∈ [0_{T}] or x ∈ [1_{T}]. □

Now we construct another model in which the function F (κ) = cov(M_{κ}) falls only once but rapidly.

Theorem 5.7. Let κ be an uncountable cardinal number and assume that GCH holds in the ground model. We define a forcing notion ℙ_{κ} = {p : p is a function & dom(p) ∈ [κ ]^{≤ω} & rng(p) ⊆ ω^{<ω}}. For p, q ∈ ℙ_{κ} we define

p ≤ q ⟺ dom(p) ⊇ dom(q) & (∀α ∈ dom(q))(p(α) ⊇ q(α)).

If G is a ℙ_{κ}-generic filter over V then V[G] ⊧ cov(M) = 𝔠 & cov(M_{ω_{1}} ) = ω_{1}.

Proof. This notion of forcing was introduced in a special case κ = ω_{3} by Miller in [15].

As GCH holds in V then a special case of the Δ–lemma is true in V, namely every family of ω_{2} countable sets contains a Δ–system of cardinality ω_{2} (see [12] for more details). This fact implies that ℙ_{κ} has the ω_{2}–chain condition so cardinal numbers bigger then ω_{1} are preserved, while ω_{1}^{V} is collapsed, which was all shown in [15]. Hence V[G] ⊧ |ω_{k+1}^{V}| = ω_{k} for every natural number k and V[G] ⊧ |ω_{ξ}^{V}| = ω_{ξ} for ξ ≥ ω.

Let us take g = ⋃ G : κ → ω^{ω}. One can show using a typical argument from the forcing theory that g(α) is a Cohen real for each ordinal number α < κ (in a sense that g(α) does not belong to any meager subset of ω^{ω} coded in V). Thus ℙ_{κ} adds |κ| many Cohen reals, which form (in V[G]) a (|κ|, ω_{2}^{V})–Luzin set for M (because ℙ_{κ} satisfies the ω_{2}–chain condition).

It is well-known that in V[G] we have 𝔠 = |κ| if cf(κ) > ω and 𝔠 = |κ|^{+} otherwise. Similarly as in the proofs of Theorems 5.4 and 5.5 the existence of the (|κ|, ω_{1})–Luzin set for M (our Luzin set is a subset of ω^{ω} instead of 2^{ω}, but cardinal coefficients of M are the same for any uncountable Polish space without isolated points) implies that V[G] ⊧ cov(M) = 𝔠. To complete the proof we have to show that V[G] ⊧ cov(M_{ω_{1}}) = ω_{1}. In order to do this we need the following claim:

Claim. V[G] ⊧ ( |•).

Let V[G] ⊧ f : ω_{1} → ω_{1} & f is an injection and take p_{0} ∈ G ∩ ℙ_{κ} such that p_{0} ∥−“ḟ : ω_{2}^{V} → ω_{2}^{V} & ḟ is an injection”. In V we define a set

D_{f} = {q ∈ ℙ_{κ} : q ≤ p_{0} & (∃Z ∈ [ω_{2}]^{ω})q ∥−“Z ⊆ rng(ḟ)”}.

We show that D_{f} is dense in ℙ_{κ} under p_{0}.

Fix any p ≤ p_{0}. For every α ∈ ω_{2} there exists q_{α} ≤ p and β_{α} ∈ ω_{2} such that q_{α} ∥−“ḟ(α) = β_{α}”. Using the same version of the Δ–lemma as above we get a set Y ∈ [ω_{2}]^{ω_{2}} such that functions {q_{α} : α ∈ Y } form a Δ–system (i.e. their domains form a Δ–system and they are equal on its root). Then for any set T ∈ [Y]^{ω} we have

q_{T} = ⋃_{α∈T} q_{α} ∈ ℙ_{κ} .

This implies that for each α ∈ T we have q_{T} ∥−“ḟ(α) = β_{α}”. Thus

q_{T} ∥−“{β_{α} : α ∈ T } ⊆ rng(ḟ)”.

So q = q_{T} and Z = {β_{α} : α ∈ T }. The set Z is infinite as the function f is an injection. Hence q ≤ p_{0} and q ∈ D_{f}.

As D_{f} is dense in ℙ_{κ} under p_{0} then there exists p′ ≤ p_{0} such that p′ ∈ G ∩ D_{f}. Then there exists Z ∈ [ω_{2}^{V}]^{ω} such that p′ ∥−“Z ⊆ rng(ḟ)” and, consequently,

V[G] ⊧ (∃Z ∈ [ω_{1}]^{ω} ∩ V) Z ⊆ rng(f).

To finish the proof of the claim it is enough to observe that if V[G] ⊧ B ∈ [ω_{1}]^{ω_{1}} then there exists in V[G] an injection f : ω_{1} → B ⊆ ω_{1}. Thus there exists a set Z ∈ [ω_{1}]^{ω} ∩ V such that Z ⊆ rng(f) ⊆ B. Hence

V[G] ⊧ |• ≤ |[ω_{1}]^{ω} ∩ V |.

But the cardinality of the family [ω_{1}]^{ω} ∩ V is equal to the cardinality of the family [ω_{2}^{V}]^{ω} counted in the ground model, which is equal to ω_{2}^{V} (as V ⊧ GCH).

To finish the proof of the theorem it is enough to use Fact 5.6. □

Remark. Miller in his paper [15] proved a slightly weaker property of the forcing ℙ_{ω_{3}}. We proved that in the generic extension obtained by this forcing the principle ( |•) holds.

As we show in Corollary 4.9 under certain assumptions the range of a function F (κ) = cov(M_{κ}) has at most two elements. The same situation is in the models from Theorems 5.4, 5.5 and 5.7. The problem is how to construct a model in which the range of the function F has three elements. Question. Is it consistent with ZFC that cov(M) = ω_{4} = 𝔠 & cov(M_{ω_{1}}) = ω_{3} & (∀κ ≥ ω_{2}) cov(M_{κ}) = ω_{2}?

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Institute of Mathematics, University of Wrocław, pl. Grunwaldzki 2/4, 50-156 Wrocław, Poland

E-mail address: kraszew@math.uni.wroc.pl