AMS JOURNAL STYLE Volume 00, 0000

ON SOME NEW IDEALS ON THE CANTOR AND BAIRE SPACES.

JACEK CICHOŃ AND JAN KRASZEWSKI

Abstract. We define and investigate some new ideals of subsets of the Cantor space and the Baire space. We show that combinatorial properties of these ideals can be described by the splitting and reaping cardinal numbers. We show that there exist perfect Luzin sets for these ideals on the Baire space.

0. Introduction

For each infinite subset T of the set ω of all natural numbers let us denote by K(T) the σ-ideal of meagre subsets of the space 2^{T} with the canonical product topology. By L(T) we denote the σ-ideal of Lebesgue measure zero subsets of 2^{T} with respect to the canonical product measure.

Notice that if T is a subset of ω then we can indentify the spaces 2^{T} × 2^{ω\T} and the Cantor space 2^{ω} using the canonical homeomorphism π_{T} defined by π_{T} (x) = (x|T, x|(ω \ T)). Directly from the definition of meagre sets it follows that if A ∈ K(T) then A × 2^{ω\T} ∈ K(ω). The same observation is also true for the ideal L(ω) but it is evidently false for the σ-ideal of all countable subsets of the Cantor space. We call this property of the ideals K(ω) and L(ω) productivity.

There are other natural productive σ-ideals of subsets of the Cantor space, e.g.

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society 0000-0000/00 $1.00 + $.25 per page

Received by the editors.

1991 Mathematics Subject Classification. 04A20, 28A05.

Key words and phrases. Set Theory, ideals, Borel sets, Cantor space, Baire space, cardinal functions.

Research of the second author supported by a grant 2149/W/IM/96 from University of Wrocław.

the σ-ideal K(ω) ∩ L(ω). It is interesting that among them there exists the least productive σ-ideal which contains all points. We call this ideal S_{2}. There exists also the least productive ideal of subsets of 2^{ω} and we call it I_{2}. These ideals have Borel bases but they do not satisfy the countable chain condition - there exists a family of continuum many pairwise disjoint Borel sets outside the ideal S_{2}. The ideal I_{2} is not σ-additive and the ideal S_{2} is precisely σ-additive. The minimum cardinality of bases of these ideals is continuum. The covering number cov of both ideals is equal to the reaping cardinal 𝔯 and the last basic combinatorial invariants (cardinal numbers non) are described in terms of splitting cardinal numbers. These results show that both splitting and reaping cardinals are closely connected with natural mathematical objects on the classical Cantor space.

We also consider the minimal productive σ-ideal S_{ω} of subsets of the Baire space ω^{ω}. We show that there exists an uncountable closed subset P of ω^{ω} which is a Luzin set for S_{ω}, i.e. the intersection of the set P with any set from S_{ω} is countable. This fact completely determines the basic combinatorial invariants of this ideal.

1. Notation, definitions and basic observations

In this paper we use the standard set theoretical notation. For example, ω denotes the first infinite cardinal number which we shall identify with the set of all natural numbers. The cardinality of the set of all real numbers is denoted by 𝔠. If κ is a cardinal number then [X]^{κ} denotes the family of all subsets of X of cardinality κ and [X]^{<κ} denotes the family of all subsets of X of cardinality strictly less then κ. X^{<ω} denotes the set of all finite sequences of elements of the set X. The power set of a set X is denoted by P (X). For A, B ⊆ ω we put A ⊆^{∗} B if and only if card (A \ B) < ω.

If X is a discrete topological space then we endow X^{ω} with the standard product topology. In paricular, for X = 2 and X = ω we get the Cantor space and the Baire space, respectively.

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

add(J) = min{card (A) : A ⊆ J & ⋃ A ∉ J },
cov(J) = min{card (A) : A ⊆ J & ⋃ A = X},
non(J) = min{card (B) : B ⊆ X & B ∉ J },
cof(J) = min{card (A) : A ⊆ J & (∀A ∈ J)(∃B ∈ A)(A ⊆ B)}.

