FILTER DESCRIPTIVE CLASSES OF BOREL FUNCTIONS

GABRIEL DEBS AND JEAN SAINT RAYMOND

Abstract. We first prove that given any analytic filter F on ω the set of all functions f on 2^{ω} which can be represented as the pointwise limit relatively to F of some sequence (f_{n})_{n∈ω} of continuous functions (f = lim_{F} f_{n}), is exactly the set of all Borel functions of class ξ for some countable ordinal ξ that we call the rank of F. We discuss several structural properties of this rank. For example we prove that any free Π_{4}^{0} filter is of rank 1.

The idea of associating to a given filter F a class of functions is due to Katětov (see [5] and [6]). In this work we restrict our study to classes of Borel functions on Polish spaces and we shall assume the reader to be familiar with Descriptive Set Theory in this context. Nevertheless to avoid any ambiguity we shall recall explicitly in this introduction a number of elementary notions.

Descriptive pointclasses : Informally speaking a descriptive pointclass Γ is a general assignment to any Polish space X a family Γ(X) of subsets of X with the property that for any continuous mapping φ : X → Y between two Polish spaces X and Y, if A ∈ Γ(Y) then φ^{−1}(A) ∈ Γ(X). More formally Γ should be viewed as a formula in the language of Set Theory and Γ(X) the set obtained by the relativization of this formula to the structure induced by the Polish topology on X. Still we shall very often refer to Γ(X) as a “class of sets on X”. Typical examples of such classes are the (additive, multiplicative, ambiguous) Borel classes Σ_{ξ}^{0}, Π_{ξ}^{0}, Δ_{ξ}^{0} or the classes Δ_{1}^{1}, Σ_{1}^{1},Π_{1}^{1} (of Borel, analytic, coanalytic sets).

In all the sequel by a function we shall always mean a real valued mapping. Descriptive classes of functions : In the same way a descriptive class D

of functions is a general assignment to any Polish space X a family D(X) of functions on X with the property that for any continuous mapping φ : X → Y between two Polish spaces X and Y, if f ∈ D(Y) then φ∘f ∈ D(X). We shall also refer to D(X) as a “class” of functions.

2000 Mathematics Subject Classification. Primary 03E15; Secondary 03E45, 54A20, 54H05.

Key words and phrases. analytic filters, Borel functions.

In this work we shall mainly be interested in the class B of all Borel functions and for any countable ordinal ξ, by the Borel class B_{ξ} and the Baire class B^{(ξ)}. We recall next briefly these latter notions.

The Borel classes B_{ξ}: For any Polish space X, B_{ξ}(X) is the set of all functions f : X → ℝ such that the inverse image of any open subset of ℝ is a Σ_{1+ξ}^{0} subset of X. In particular B_{0}(X) is just the set of all continuous functions on X.

The Baire classes B^{(ξ)}: For any Polish space X the sets B^{(ξ)}(X) are defined inductively by :

– B^{(0)}(X): the set of all continuous functions on X; and for any countable ordinal ξ > 0:

– B^{(ξ)}(X): the set of all pointwise limits of sequences from ⋃_{η<ξ} B^{(η)}(X).

We recall that (in the context of functions of real variables) the classes B^{(ξ)} were introduced first, and this was by Baire in his Thesis ([1]), while the Borel classes (of successor rank B_{ξ+1}) were introduced later on by Lebesgue in his famous memoir [8] where he also proved the fundamental result:

B^{(1+ξ)}(X) = B_{ξ+1}(X) .

And all these results were then generalized by Hausdorff for arbitrary Polish spaces. So the Baire classes are just the Borel classes of successor rank.

We now come to the central concept of “filter descriptive class of functions”. This notion was initially introduced by Katětov for filters with arbitrary domain. However all filters we shall consider in this work will be with countable domain. And in all the sequel when no precision is given “filter” will mean “filter on ω”. As usual when we speak about the descriptive complexity of such a filter F we view F as a subset of the Cantor space via the canonical identification P(ω) ≈ 2^{ω}.

Definition. (Katětov) Let F be a filter. For any topological space X we denote by C_{F}(X) the set of all functions f = lim_{F} f_{n} on X which can be represented as the pointwise limit relatively to F of some sequence (f_{n})_{n∈ω} of continuous functions on X.

A class D of functions is said to be a filter descriptive class if there exists a filter F such that for any Polish space X, D(X) = C_{F}(X). The class D is then said to be generated by the filter F.

The main examples of filter descriptive classes are the Borel classes. More precisely:

Any Borel class B_{ξ} of functions is generated by a Borel filter.

This was proved by Katětov for Baire classes, that is when ξ is successor, but as we shall see this is also true for ξ limit. This additional information is of some importance since our first goal is to prove the following converse.

Theorem A. For any Σ_{1}^{1} filter F there exists a countable ordinal ξ (called the rank of F) such that for any zero-dimensional Polish space X, C_{F}(X) is precisely the Borel class B_{ξ}(X).

We conjecture that in Theorem A the zero-dimensional assumption on X can be dropped, and we shall prove that this is indeed the case when ξ ≤ 2. But we point out that the Σ_{1}^{1} assumption on the filter F cannot be released since by Theorem 2 of [9]:

(Louveau) The class B of all Borel functions is generated by a Π_{1}^{1} filter.

In fact Theorem A will be derived from the combination of two more general results, characterizing each of the two inclusions: C_{F}(X) ⊂ B_{ξ}(X) and C_{F}(X) ⊃ B_{ξ}(X), from which we shall also derive the following:

Theorem B. Any Π_{ξ}^{0} free filter F is separated from its dual ideal by a Σ_{η}^{0} set for some η < ξ.

This latter result answers a question asked by Solecki in [15] where he proved the particular case ξ = 3. It is interesting to point out that Solecki’s motivation in considering these separation conditions was far away from ours and was actually dictated by some complexity computation (see Remark 2.11 below).

We shall also study several closure properties of this rank and some canonical constructions of filters of arbitrary rank.

In the last part of the paper we shall discuss two conjectures on the structure of Σ_{1}^{1} filters of rank ≥ ξ, and that we shall prove for ξ ≤ 2. In particular we obtain the following two results:

Theorem C. There exists a Σ_{4}^{0} filter of rank 2 which embeds in any Σ_{1}^{1} filter of rank ≥ 2.

Here by “embedding” a filter F in a filter G we mean that up to a bijection between their domains the filter F is coarser than the filter G.

Theorem D. Any Π_{4}^{0} filter is of rank ≤ 1.

1. Preliminaries

1.1. The canonical involution on P(E):

Given any set E the canonical involution σ : A ↦ E \ A on P(E) will play a fundamental role in our study. Also given any subset A of P(E) we define:

A^{∗} := σ(A) = {E \ A; A ∈ A} .

Observe that since σ is a homeomorphism, A is in some pointclass Γ if and only if A^{∗} is in Γ. If F is a filter on E then F^{∗} is its associated ideal.

1.2. Separation properties:

Let Γ be a pointclass, A and B two subsets of a Polish space X. We shall say that A is Γ-separated from B if there exists a subset S of X in Γ such that A ⊂ S and S ∩ B = ∅.

