On invariant CCC σ−ideals.

Jan Kraszewski Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland (e-mail: kraszew@math.uni.wroc.pl)

Abstract

We re-read Recław’s proof from [6] on invariant CCC σ−ideals of subsets of reals and obtain a reasonably stronger corollary for such ideals on the Cantor space.

1. Preliminaries. In 1998 Recław in [6] investigated cardinal invariants of CCC σ−ideals of subsets of reals. In particular, he showed that if such a σ−ideal J is invariant, then 𝔭 ≤ non(J), where 𝔭 is a pseudointersection number (cf. [8] for more details). In this paper we analyze his proof and get an apparently stronger result for σ −ideals of subsets of the Cantor space 2^{ω}.

We use standard set-theoretical notation and terminology derived from [1]. 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|. By [ω]^{ω} we denote the family of all infinite subsets of ω. If φ : X → Y is a function then rng(φ) denotes the range of φ.

Let (G, +) be an abelian Polish (i.e. separable, completely metrizable, without isolated points) group and let J be a σ −ideal of subsets of G (we assume from now on that J is proper and contains all singletons). We will consider that J is invariant, that is for every A ⊆ G and g ∈ G we have A + g = {a + g : a ∈ A} ∈ J and −A = {−a : a ∈ A} ∈ J). Moreover, we will assume that the σ−ideal J has a Borel basis i.e. every set from J is contained in a certain Borel set from the ideal.

We say that J is CCC (countable chain condition) if the quotient Boolean algebra B(G)/J is CCC, where B(G) is the σ−algebra of all Borel subsets of G.

We define the following cardinal invariants of J.

non(J) = min{|B| : B ⊆ G ∧ B ∉ J },
cov_{t}(J) = min{|T | : T ⊆ G ∧ (∃A ∈ J) A + T = G},

2000 Mathematics Subject Classification: 03E05, 03E17.

Key words and phrases: invariant σ−ideal, CCC, cardinal invariant.

We define also an operation on the σ−ideal J (it was introduced by Seredyński in [7], who denoted it by J^{∗})

s(J) = {A ⊆ G : (∀B ∈ J)(∃g ∈ G) (A + g) ∩ B = ∅}.

If we apply these operations to the σ − ideals of meagre sets M and of null sets N we obtain strongly null sets s(M) and strongly meager sets s(N). The following is well-known

non(s(J)) = cov_{t}(J).

We define

P if = {f : f is a function ∧ dom(f) ∈ [ω]^{ω} ∧ rng(f) ⊆ 2}.

If f ∈ P if then we put

[f] = {x ∈ 2^{ω} : f ⊆ x}.

Let 𝕊_{2} denotes the σ-ideal of subsets of the Cantor space 2^{ω}, which is generated by the family {[f] : f ∈ P if}. It was thoroughly investigated in [2] and [4]. We recall some properties of 𝕊_{2}, which were proved in [2].

Fact 1.1 𝕊_{2} is a proper, invariant σ-ideal which contains all singletons and has a Borel basis. Every A ∈ 𝕊_{2} is both meager and null. Moreover, there exists a family of size 𝔠 of pairwise disjoint Borel subsets of 2^{ω} that do not belong to 𝕊_{2}. Hence 𝕊_{2} is not CCC. □

Let A, S be two infinite subsets of ω. We say that S splits A if |A ∩ S| = |A \ S| = ω. Let us recall a cardinal number related with a notion of splitting, introduced by Malychin in [5], namely

ℵ_{0}-𝔰 = min{|S| : S ⊆ [ω]^{ω} ∧ (∀A ∈ [[ω]^{ω}]^{ω})(∃S ∈ S)(∀A ∈ A) S splits A}.

More about cardinal numbers connected with the relation of splitting can be found in [3].

2. Recław’s proof revisited. In [6] Recław proved a theorem, which can be generalized as follows.

Theorem 2.1 Let I and J be two σ−ideals of subsets of an abelian Polish group G, which are invariant and have Borel bases. If I is CCC then

J ∩ s(J) ⊆ I.

Proof. (Recław) Let X ∈ J ∩ s(J). Assume that X ∉ I. We construct a sequence {F_{α} : α < ω_{1}} of Borel sets from J and a sequence {t_{α} : α < ω_{1}} of elements of G. Let t_{0} = 0 and F_{0} be any Borel set from J containing X. Suppose that we have constructed F_{β} and t_{β} for β < α. Then from the definition of s(J) there exists t_{α} ∈ G such that

(X + t_{α}) ∩ ⋃_{β<α} F_{β} = ∅.

As F_{α} we take any Borel set from J containing ⋃_{β<α} F_{β} ∪ (X + t_{α}).

Let G_{α} = F_{α} \ ⋃_{β<α} F_{β}. Thus {G_{α} : α < ω_{1}} is a family of pairwise disjoint Borel sets such that none of them belongs to I, as G_{α} ⊇ X + t_{α} and I is invariant. Hence I is not CCC, a contradiction. □

Corollary 2.2 Let I and J be as above. If I is CCC then

min{non(J), cov_{t}(J)} ≤ non(I).

Proof. It is enough to observe that J ⊆ I implies non(J) ≤ non(I). □

Corollary 2.3 Let I be a σ−ideal of subsets of the Cantor space 2^{ω} (endowed with a standard group structure), which is invariant and has a Borel basis. If I is CCC then

ℵ_{0}-𝔰 ≤ non(I).

Proof. In [2] it was proved that non(𝕊_{2}) = ℵ_{0}-𝔰 and in [4] it was proved that cov_{t}(𝕊_{2}) = 𝔠. So it is enough to apply Corollary 2.2 for G = 2^{ω} and J = 𝕊_{2}. □

Question. Let I be an invariant CCC σ −ideal of subsets of the real line ℝ. Is the inequality ℵ_{0}-𝔰 ≤ non(I) still true?

References

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