Note that if J is a proper ideal and ⋃ J = X then the following relations hold:

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

Suppose that J is an ideal of subsets of X and A ⊆ P (X). We say that J has an A-base if for each A ∈ J there exists such B ∈ A ∩ J that A ⊆ B. Hence, in particular, if X is a topological space then J has an F_{σ}-base if each element from J can be covered by some F_{σ} subset of X from J.

Let K and L denote the σ-ideals of meagre subsets and of Lebesgue measure zero subsets of the Cantor space 2^{ω}, respectively. The ideal K has an F_{σ}-base and the ideal L has a G_{δ}-base.

From now on let us assume that X has at least two elements. We define

P if(X) = {φ : φ is a function & dom(φ) ∈ [ω]^{ω} & rng(φ) ⊆ X}.

If φ ∈ P if(X) then we put

\begin{matrix}[φ]_{X} = & \{x ∈ X^{ω} : φ ⊆ x\}, \\ [φ]_{X}^{∗} = & \{x ∈ X^{ω} : φ ⊆^{∗} x\}. \end{matrix}

If we treat X as a discrete topological space then [φ]_{X} is a closed and [φ]_{X}^{∗} is an F_{σ} subset of X^{ω} for each φ ∈ P if(X).

Now we are able to define the ideals we are going to deal with. Let I_{X} and I_{X}^{∗} denote the ideals generated by families {[φ]_{X} : φ ∈ P if(X)} and {[φ]_{X}^{∗} : φ ∈ P if(X)}, respectively. Then [X]^{<ω} ⊆ I_{X} ⊆ I_{X}^{∗} and I_{X}^{∗} is a proper ideal of subsets of X^{ω}. The first ideal has a closed base and the other one has an F_{σ}-base.

Similarly, let S_{X} and S_{X}^{∗} denote the σ-ideals generated by families {[φ]_{X} : φ ∈ P if(X)}, {[φ]_{X}^{∗} : φ ∈ P if(X)}, respectively. It is easy to observe that S_{X} = S_{X}^{∗}. Hence we have I_{X} ⊆ I_{X}^{∗} ⊆ S_{X}. The ideal S_{X} has an F_{σ}-base and is proper. Directly from the definition of the ideals we can deduce that

cov(I_{X}) = cov(I_{X}^{∗}) = cov(S_{X}).

If j : ω → ω is an injection and A ⊆ 2^{ω} then we define

j ∗ A = {x ∈ 2^{ω} : x ∘ j ∈ A}.

We say that an ideal J of subsets of 2^{ω} is productive if j ∗ A ∈ J for each A ∈ J and any injection j : ω → ω. As we have mentioned in the introduction, the ideals K and L are productive.

Let φ ∈ P if(2) and let j : ω → dom(φ) be any bijection. Then j ∗{φ∘j} = [φ]_{2}. Conversely, if j : ω → ω is an injection and x ∈ 2^{ω}, then x ∘ j^{−1} ∈ P if(2) and

j ∗ {x} = [x ∘ j^{−1}]_{2}.

These observations imply that I_{2} and S_{2} are the least productive ideal and σ-ideal of subsets of 2^{ω} (containing all points), respectively. Since K and L are productive, we see that S_{2} ⊆ K ∩ L.

Let us recall that σ-ideals of subsets of a Polish space with Borel bases have plenty of interesting properties (see e.g. [2]). The next result shows the main difference between the ideal S_{2} and the ideals K and L.

Theorem 1.1. There exists a family F of pairwise disjoint Borel subsets of 2^{ω} such that card (F) = 𝔠 and none of elements of F belongs to S_{2}.

Proof. For each real number α ∈ [0, 1] we put

A_{α} = \{x ∈ 2^{ω} : lim_{n→∞} \frac{1}{n} ∑_{i=1}^{n} x(i) = α\}.