As we shall see behind the problem under the study lies a separation problem between the given filter F and its associated ideal F^{∗}. But before going into further details let us point out two general elementary properties.

Lemma 1.3. For any A ⊂ P(ω) the following are equivalent:

(i) A is Σ_{ξ}^{0}-separated from A^{∗}.
(ii) A is Π_{ξ}^{0}-separated from A^{∗}.
(iii) A and A^{∗} are contained in two disjoint Σ_{ξ}^{0}-sets.

Proof. For the equivalence (i) ⇔ (ii) observe that if B separates A from A^{∗} then P(ω) \ B^{∗} also separates A from A^{∗}.

The implication (i) ⇒ (iii) follows from the reduction property of the class Σ_{ξ}^{0} and the converse is obvious. □

Lemma 1.4. For any A ⊂ P(ω) and the following are equivalent:

(i) A is Δ_{ξ}^{0}-separated from A^{∗}.
(ii) A and A^{∗} are contained in two disjoint Π_{ξ}^{0}-sets.

Proof. If B is a Δ_{ξ}^{0}-set separating A from A^{∗}, then B and B^{∗} are disjoint Π_{ξ}^{0} sets containing A and A^{∗}. Conversely, if B_{0} and B_{1} are two disjoint Π_{ξ}^{0} set containing A and A^{∗}, it follows from the separation property of the class Π_{ξ}^{0} that there exists a Δ_{ξ}^{0} set B separating B_{0} from B_{1}, hence A from A^{∗}. □

2. Filter descriptive classes versus Borel classes

Our goal in this section is to characterize each of the two following inclusions: C_{F}(X) ⊂ B_{ξ}(X) and C_{F}(X) ⊃ B_{ξ}(X). For this we need to prove a number of closure properties of the set C_{F}(X).

In all the sequel X denotes a Polish space, Γ a pointclass, F a filter on ω, and ξ a countable ordinal. When no other indication is explicited, limits in ℝ^{X} are relatively to the pointwise convergence (product) topology.

Whenever A is a subset of X, 1l_{A} : X → {0, 1} will denote its characteristic function. And if f is a real-valued function on X and t ∈ ℝ, we will denote by [f ≤ t] (resp. [f ≥ t]) the set {x ∈ X : f(x) ≤ t} (resp. {x ∈ X : f(x) ≥ t}).

Lemma 2.1. If 1l_{A} ∈ C_{F}(X) for all A ∈ Δ_{1+ξ}^{0} then B_{ξ}(X) ⊂ C_{F}(X).

Proof. Let f ∈ B_{ξ}(X); we have to show that f ∈ C_{F}(X).

Consider first the particular case where f is valued in the unit interval

[0, 1] and fix \frac{1}{3} < s < t < \frac{2}{3}. Starting from f_{0} = f define inductively for

all n ∈ ω: f_{n} ∈ B_{ξ} and A_{n} ∈ Δ_{1+ξ}^{0} such that: A_{n} separates [f_{n} ≤ s] from

[f_{n} ≥ t] and f_{n+1} = \frac{3}{2}(f_{n} − \frac{1}{3}1l_{A_{n}}). This construction is possible since the

disjoint sets [f_{n} ≤ s] and [f_{n} ≥ t] are in Π_{1+ξ}^{0}. Then one easily checks then

that f = ∑_{n=0}^{∞} α_{n}1l_{A_{n}} with α_{n} = \frac{2^{n}}{3^{n+1}} . Now since each 1l_{A_{n}} ∈ C_{F}(X) we can fix a sequence (g_{⟨n,k⟩})_{k∈ω} of continuous functions on X such that 1l_{A_{n}} = lim_{F}(g_{⟨n,k⟩})_{k}. Also replacing if necessary g_{⟨n,k⟩} by min(1, max(0, g_{⟨n,k⟩})) we can assume that each g_{⟨n,k⟩} is also valued in the unit interval. Then the functions f_{k} = ∑_{n=0}^{∞} α_{n}g_{⟨n,k⟩} are well defined and continuous on X, and we next check that f = lim_{F} f_{k}, which will prove that f ∈ C_{F}(X).

So let x ∈ X and ε > 0, and fix N such that ∑_{n>N} α_{n} < \frac{ε}{3}. Then the

set M = ⋂_{n=0}^{N}\{k : |1l_{A_{n}}(x) − g_{⟨n,k⟩}(x)| < \frac{ε}{3Nα_{n}} \} is in F and for all k ∈ M

we have:

|f(x) − f_{k}(x)| < ∑_{n=0}^{N} α_{n}|1l_{A_{n}}(x) − g_{⟨n,k⟩}(x)| + 2 ∑_{n>N}^{∞} α_{n} < ε.

In the general case of an arbitrary function f ∈ B_{ξ} pick any homeomorphism φ : ℝ →]0, 1[ and consider the bounded function g = φ∘f ∈ B_{ξ}. By the previous case g ∈ C_{F}(X); and since C_{F}(X) is closed under composition on the left with continuous mappings then f = φ^{−1} ∘ g ∈ C_{F}(X). □

Lemma 2.2. If a subset A of X is such that A = ⋃_{n∈ω} A_{n} and X \ A = ⋃_{n∈ω} B_{n} with 1l_{A_{n}} and 1l_{B_{n}} ∈ C_{F}(X) for all n, then 1l_{A} ∈ C_{F}(X).

Proof. Fix for all n two sequences (f_{⟨n,k⟩})_{k} and (g_{⟨n,k⟩})_{k} of continuous functions such that 1l_{A_{n}} = lim_{F}(f_{⟨n,k⟩})_{k} and 1l_{B_{n}} = lim_{F}(g_{⟨n,k⟩})_{k}. As in the proof of Lemma 2.1 we may assume that all these functions are valued in [0, 1]. Then for all k ∈ ω the functions f_{k} = ∑_{n=0}^{∞} 2^{−n}f_{⟨n,k⟩} and g_{k} = ∑_{n=0}^{∞} 2^{−n}g_{⟨n,k⟩} are well defined and continuous; and letting f = ∑_{n=0}^{∞} 2^{−n}1l_{A_{n}} and g = ∑_{n=0}^{∞} 2^{−n}1l_{B_{n}} one checks in the lines of the computations in the proof of Lemma 2.1 that f = lim_{F} f_{k} and g = lim_{F} g_{k}.

Now since A = [f > 0] and X \ A = [g > 0] then we clearly have

1l_{A} = lim_{F} \frac{f_{k}}{f_{k} + g_{k} + 2^{−k}} . □

Lemma 2.3. Suppose that X is zero-dimensional. If a subset A of X is such that 1l_{A} ∈ C_{F}(X) then there exists a sequence (A_{n})_{n∈ω} of clopen subsets of X such that 1l_{A} = lim_{F} 1l_{A_{n}}.