It is easy to check that {A_{α} : α ∈ [0, 1]} is the required family. □

Let A, S be two infinite subsets of ω. We say that S splits A if card (A ∩ S) = card (A \ S) = ω. Let us recall the following three cardinal numbers (see e.g. [3]):

𝔰 = min{card (S) : S ⊆ [ω]^{ω} & (∀A ∈ [ω]^{ω})(∃S ∈ S)(S splits A)},
ℵ_{0}-𝔰 = min{card (S) : S ⊆ [ω]^{ω} & (∀A ∈ [[ω]^{ω}]^{ω})(∃S ∈ S)(∀A ∈ A)(S splits A)},
𝔯 = min{card (R) : R ⊆ [ω]^{ω} & (∀S ∈ [ω]^{ω})(∃R ∈ R)(S does not split R)}.

The cardinal numbers 𝔰 and 𝔯 are called splitting and reaping, respectively. It is easy to check that ω_{1} ≤ 𝔰 ≤ ℵ_{0}-𝔰 ≤ 𝔠 and ω_{1} ≤ 𝔯 ≤ 𝔠. It is an open problem now if 𝔰 = ℵ_{0}-𝔰 can be proved in ZFC. An easy reformulation of the definition of the reaping number 𝔯 gives us the following description

𝔯 = min{card (R) : R ⊆ [ω]^{ω} & (∀S ∈ [ω]^{ω})(∃R ∈ R)(R ⊆ S ∨ R ⊆ ω \ S)}.

Let us introduce an auxiliary cardinal number. Namely, we define

fin-𝔰 = min{card (S) : S ⊆ [ω]^{ω} & (∀A ∈ [[ω]^{ω}]^{<ω})(∃S ∈ S)(∀A ∈ A)(S splits A)}.

Lemma 1.2. 𝔰 = fin-𝔰 .

Proof. It is clear that 𝔰 ≤ fin-𝔰. Suppose now that S ⊆ [ω]^{ω} is such that card (S) = 𝔰 and every infinite subset A ⊆ ω is split by some element of S. We may assume that S is a field of subsets of ω containing all finite sets. One can show by an easy induction on n ∈ ω that

(∀A ∈ [[ω]^{ω}]^{n})(∃S ∈ S)(∀A ∈ A)(S splits A)}.
□

2. Basic properties

Assume that X ⊆ Y, card (X) ≥ 2. Then it is easy to notice that

I_{X} = I_{Y} ∩ P (X^{ω}), I_{X}^{∗} = I_{Y}^{∗} ∩ P (X^{ω}) and S_{X} = S_{Y} ∩ P (X^{ω}).

Let us recall a well-known fact.

Lemma 2.1. Suppose that I is an ideal of subsets of X, J is an ideal of subsets of Y, X ⊆ Y and I = J ∩ P (X). Then cov(I) ≤ cov(J) and non(J) ≤ non(I).

For each set X we have

Lemma 2.2. add(I_{X}) = non(I_{X}) = ω.

Proof. Lemma 2.1 implies that we have to prove non(I_{2}) = ω only. Let

A = {x ∈ 2^{ω} : (∀^{∞}n ∈ ω) x(n) = 0}.

Then card (A) = ω and it is easy to check that A ∉ I_{2}. □

We call a family F ⊆ P if(X) normal if for each two different φ_{1}, φ_{2} ∈ F we have dom(φ_{1}) ∩ dom(φ_{2}) = ∅. Notice that if {φ_{i} : i ∈ I} ⊆ P if(X) and card (X) ≤ ω then there exists such a normal family {ψ_{i} : i ∈ I} that ψ_{i} ⊆ φ_{i} for each i ∈ I. Notice also that if φ, ψ ∈ P if(X) and ψ ⊆ φ then [φ]_{X} ⊆ [ψ]_{X} and [φ]_{X}^{∗} ⊆ [ψ]_{X}^{∗}. Therefore for every A ∈ X^{ω} we have

1) A ∈ I_{X} ⟺ A ⊆ ⋃_{φ∈F}[φ]_{X} for some finite normal family F ⊆ P if(X),
2) A ∈ I_{X}^{∗} ⟺ A ⊆ ⋃_{φ∈F}[φ]_{X}^{∗} for some finite normal family F ⊆ P if(X),
3) A ∈ S_{X} ⟺ A ⊆ ⋃_{φ∈F}[φ]_{X} for some countable normal family F ⊆ P if(X).

Lemma 2.3. Suppose that {φ_{i} : i ∈ I} is a normal family of functions from P if(X), φ ∈ P if(X) and

[φ]_{X} ⊆ ⋃_{i∈I}[φ_{i}]_{X} ([φ]_{X}^{∗} ⊆ ⋃_{i∈I}[φ_{i}]_{X}^{∗}).

Then [φ]_{X} ⊆ [φ_{i}]_{X} ([φ]_{X}^{∗} ⊆ [φ_{i}]_{X}^{∗}) for some i ∈ I.

Proof. We shall prove only the second case of the lemma. Suppose that [φ]_{X}^{∗} ⊈ [φ_{i}]_{X}^{∗} for each i ∈ I, i.e φ_{i} ⊈^{∗} φ for each i ∈ I. Hence, we may find a family {K_{i} : i ∈ I} of infinite subsets of ω such that

\begin{matrix}1) & K_{i} ⊆ dom(φ_{i}), \\ 2) & (∀k ∈ K_{i})(k ∉ dom(φ) ∨ φ_{i}(k) ≠ φ(k)) \end{matrix}

for each i ∈ I. Notice that elements from the family {K_{i} : i ∈ I} are pairwise disjoint. We consider an arbitrary function f ∈ X^{ω} such that φ ⊆ f and f(k) ≠ φ_{i}(k) for each k ∈ K_{i} \ dom(φ) and i ∈ I. One can show with ease that f ∈ [φ]_{X}^{∗} \ ⋃_{i∈I}[φ_{i}]_{X}^{∗}, which leads to contradiction. The proof of the first case of the lemma is similar to the presented one: infinite sets K_{i} should be replaced by singletons. □

Let I be an arbitrary ideal and let κ ≤ λ be two infinite cardinals. A family A ⊆ I is called a (κ, λ)-family for I if card (A) = λ and card ({A ∈ A : A ⊆ S}) < κ for each S ∈ I.

Lemma 2.4. Let card (X) ≥ 2. Then there exists a family A ⊆ I_{X} which is an (ω, 𝔠)-family for I_{X}^{∗} and an (ω_{1}, 𝔠)-family for S_{X}.

Proof. We may assume that {0, 1} ⊆ X. Let us fix a family F ⊆ [ω]^{ω} of cardinality 𝔠 such that card (A∩B) < ω for any two different A, B ∈ F. Let A = {[A×{1}]_{X} : A ∈ F}. It is clear that A ⊆ I_{X}. Suppose now that S ∈ I_{X}^{∗}. Then for some k ∈ ω and a normal family {φ_{1}, …, φ_{k}} ⊆ P if(X) we have S ⊆ [φ_{1}]_{X}^{∗} ∪ ⋯ ∪ [φ_{k}]_{X}^{∗}. Let Y = {A ∈ F : [A × {1}]_{X}^{∗} ⊆ [φ_{1}]_{X}^{∗} ∪ ⋯ ∪ [φ_{k}]_{X}^{∗}}. We claim that card (Y) ≤ k. Suppose otherwise. Notice that Lemma 2.3 implies

Y = {A ∈ F : (∃i ∈ {1, …, k})([A × {1}]_{X}^{∗} ⊆ [φ_{i}]_{X}^{∗})}.

Therefore, there are two different A, B ∈ F and i ∈ {1, …, k} such that [A × {1}]_{X}^{∗} ⊆ [φ_{i}]_{X}^{∗}, [B × {1}]_{X}^{∗} ⊆ [φ_{i}]_{X}^{∗}. But then φ_{i} ⊆^{∗} A × {1} and φ_{i} ⊆^{∗} B × {1} hence card (A ∩ B) = ω. So we obtained a contradiction. The proof of the other part of the lemma is similar. □

It is easy to check (see e.g. [1]) that if there exists a (κ, λ)-family for an ideal I and κ < λ (or κ = λ and κ is regular) then add(I) ≤ κ and cof(I) ≥ λ. Hence we obtained the following result.