Proof. Let (f_{n})_{n∈ω} be a sequence of continuous functions such that 1l_{A} = lim_{F} f_{n}. Since X is zero-dimensional, we can find for all n a continuous func-

tion g_{n} on X with values in ℝ \ \{\frac{1}{2}\} and such that sup_{x∈X} |f_{n}(x) − g_{n}(x)| < 2^{−n}; then clearly 1l_{A} = lim_{F} g_{n}. In particular for all x ∈ A (x ∉ A) the set

\{n : g_{n}(x) > \frac{1}{2}\} (\{n : g_{n}(x) < \frac{1}{2}\}) is in F. Then for all n, A_{n} = [g_{n} > \frac{1}{2}]

is a clopen subset of X, and it follows from the previous observation that 1l_{A} = lim_{F} 1l_{A_{n}}. □

We recall that a pointclass Γ is said to be a Wadge class if there exists some set A_{0} in Γ(2^{ω}) such that for any set A in Γ(2^{ω}) there exists a continuous mapping φ : 2^{ω} → 2^{ω} such that A = φ^{−1}(A_{0}).

Lemma 2.4. Suppose that X is zero-dimensional. If F is Γ-separated from F^{∗} and 1l_{A} ∈ C_{F}(2^{ω}) for all A ∈ Γ, then Γ is a Wadge class.

Proof. Suppose that F is separated from F^{∗} by some G in Γ. Consider now an arbitrary subset A of 2^{ω} in Γ. Since 1l_{A} ∈ C_{F}(2^{ω}) then by Lemma 2.3 we can find a sequence (φ_{n}) of characteristic functions of clopen subsets of 2^{ω} such that 1l_{A} = lim_{F} φ_{n}. If Φ : 2^{ω} → 2^{ω} denotes the mapping whose n^{th} coordinate is φ_{n} then Φ is clearly continuous and

\{ \begin{matrix}x ∈ A & ⇒ & Φ(x) ∈ F & ⇒ & Φ(x) ∈ G , \\ x ∉ A & ⇒ & Φ(x) ∈ F^{∗} & ⇒ & Φ(x) ∉ G , \end{matrix}

hence A = Φ^{−1}(G). This proves that Γ is a Wadge class. □

Lemma 2.5. Suppose that X is zero-dimensional and F is Σ_{1}^{1}. If Γ is a Wadge class and F is not Γ-separated from F^{∗} then 1l_{A} ∈ C_{F}(X), for any A ∈ Γ(X).

Proof. Fix A ∈ Γ(X). Since F is not Γ-separated from F^{∗} then by [10], Theorem 3 (see also [11], Corollary 9), there exists a continuous mapping Φ : X → 2^{ω} ≈ P(ω) such that for all x ∈ X:

\{ \begin{matrix}x ∉ A & ⇒ & Φ(x) ∈ F , \\ x ∈ A & ⇒ & Φ(x) ∈ F^{∗} . \end{matrix}

Hence if φ_{n} : X → {0, 1} denotes the n^{th} coordinate of the mapping Φ : X → 2^{ω} then (φ_{n}) is a sequence of continuous functions on X and clearly lim_{F} φ_{n} = 1l_{B} where B = 2^{ω} \ A. Hence 1l_{B} ∈ C_{F}(X) and consequently 1l_{A} ∈ C_{F}(X) too. □

Theorem 2.6. For any Σ_{1}^{1} filter F and any countable ordinal ξ, the following conditions are equivalent:

(i) C_{F}(2^{ω}) ⊂ B_{ξ}(2^{ω}).
(ii) C_{F}(X) ⊂ B_{ξ}(X) for any zero-dimensional Polish space X.
(iii) C_{F}(X) ⊂ B_{ξ}(X) for any Polish space X.
(iv) F is Σ_{1+ξ}^{0}-separated from F^{∗}.

Proof. (iii) ⇒ (i) is obvious.

(i) ⇒ (iv): If F were not Σ_{1+ξ}^{0}-separated from F^{∗}, consider any Σ_{1+ξ}^{0} non Δ_{1+ξ}^{0} subset A of 2^{ω}. Then by Lemma 2.5 applied to the Wadge class Σ_{1+ξ}^{0} we would have that 1l_{A} ∈ C_{F}(2^{ω}) which is impossible since A ∉ Π_{1+ξ}^{0}.

(iv) ⇒ (ii): Fix a Σ_{1+ξ}^{0} set G containing F and disjoint from F^{∗} and let X be a zero-dimensional Polish space. Consider a function f ∈ C_{F}(X) and fix some sequence of continuous functions f_{n} : X → ℝ such that f = lim_{F} f_{n}. Let a < b; we have to prove that f^{−1}(]a, b[) ∈ Σ_{1+ξ}^{0}(X).

Let Q_{(a,b)} denote the set of all pairs (r, s) of rational numbers r, s such that a < r < s < b. For any (r, s) ∈ Q_{(a,b)} pick some (r′, s′) such that a < r′ < r < s < s′ < b. Then for any n ∈ ω, f_{n}^{−1}([r, s]) is a closed set which is contained in the open set f_{n}^{−1}(]r′, s′[), and since X is zero-dimensional, we can find a clopen set A_{n}^{(r,s)} such that

f_{n}^{−1}([r, s]) ⊂ A_{n}^{(r,s)} ⊂ f_{n}^{−1}(]r′, s′[).

Let Φ_{(r,s)} = X → 2^{ω} denote the mapping whose n^{th} coordinate is the characteristic function of A_{n}^{(r,s)}. Since each Φ_{(r,s)} is continuous then (ii) follows from the following equality: f^{−1}(]a, b[) = ⋃_{(r,s)∈Q_{(a,b)}} Φ_{(r,s)}^{−1}(G) that we shall prove by checking the two corresponding inclusions.

– If x ∈ f^{−1}(]a, b[) then we can find (r, s) ∈ Q_{(a,b)} such that x ∈ f^{−1}(]r, s[), and since f = lim_{F} f_{n} then the set {n : x ∈ f_{n}^{−1}(]r, s[)} is in F and a fortiori the larger set {n : x ∈ A_{n}^{(r,s)}} is in F hence in G, which proves that x ∈ Φ_{(r,s)}^{−1}(G).

– Conversely suppose that x ∈ Φ_{(r,s)}^{−1}(G) for some (r, s) ∈ Q_{(a,b)} and pick

(r′, s′) ∈ Q_{(a,b)} such that f(A_{n}^{(r,s)}) ⊂]r′, s′[ for all n.

Then the set {n : x ∈ A_{n}^{(r,s)}} which is in G is not in F^{∗}, and a fortiori the larger set {n : x ∈ f_{n}^{−1}(]r′, s′[)} is not in F^{∗} too. This proves that x ∈ f^{−1}([r′, s′]) for otherwise the set {n : x ∉ f_{n}^{−1}([r′, s′])} would be in F or equivalently the set {n : x ∈ f_{n}^{−1}([r′, s′])} would be in F^{∗} which contradicts the previous conclusion. Hence x ∈ f^{−1}([r′, s′]) ⊂ f^{−1}(]a, b[) which finishes the proof of the equality above.

(ii) ⇒ (iii): Let X be an arbitrary Polish space and fix any continuous surjection π̂ : 2^{ω} → X̂ from 2^{ω} onto some compactification X̂ of X. Then Y = π̂^{−1}(X) is a zero-dimensional Polish space and the mapping π : Y → X obtained by the restriction of π̂ is perfect.

Consider now any f ∈ C_{F}(X) and fix a sequence of continuous functions f_{n} : X → ℝ such that f = lim_{F} f_{n}. Then the functions g_{n} = π∘f_{n} are continuous on Y and by the continuity of π, g := π∘f = lim_{F} g_{n} which proves that g ∈ C_{F}(Y), hence by (ii) g = π∘f ∈ B_{ξ}. It follows then from [14] that f ∈ B_{ξ}. □

Remark 2.7. At the first glance condition (ii) of Theorem 2.6 might appear as insignificant since trivially sitting in between conditions (i) and (iii). However it is interesting to point out that the role of this condition in the proof above is crucial. Indeed because of the compactness of 2^{ω} there is no direct way of deriving (iii) from (i) and the only proof we can produce for the (most significant) implication “(iv) ⇒ (iii)” is via (ii).

Theorem 2.8. For any Σ_{1}^{1} filter F and any countable ordinal ξ, the following conditions are equivalent:

(i) B_{ξ}(X) ⊂ C_{F}(X) for any zero-dimensional Polish space X.
(ii) For all A ∈ ⋃_{η<ξ} Σ_{1+η}^{0}(2^{ω}), 1l_{A} ∈ C_{F}(2^{ω}).
(iii) For all η < ξ, F is not Σ_{1+η}^{0}-separated from F^{∗}.
(iv) ξ = 0 or F is not Δ_{1+ξ}^{0}-separated from F^{∗}.

Proof. Since for ξ = 0 all four conditions are trivially true we may suppose that ξ > 0. Observe first that the implications (i) ⇒ (ii) and (iv) ⇒ (iii) are obvious.

(ii) ⇒ (iv): Observe that it follows from (ii) and Lemma 2.2 that 1l_{A} ∈ C_{F}(2^{ω}) for all A ∈ Δ_{1+ξ}^{0}. Hence by Lemma 2.4 if F were Δ_{1+ξ}^{0}-separated from F^{∗} then Δ_{1+ξ}^{0} should be a Wadge class which is not the case since ξ > 0.

(iii) ⇒ (i): By Lemma 2.1 and Lemma 2.2 to prove (i) it is enough to prove that 1l_{A} ∈ C_{F}(X) for all A ∈ ⋃_{η<ξ} Σ_{1+η}^{0}(X); and this follows from Lemma 2.5 applied to the Wadge classes Σ_{1+η}^{0}. □

Remark 2.9. It will follow from the main result of Section 7 that if ξ ≤ 2 then the zero-dimensional restriction on X in condition (i) of Theorem 2.8 can be dropped. We conjecture that this is the case for any ordinal ξ. As we shall see this seemingly soft topological problem is related to other deep descriptive problems (see Conjecture 7.8) .

The following result answers a question asked by Solecki in [15] (see the Remark preceeding Theorem 1.4 where the particular case ξ = 3 is proved). Recall that a filter F is said to be free if ⋂ F = ∅.

Corollary 2.10. Any Π_{ξ}^{0} free filter F is separated from its ideal F^{∗} by a Σ_{η}^{0} set for some η < ξ.

Proof. Observe first that since F is free then necessarily ξ ≥ 3. Now since F is Π_{ξ}^{0} then so is its ideal F^{∗}; it follows then from Lemma 1.4 that F is Δ_{ξ}^{0}- separated from F^{∗} and since ξ ≥ 3 then by Theorem 2.8 F is Σ_{η}^{0}-separated from F^{∗} for some η < ξ. □

Remark 2.11. It follows from Lemma 1.3 of [15] and Corollary 2.10 that if F is a Π_{ξ}^{0} filter then the set

Conv(F) := {(x_{n})_{n∈ω} ∈ ℝ^{ω} : converging relatively to F}

is itself a Π_{ξ}^{0} subset of ℝ^{ω}, which solves a problem raised by Dobrowolski and Marciszewski in [3].

3. The Borel separation rank of a filter

Definition 3.1. The Borel separation rank (or more simply the rank) of a filter F is the unique ordinal:

rk(F) = min{ξ < ω_{1} : F is Σ_{1+ξ}^{0}-separated from F^{∗}} .

So rk(F) < ω_{1} if and only if F is Δ_{1}^{1}-separated from F^{∗}. In particular by Suslin separation Theorem any Σ_{1}^{1} filter admits a countable rank.

It follows readily from the definition above that any of the four conditions in Theorem 2.6 is equivalent to say that rk(F) ≤ ξ. Similarly any of the four conditions in Theorem 2.8 is equivalent to say that rk(F) ≥ ξ, and combining these two results one gets:

Theorem 3.2. For any Σ_{1}^{1} filter F and any countable ordinal ξ > 0, the following conditions are equivalent:

(i) rk(F) = ξ.
(ii) C_{F}(X) = B_{ξ}(X) for any zero-dimensional Polish space X.
(iii) F is Σ_{1+ξ}^{0}-separated but not Δ_{1+ξ}^{0}-separated from F^{∗}.
(iv) ξ = min{η : F is Σ_{1+η}^{0}-separated from F^{∗}}.
(v) ξ = max{η : F is not Δ_{1+η}^{0}-separated from F^{∗}}.

For ξ = 0 we have the following simple characterization which does not necessitate any descriptive regularity on the filter.

Proposition 3.3. A filter F is of rank 0 if and only if F is not free.

Proof. If F is free then for all m ∈ ω the singleton {m} belongs to F^{∗} and so does the finite set J_{m} = {p : p ≤ m}. But since we have in P(ω): lim_{m→∞} J_{m} = ω ∈ F, we conclude that F meets the closure of F^{∗}, and F is not Σ_{1}^{0}-separated from F^{∗}, hence rk(F) > 0.

Conversely if F is not Σ_{1}^{0}-separated from F^{∗} then there exists some M ∈ F which is in the closure of F^{∗}, and we can find a sequence (M_{n}) in F^{∗} which converges to M. Thus defining M′_{n} by M′_{n} = (ω \ M) ∪ M_{n}, we get a sequence of elements of F^{∗} which converges to ω in P(ω) ≈ 2^{ω}. Choose for every n an integer k_{n} ∉ ⋃_{p≤n} M′_{p} and define the continuous function φ_{n} : P(ω) → {0, 1} by φ_{n}(M) = 1 ⟺ ∀p ≤ n k_{p} ∈ M. Then the sequence (φ_{n}) is non-increasing and converges to some function φ along the Fréchet filter N, hence along F since N ⊂ F. Clearly we have φ(ω) = 1 and φ_{k}(M′_{n}) = 0 for all k > n hence φ(M′_{n}) = 0 for all n, which show that φ is not continuous. This proves that F is free: for if there were some integer k in ⋂ F then we would necessarily have φ = φ_{k} and φ would be continuous. □

In the two coming sections we shall study the behaviour of the rank function relatively to some natural operations on filters. Let us only point here the following simple property:

Proposition 3.4. Given two filters F and G if F ⊂ G then rk(F) ≤ rk(G).

Proof. Observe that if F ⊂ G then F^{∗} ⊂ G^{∗} and any set separating G from G^{∗} also separates F from F^{∗}. □

4. Sum operations of filters

The operations considered in this section are quite standard. Since there is no universal terminology in this area we shall fix one for our own use.