Theorem 2.5. Let card (X) ≥ 2. Then

add(I_{X}^{∗}) = ω, add(S_{X}) = ω_{1}.

If moreover card (X) ≤ ω then

cof(I_{X}) = cof(I_{X}^{∗}) = cof(S_{X}) = 𝔠.

3. Ideals on the Baire space

In this part we shall discuss the properties of ideals I_{ω}, I_{ω}^{∗} and S_{ω}. Hence we shall work now on the classical Baire space of infinite sequences of natural numbers. Let us recall that if I is an ideal of subsets of X then a set L ⊆ X is called a Luzin set for I if card (L) = card (X) and card (A ∩ L) ≤ ω for each A ∈ I. Notice if there exists a Luzin set for an ideal I then non(I) ≤ ω_{1} and cov(I) = card (X).

A subset of a topological space is perfect if it is closed and contains no isolated points.

Theorem 3.1. There exists a perfect Luzin set for the ideal S_{ω}.

Proof. Fix a bijection b : 2^{<ω} → ω. To each f ∈ 2^{ω} associate f̃: ω → ω defined by f̃(n) = b(f(0), …, f(n − 1)). Then {f̃: f ∈ 2^{ω}} is a perfect set and it is Luzin for S_{ω} because no two of its members agree infinitely often. □

Putting together Lemma 2.2, Theorem 2.5, Theorem 3.1 and the observations from the beginning of this part we are able to describe cardinal coefficients add, non, cov, cof of the ideals I_{ω}, I_{ω}^{∗}, S_{ω}.

Theorem 3.2.

1) add(I_{ω}) = add(I_{ω}^{∗}) = ω < add(S_{ω}) = ω_{1};
2) non(I_{ω}) = non(I_{ω}^{∗}) = ω < non(S_{ω}) = ω_{1};
3) cov(I_{ω}) = cov(I_{ω}^{∗}) = cov(S_{ω}) = 𝔠;
4) cof(I_{ω}) = cof(I_{ω}^{∗}) = cof(S_{ω}) = 𝔠.

4. Ideals on the Cantor spaces

In this section we shall discuss the ideals I_{n}, I_{n}^{∗} and S_{n} for natural numbers n ≥ 2. Let us recall we indentify a number n with the set {0, …, n − 1}. Hence, for example, the ideals I_{2}, I_{2}^{∗} and S_{2} are ideals of subsets of the classical Cantor space.

Lemma 4.1. Let 2 ≤ n ≤ ω. Then cov(I_{n}) = cov(I_{n}^{∗}) = cov(S_{n}) = 𝔯.

Proof. As we have observed in the introduction the cardinal numbers cov(I_{n}), cov(I_{n}^{∗}) and cov(S_{n}) are equal. Hence we shall prove cov(S_{n}) = 𝔯.

Suppose first F ⊆ P if(2) is such a family that

⋃_{φ∈F}[φ] = 2^{ω}.

Let R = {φ^{−1}[{i}] : i ∈ {0, 1} & φ ∈ F} ∩ [ω]^{ω}. Then

(∀A ∈ [ω]^{ω})(∃R ∈ R)(R ⊆ A ∨ R ⊆ A^{c}),

so cov(S_{2}) ≥ 𝔯. We get from Lemma 2.1 that cov(S_{n}) ≥ 𝔯 for each n ≥ 2. Let us fix now such a family R ⊆ [ω]^{ω} of cardinality 𝔯 that for each A ∈ [ω]^{ω} there exists R ∈ R which is contained either in A or in its complement. We can inscribe an isomorphic copy of the whole family R into each of its elements and repeat this process countably many times. Then we obtain a family R^{∗} ⊆ [ω]^{ω} of cardinality 𝔯 such that for each partition of ω into finitely many pieces there exists R ∈ R^{∗} which is contained in some element of the partition. Let

F = {[R × {i}] : i ∈ {1, …, n} & R ∈ R^{∗}}.