For any family (E_{i})_{i∈I} of sets we denote by ∑_{i∈I} E_{i} its disjoint union.

4.1. Fubini sums of filters: If (F_{i})_{i∈I} is a family of filters and A is a filter on I then the set of all sets of the form:

{∑_{i∈J} M_{i} : ∀i ∈ J, M_{i} ∈ F_{i}}

for J ∈ A is the basis of a filter F on ∑_{i∈I} dom (F_{i}) that we call the A-Fubini sum of the family (F_{i})_{i∈I}.

Of most importance for the sequel is the case where I is a directed set and A is the canonical filter generated by the sets of the form J_{i} = {j ∈ I : j ≥ i}. In this case we shall say more simply that F is the Fubini sum of the family (F_{i})_{i∈I}.

We now point out two particular instances of the Fubini sum operation.

4.2. Complete sums of filters: Observe that in the particular case where A = {I} is the trivial filter on I then the set of all sets of the form:

{∑_{i∈I} M_{i} : ∀i ∈ I, M_{i} ∈ F_{i}}

is already a filter on ∑_{i∈I} dom (F_{i}) that we call the complete sum (or simply the sum) filter of the family (F_{i})_{i∈I}.

4.3. Extensions of filters: Given a filter F we shall say that a filter G is an A-extension of F if G is the A-Fubini sum of the family (F_{i})_{i∈I} with F_{i} = F for all i. If moreover A is the Fréchet filter then we shall say that G is a Fréchet extension of F.

Proposition 4.4. Suppose that G is the A-Fubini sum of the family (F_{i})_{i∈I} and J ⊂ I be an element of A.

a) If rk(A) ≥ α and rk(F_{i}) ≥ ξ for all i ∈ J then rk(G) ≥ ξ + α.

b) If rk(A) ≤ α and rk(F_{i}) ≤ ξ for all i ∈ J then rk(G) ≤ ξ + 1 + α.

c) If A is the Fréchet filter and rk(F_{i}) ≤ ξ for all i ∈ J then rk(G) ≤ ξ+1.

Proof. Let D_{i} = dom (F_{i}) and D = ∑_{i∈I} D_{i} be the domain of G.

a) Let τ_{0} denote the standard topology on 2^{ω}. It is well known then that given any function g ∈ B_{ξ+α}(2^{ω}) one can find a zero-dimensional Polish topology τ on 2^{ω}, with Σ_{1}^{0}(τ_{0}) ⊂ Σ_{1}^{0}(τ) ⊂ Σ_{1+ξ}^{0}(τ_{0}) and such that g is of class α relatively to τ. So setting X = (2^{ω}, τ) the identity mapping h : 2^{ω} → X is of class ξ and g = f ∘h with f : X → ℝ of class α. Since rk(A) ≥ α, there exists a family (f_{i})_{i∈I} of continuous functions on X converging to f along A. Then for each i ∈ J the function f_{i}∘h is of class ξ on 2^{ω}, hence is the limit along F_{i} of a sequence of continuous functions f_{i,n} : 2^{ω} → ℝ. Then setting f_{i,n} = 0 for n ∈ ω and i ∈ I \ J one easily checks that the family (f_{i,n})_{(i,n)∈I×ω} converges to g along G.

b) For all i ∈ J since rk(F_{i}) ≤ ξ we can pick a Σ_{1+ξ}^{0} set S_{i} ⊂ P(D_{i}) separating F_{i} from F_{i}^{∗}. Then each mapping ψ_{i} : P(D) → {0, 1} defined by:

ψ_{i}(M) = \begin{cases}1 & if i ∈ J and M ∩ D_{i} ∈ S_{i} \\ 0 & if not. \end{cases}

is of class ξ + 1, and so is the mapping ψ : P(D) → {0, 1}^{I} ≈ P(I) defined by ψ(M) = (ψ_{i}(M))_{i∈I}.

Since rk(A) ≤ α we can also fix a Σ_{1+α}^{0} set S ⊂ P(I) separating A from A^{∗}. Then S̃ = ψ^{−1}(S) ∈ Σ_{1+ξ+1+α}^{0} and one readily checks that S̃ separates G from G^{∗}. Hence rk(G) ≤ ξ + 1 + α.

c) If A = N is the Fréchet filter then S := {M ⊂ ω : ∀i ∃j ≥ i j ∈ M} is a Π_{2}^{0} set separating N from N^{∗} and as above S̃ := ψ^{−1}(S) separates G from G^{∗}. But since S̃ = ⋂_{i} ⋃_{j≥i}{M ∈ P(D) : M ∩D_{i} ∈ S_{i}} then S̃ is clearly a Π_{1+ξ+1}^{0} set which shows that rk(G) ≤ ξ + 1. □

Proposition 4.5. If G is the complete sum filter of the family (F_{i})_{i∈I} then rk(G) = min_{i∈I} rk(F_{i}).

Proof. One easily checks that a function f : X → ℝ is the limit along G of a family (f_{i,n})_{i∈I,n∈dom (F_{i})} of functions if and only if, for all i ∈ I, f is the limit along F_{i} of (f_{i,n})_{n∈dom (F_{i})}. And for a function f : X → ℝ of Borel class ξ on a zero-dimensional space X this is possible if and only if ξ ≤ rk(F_{i}) for all i ∈ I. □

Proposition 4.6. Let ξ be an ordinal, A be a filter with countable basis, and let G be the A-Fubini sum of the family (F_{i})_{i∈I}. If for all i ∈ I F_{i} is a Σ_{η}^{0} subset of P(dom (F_{i})) for some η < ξ then G is a Σ_{ξ+1}^{0} set.

Proof. Denote by (A_{k}) a basis of A, by D_{i} the domain of F_{i} and D = ∑ D_{i} the domain of G. Then M ∈ G if and only if there is some A ∈ A such that M ∩ D_{i} ∈ F_{i} for all i ∈ A, hence

G = ⋃_{k} ⋂_{i∈A_{k}}{M ∈ P(D) : M ∩ D_{i} ∈ F_{i}} .

And since the mapping M ↦ M ∩ D_{i} is continuous, each set {M ∈ P(D) : M ∩ D_{i} ∈ F_{i}} is Π_{ξ}^{0}. □

Notice that in the particular case of complete sums we have the following stronger conclusion obtained by a straightforward complexity computation.

Proposition 4.7. Let G be the complete sum filter of the family (F_{i})_{i∈I}. If for all i ∈ I F_{i} is a Π_{ξ}^{0} subset of P(dom (F_{i})) then G is a Π_{ξ}^{0} set.

5. Inductive limits

We shall define in this section a kind of “union” operation on families of filters with variable domains. For this we need the following slight modification of the notion of morphism or homomorphism between filters introduced by Katětov (see [4] or [5]).

Definition 5.1. A quasi-homomorphism from the filter F to the filter G is a mapping π : F → dom (G) with domain F ∈ F and such that for all M ∈ G, π^{−1}(M) ∈ F.

Lemma 5.2. If there exists a quasi-homomorphism from the filter F to the filter G then rk(G) ≤ rk(F).