Then F ⊆ S_{n} and ⋃ F = n^{ω}. Hence cov(S_{n}) ≤ 𝔯. □

Lemma 4.2. Let 2 ≤ n ≤ ω. Then non(I_{n}^{∗}) = 𝔰 and non(S_{n}) = ℵ_{0}-𝔰.

Proof. Suppose that T ⊆ n^{ω}, card (T) = non(I_{n}^{∗}) and T ∉ I_{n}^{∗}. Let

S = {x^{−1}[{i}] : i ∈ {1, …, n} & x ∈ T } ∩ [ω]^{ω}.

We claim that for each A ∈ [ω]^{ω} there exists S ∈ S such that card (A ∩ S) = card (A \ S) = ω and so 𝔰 ≤ card (S) = non(I_{n}^{∗}). Indeed, let A ∈ [ω]^{ω}. Then the set A^{∗} = ⋃{[A × {i}]^{∗} : i ∈ {1, …, n}} belongs to the ideal I_{n}^{∗}, so for some x ∈ T we have x ∉ A^{∗}. We consider such i < ω that card (A ∩ x^{−1}[{i}]) = ω. Note that the relation card (A \ x^{−1}[{i}]) < ω is impossible, since x ∉ A^{∗}. Hence x^{−1}[{i}] splits the set A. Therefore the inequality 𝔰 ≤ non(I_{n}^{∗}) holds. We shall prove now that non(I_{2}^{∗}) ≤ fin-𝔰. Suppose that S ⊆ [ω]^{ω} is such a family that card (S) = fin-𝔰 and for each A ∈ [[ω]^{ω}]^{<ω} there exists S ∈ S which splits each element of A. Then one can check that the set

Y = {(S × {1}) ∪ ((ω \ S) × {0}) : S ∈ S} ⊆ n^{ω}

does not belong to I_{2}^{∗}. Hence non(I_{n}^{∗}) ≤ non(I_{2}^{∗}) ≤ fin-𝔰 ≤ 𝔰. Similarly, non(S_{2}) = ℵ_{0}-𝔰. According to Lemma 2.1, we have

non(S_{2}) ≥ non(S_{3}) ≥ …,

and hence it is sufficient to prove that non(S_{n}) ≤ non(S_{n+1}) for each natural n. This is easily established by a standard technique from partition calculus, temporarily identifying two elements of n + 1 to obtain a nearly homogeneous set and then un-identifying those two elements to get homogeneity. □

Our last theorem summarizes properties of ideals I_{n}, I_{n}^{∗} and S_{n} for finite n proved in this paper.

Theorem 4.3. Let 2 ≤ n ≤ ω. Then

1) add(I_{n}) = add(I_{n}^{∗}) = ω < add(S_{n}) = ω_{1};
2) non(I_{n}) = ω < non(I_{n}^{∗}) = 𝔰 ≤ non(S_{n}) = ℵ_{0}-𝔰;
3) cov(I_{n}) = cov(I_{n}^{∗}) = cov(S_{n}) = 𝔯;
4) cof(I_{n}) = cof(I_{n}^{∗}) = cof(S_{n}) = 𝔠.

Acknowledgements. The authors are grateful to Professor Andreas Blass for useful comments on the original version of the paper.

References

[1]. Cichoń J., On two-cardinal properties of ideals, Trans. Amer. Math. Soc. 314 (1989), 693–708. [2]. Cichoń J., Kharazishvili A.B., On ideals with projective bases (1995) (to appear). [3]. van Douwen E.K., The integers and topology in Handbook of Set Theoretical Topology, K.

Kunen and J. Vaughan, eds., North-Holland, Amsterdam (1984), 111-167.

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

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