Proof. Let ξ be the rank of F, π : F → dom (G) be a quasi-homomorphism and S ⊂ P(dom (F)) be a Σ_{1+ξ}^{0} set separating F from F^{∗}. Then the set

S′ := {M ∈ P(dom (G)) : π^{−1}(M) ∈ S}

is Σ_{1+ξ}^{0}. If M ∈ G then π^{−1}(M) ∈ F ⊂ S.

And if M ∈ G^{∗} then its complement N := dom (G) \ M belongs to G. Hence π^{−1}(N) ∈ F and π^{−1}(M) = F \ π^{−1}(N) ∈ F^{∗} ⊂ P(dom (F)) \ S. Thus S′ separates G from G^{∗}, and rk(G) ≤ ξ. □

Definition 5.3. Let (I, ≤) be a directed set and (F_{i})_{i∈I} be a family of filters. We shall say that the family (π_{i,j})_{i≤j} is a coherent system of quasihomomorphisms for (F_{i})_{i∈I} if for all i ≤ j ≤ k in I:

1) π_{i,j} is a quasi-homomorphism from F_{j} to F_{i}.

2) For any a ∈ dom (π_{i,k}) ∩ dom (π_{j,k}) ∩ π_{j,k}^{−1}(dom (π_{i,j})) we have:

π_{i,k}(a) = π_{i,j}(π_{j,k}(a)) .

We shall then say that (F_{i}, π_{i,j})_{i≤j} is a quasi-inductive system of filters.

Proposition-Definition 5.4. Let (π_{i,j})_{i≤j} be a coherent system of quasihomomorphisms for the family (F_{i})_{i∈I} of filters. Then the set of all sets of the form ⋃_{j≥i} π_{i,j}^{−1}(M) for some i and M ∈ F_{i} constitutes a basis of filter on ∑_{i∈I} dom (F_{i}).

The generated filter that we denote by lim_{←} F_{i} (when there is no ambiguity on the π_{i,j}) will be called the inductive limit of the system (F_{i}, π_{i,j})_{i≤j}.

Moreover for every i ∈ I there is a quasi-homomorphism π_{i} from lim_{←} F_{i} to

F_{i} such that for all i ≤ j, π_{i} and π_{i,j}∘π_{j} agree on dom (π_{j})∩π_{j}^{−1}(dom (π_{i,j})).

Proof. For i ∈ I and M ∈ F_{i} define M̂ = ⋃_{j≥i} π_{i,j}^{−1}(M). We first prove that the family {M̂ : i ∈ I, M ∈ F_{i}} is a basis of filter. Indeed let i, j ∈ I, M ∈ F_{i}, N ∈ F_{j}. Choose k ∈ I such that i ≤ k and j ≤ k and define

P := π_{i,k}^{−1}(M) ∩ π_{j,k}^{−1}(N) ∈ F_{k} .

We show that P̂ ⊂ M̂ ∩ N̂. So let x ∈ P̂. There exists ℓ ≥ k such that x ∈ dom (π_{k,ℓ}) ⊂ dom (F_{ℓ}) and y := π_{k,ℓ}(x) ∈ P. By coherence condition we conclude that π_{i,k}(y) ∈ M and π_{j,k}(y) ∈ N hence x ∈ M̂ ∩ N̂.

Let i ∈ I and define F_{i} := ⋃_{j≥i} dom (π_{i,j}) ⊂ ∑_{j∈I} dom (F_{j}).

Since M_{0} = dom (F_{i}) ∈ F_{i}, we have F_{i} = M̂_{0}, hence F_{i} ∈ F := lim_{←} F_{j}. Then define π_{i} : F_{i} → dom (F_{i}) by

π_{i}(a) = π_{i,j}(a) if j ≥ i and a ∈ dom (π_{i,j}) ⊂ dom (F_{j}) .

It is then easily checked that π_{i} is a quasi-homomorphism from F to F_{i}. Moreover if i ≤ j and a ∈ F_{j} ⊂ F_{i} there is a k ≥ j ≥ i such that a ∈ dom (π_{j,k}) and π_{j}(a) = π_{j,k}(a). Then the conclusion follows from the coherence condition. □

Proposition 5.5. Let F be the limit of the quasi-inductive system (F_{i}, π_{j,i})_{i≤j} of filters.

a) If rk(F_{i}) ≥ ξ for some i then rk(F) ≥ ξ.

b) If rk(F_{i}) ≤ ξ for all i then rk(F) ≤ ξ + 1.

Proof. Statement a) immediately follows from Lemma 5.2 and Proposition 5.4.

Denote by D_{i} the domain of F_{i} and by D = ∑_{i∈I} D_{i} the domain of F. Choose for each i ∈ I a Π_{1+ξ}^{0} set S_{i} separating F_{1} from F_{i}^{∗} in P(D_{i}). And define

S := {M ∈ P(D) : ∃i ∀j ≥ i M ∩ D_{j} ∈ S_{j}} .

Since the mapping M ↦ M ∩ D_{i} is continuous from P(D) to P(D_{i}) for every i, it is clear that S is Σ_{1+ξ+1}^{0} in P(D).

If M ∈ F there are an i ∈ I and some N ⊂ D_{i} in F_{i} such that N̂ ⊂ M. Then for j ≥ i we have M ∩ D_{j} ⊃ N̂ ∩ D_{j} = π_{i,j}^{−1}(N) ∈ F_{j}, hence M ∩ D_{j} ∈ F_{j} ⊂ S_{j}. And this shows that M ∈ S.

Conversely, if M ∈ F^{∗}, there are k ∈ I and N ⊂ D_{k} in F_{k} such that M and N̂ are disjoint. Then for every i in I there is a j ∈ I such that j ≥ i and j ≥ k and since M ∩ D_{j} is disjoint from N̂ ∩ D_{j} = π_{k,j}^{−1}(N) ∈ F_{j} we conclude that M ∩ D_{j} ∈ F_{j}^{∗} ⊂ P(D_{j}) \ S_{j} hence M ∉ S.

So S separates F from F^{∗}, and rk(F) ≤ ξ + 1. □

Remark 5.6. The limit of a quasi-inductive system of filters of rank ξ might be of rank ξ + 1. To see this consider for all i ∈ I = ω the filter

F_{i} = {M ⊂ ω × ω : ∀p ≥ i {q : (p, q) ∈ M} is cofinite}

with domain D_{i} = ω ×ω and as system of quasi-homomorphims the identity mappings π_{i,j}(p, q) = (p, q) for all i ≤ j, which is clearly coherent.

Then each F_{i} is of rank 1: indeed a family (f_{p,q}) of continuous functions converges to a function f along F_{i} only if f = lim_{q}(f_{i,q}) hence f is of Baire class 1.

To see that F is of rank ≥ 2 consider the mapping ψ : (p, q, i) → (p, q) from ∑_{i∈ω} D_{i} ≈ ω × ω × ω to ω × ω: one easily checks that ψ is a quasihomomorphism from N_{2} to F := lim_{←} F_{i}, where N_{2} is the Fréchet extension of the Fréchet filter, hence by Lemma 5.2 and Proposition 4.4 a), rk(F) ≥ rk(N_{2}) ≥ 2.

Proposition 5.7. Let F be the limit of the quasi-inductive system (F_{i}, π_{j,i})_{i≤j} of filters. If each filter F_{i} is Σ_{η_{i}}^{0} in P(dom (F_{i})) for some η_{i} < ξ then F is Σ_{ξ}^{0}.

Proof. Let D_{i} be the domain of F_{i} and D = ∑ D_{i} the domain of F. Define for i ∈ I the mapping ψ_{i} : P(D) → P(D_{i}) by

ψ_{i}(M) = {a ∈ D_{i} : ∀j ≥ i ∀b ∈ π_{i,j}^{−1}(a) b ∈ M} .

For every a ∈ D_{i} the set {M : a ∈ ψ_{i}(M)} is Π_{1}^{0}. Thus ψ_{i} is of Baire class 1. Moreover a set M is in F if and only if there is an i and a N ⊂ D_{i} in F_{i} such that M ⊃ N̂, i.e. ψ_{i}(M) ∈ F_{i}. And since ψ_{i}^{−1}(F_{i}) ∈ Σ_{η_{i}+1}^{0} ⊂ Σ_{ξ}^{0}, F = ⋃_{i∈I} ψ_{i}^{−1}(F_{i}) ∈ Σ_{ξ}^{0}. □

Proposition 5.8. Any quasi-inductive system (F_{i}, π_{j,i})_{{i≤j \atop i,j∈I}} of filters admits a canonical extension to a quasi-inductive system (F_{i}, π_{j,i})_{{i≤j \atop i,j∈Î}} of filters,

where Î = I ∪ {∞} is the directed set obtained by adding to I a largest element ∞ and F_{∞} is the Fubini sum of the family (F_{i})_{i∈I}.

Proof. For all i ∈ I set π_{∞,i} = ∑_{{j≥i \atop i,j∈I}} π_{j,i} (the disjoint union mapping): one easily checks that this defines a quasi-homomorphism from F_{∞} to F_{i} and that the extended system (π_{j,i})_{{i≤j \atop i,j∈Î}} is coherent. □

6. Some canonical filters

Our goal in this section is to explicit a canonical family (N_{ξ})_{ξ<ω_{1}} of Borel filters such that for each ξ the filter N_{ξ} generates the Borel class B_{ξ}, hence is of rank ξ. We recall that a similar family was already constructed by Katětov for the generation of the Baire classes B^{(ξ)} that we explicit next.

6.1. The Katětov filters N^{(ξ)}: Consider the family (N^{(ξ)})_{ξ<ω_{1}} of filters defined inductively by:

N^{(0)} = {{0}} (the unique filter on {0}) ,
N^{(ξ+1)} is the Fréchet extension of N^{(ξ)} ,
N^{(λ)} is the Fubini sum of the family (N^{(ξ)})_{ξ<λ}, if λ is limit .

Theorem 6.2 (Katětov). For all ξ, the filter N^{(ξ)} generates the Baire class B^{(ξ)}.

This result is a particular case of Theorem 6.5 below. We point out that the proof of Theorem 6.5 will be based on totally different arguments than Katětov’s original proof of Theorem 6.2.

6.3. The filters N_{ξ} : Since the Baire class B^{(1+ξ)} is just the Borel class B_{ξ+1} we shall make a shift in the notations and set:

\{ \begin{matrix}N_{0} & = & N^{(0)} , \\ N_{ξ+1} & = & N^{(1+ξ)} . \end{matrix}

To define N_{λ} for λ limit, observe that since in the Katětov family (N^{(ξ)})_{ξ<ω_{1}} each N^{(ξ)} is obtained as a Fubini sum of the previous filters then applying Proposition 5.8 inductively one gets a coherent system of quasihomomorphisms (π_{η,ξ})_{η<ξ<λ} for this family, hence (N^{(ξ)}, π_{η,ξ})_{η<ξ<λ} is a quasiinductive system of filters and we can define:

N_{λ} = lim_{\overleftarrow{ξ<λ}} N^{(ξ)} .

Proposition 6.4. For all ξ, the filter N_{ξ} is Borel. More precisely: a) if ξ = λ + n with λ limit and 1 ≤ n < ω then N_{ξ} is a Σ_{1+λ+2n−1}^{0} set.

b) if λ is limit then N_{λ} is a Σ_{λ}^{0} set.

Proof. It follows by a straightforward induction from Proposition 4.6 that if n is finite then N_{n} is a Σ_{2n}^{0} set, and if moreover λ is limit then N_{λ+n+1} = N^{(λ+n)} is a Σ_{λ+2n+1}^{0} set. This proves part a)

Part b) follows then from part a) and Proposition 5.7. □

Theorem 6.5. For all ξ, the filter N_{ξ} generates the Borel class B_{ξ}.

Proof. It follows from the inductive construction of the Katětov filters N^{(ξ)} and Proposition 4.4 a) and c) that rk(N^{(1+ξ)}) = ξ + 1. Moreover if λ is limit then by Proposition 5.5 we have λ ≤ rk(N_{λ}) ≤ λ + 1; and since by Proposition 6.4 the filter N_{λ} is itself a Σ_{λ}^{0} set then it follows from Proposition 5.5 that rk(N_{λ}) = λ.

Hence rk(N_{ξ}) = ξ for all ξ, and it follows then from Theorem 2.6 that for any Polish space X we have C_{N_{ξ}}(X) ⊂ B_{ξ}. Notice that by Theorem 2.8 if X is zero-dimensional then the converse inclusion C_{N_{ξ}}(X) ⊃ B_{ξ} also holds, but we have to prove this latter inclusion for an arbitrary Polish space.

We first treat the successor case ξ+1. Observe that by Lebesgue-Hausdorff Theorem any function f : X → ℝ of Baire class B^{(ξ)} is the is pointwise limit of a sequence (f_{k}) of functions of Baire class < ξ ; and if each f_{k} itself is the limit along a filter F_{k} of continuous functions (f_{k,n}) then clearly f itself is the limit of (f_{k,n}) along the Fubini sum of the F_{k}. It follows then by a straightforward induction on ξ that such a function f is the limit along N^{(ξ)} = N_{ξ+1} of the family (f_{k,n}). Hence C_{N_{ξ+1}}(X) ⊃ B_{ξ+1}.

Suppose now that λ is limit and observe that if A is any Π_{ξ}^{0} subset of X for some ξ < λ, then the function 1l_{A} is of Baire class < λ. Thus 1l_{A} is the limit of continuous functions along N_{η} for ξ < η < λ hence the limit of continuous functions along N_{λ}. Then by Lemmas 2.1 and 2.2, C_{N_{λ}}(X) ⊃ B_{λ}. □

Remark 6.6. We mentioned in the introduction that in [9] Louveau constructs a unique Π_{1}^{1} filter F which generates the whole class B. In fact this filter is provided with a natural Π_{1}^{1} norm ρ : F → ω_{1} with the property that for all ξ < ω_{1} the set F_{ξ} := {M ∈ F : ρ(M) < ξ} is actually a filter; and a careful analysis of the norm ρ shows that the filter F_{ξ} is also of rank ξ and has exactly the same Borel complexity as N_{ξ}.

7. A dichotomy for Σ_{1}^{1} filters

Notation 7.1. Given two filters F and G we shall write:

F ⊑ G

if there exists a bijection σ : dom (F) → dom (G) which sends any element of F onto an element of G.

Lemma 7.2. If F ⊑ G then rk(F) ≤ rk(G).

Proof. Let ξ = rk(G) and S be a Σ_{1+ξ}^{0} subset of P(dom (G)) separating G from G^{∗}. Then since σ is one-to-one the mapping M ↦ σ(M) is continuous from P(dom (F)) to P(dom (G)) and the set S′ = {M : σ(M) ∈ S} is Σ_{1+ξ}^{0} in P(dom (F)) and separates F from F^{∗}. Hence rk(F) ≤ ξ. □

Proposition 7.3. A filter F is of rank ≥ 1 if and only if N_{1} ⊑ F.

Proof. If N_{1} ⊑ F we have 1 = rk(N_{1}) ≤ rk(F). Conversely if rk(F) > 0 then F is free by Proposition 3.3. and if σ is any bijection from ω to dom (F), it is easy to see that for all cofinite subset M of ω, σ(M) is cofinite in dom (F), hence in F. So N_{1} ⊑ F. □

Lemma 7.4. Let D be a countable set, F a filter on D, and (M_{i})_{i∈I} a countable family of elements of F^{∗} satisfying the property

∀M ⊂ D (∀i ∈ I M ∩ M_{i} is finite ⟹ M ∈ F^{∗}) .  (⋆)

Then N_{2} ⊑ F.

Proof. Without loss of generality we can assume I = ω. Replacing if necessary M_{i} by M′_{i} = M_{i}\⋃_{j<i} M_{j} we also can assume that the M_{i}’s are pairwise disjoint: indeed if M ∩ M′_{i} is finite for all i, M ∩ M_{i} ⊂ ⋃_{j≤i} M ∩ M′_{j} is finite too for all i.

Let N_{0} = D \ ⋃_{i∈I} M_{i} = D \ ⋃_{i∈I} M′_{i}. Since N_{0} ∩ M_{i} = ∅ for all i, it follows clearly from (⋆) that N_{0} ∈ F^{∗}. And if J = {i : M′_{i} is finite} we see in the same way that N_{1} = ⋃_{i∈J} M′_{i} ∈ F^{∗} and that I′ := I \ J is infinite. Choose i_{0} ∈ I′ = I \ J and define for i ∈ I′ :

M′′_{i} = \begin{cases}M′_{i} & if i ≠ i_{0} , \\ M′_{i_{0}} ∪ N_{0} ∪ N_{1} & if i = i_{0} . \end{cases}

Then (M′′_{i})_{i∈I′} is a countable partition of D into infinite sets still satisfying (⋆). Choose a bijection f from ω onto I′ and for each integer p a bijection g_{p} from ω onto M′′_{f(p)}. Then define a bijection σ from ω^{2} = dom (N_{2}) onto D by σ(p, q) = g_{p}(q). If A ∈ N_{2}^{∗} there is an m such that A_{p} := {q : (p, q) ∈ A} is finite for all p > m. Then M′ = ⋃_{p>m} g_{p}(A_{p}) ∈ F^{∗} by (⋆), and so is σ(A) ⊂ M′ ∪ ⋃_{p≤m} M′′_{f(p)}. And this shows that N_{2} ⊑ F. □

Theorem 7.5. A Σ_{1}^{1} filter F is of rank ≥ 2 if and only if N_{2} ⊑ F.

Proof. If N_{2} ⊑ F, we have 2 = rk(N_{2}) ≤ rk(F).

Conversely, if rk(F) ≥ 2 we have C_{F}(2^{ω}) ⊃ B_{2}(2^{ω}). Denote by D the domain of F, by ℚ the subspace {α ∈ 2^{ω} : ∃p ∀q ≥ p α(q) = 0} of 2^{ω} which is countable and dense. Then 1l_{ℚ} ∈ B_{2}(2^{ω}) and we can find a sequence (f_{n})_{n∈D} of continuous functions : 2^{ω} → {0, 1} such that 1l_{ℚ} = lim_{F} f_{n}.

For every α ∈ 2^{ω} denote by M(α) the set {n ∈ D : f_{n}(α) = 0}. We have M(α) ∈ F^{∗} if α ∈ ℚ and M(α) ∈ F if α ∉ ℚ. We claim that the family (M(α))_{α∈ℚ} has property (⋆) from Lemma 7.4, which is enough for proving the Theorem. Indeed assume that M ∈ P(D) has finite intersection with all M(α) for α ∈ ℚ.

We want to construct β and γ ∈ 2^{ω} \ ℚ such that N = M ∩ M(β) ∩ M(γ) is finite, and this will imply that M ⊂ N ∪ (D \ M(β)) ∪ (D \ M(γ)) ∈ F^{∗}.

Define 0 ∈ 2^{ω} as the infinite null sequence (0(n) = 0 for all n), N as the finite set M ∩ M(0) and u_{n} as the sequence of length n + 1 having n zeros followed by one 1. We construct by induction on the integer k sequences s_{k} and t_{k} in 2^{<ω} such that M ∩ M(s_{k}^{⌢}0) ∩ M(t_{k}^{⌢}0) = N. Put s_{0} = t_{0} = ∅. Observe that the function α ↦ M(α) is continuous and that lim_{n→∞} s_{k}^{⌢}u_{n}^{⌢}0 = s_{k}^{⌢}0. Thus since M ∩ M(t_{k}^{⌢}0) is finite and M(s_{k}^{⌢}u_{n}^{⌢}0) → M(s_{k}^{⌢}0) there is some n_{k} large enough such that

M ∩ M(t_{k}^{⌢}0) ∩ M(s_{k}^{⌢}u_{n_{k}}^{⌢}0) = M ∩ M(t_{k}^{⌢}0) ∩ M(s_{k} ^{⌢}0) = N ,

and we define s_{k+1} = s_{k}^{⌢}u_{n_{k}}. In the same way, since M ∩ M(s_{k+1}^{⌢}0) is finite there is m_{k} large enough such that

M ∩ M(s_{k+1}^{⌢}0) ∩ M(t_{k}^{⌢}u_{m_{k}}^{⌢}0) = M ∩ M(s_{k+1}^{⌢}0) ∩ M(t_{k}^{⌢}0) = N ,

and we define t_{k+1} = t_{k}^{⌢}u_{m_{k}}. Then there are two points β and γ in 2^{ω} \ ℚ such that s_{k} ≺ β and t_{k} ≺ γ for all k. It is clear that s_{k}^{⌢}0 → β and t_{k}^{⌢}0 → γ. Thus by continuity we get

M ∩ M(β) ∩ M(γ) = lim_{k→∞} M ∩ M(s_{k}^{⌢}0) ∩ M(t_{k}^{⌢}0) = N ,

and this completes the proof. □

Corollary 7.6. If F is a Σ_{1}^{1} filter of rank ≥ 2 then C_{F}(X) ⊃ B_{2}(X) for any Polish space X.

Proof. Since N_{2} ⊑ F there exists a bijection σ from ω^{2} = dom (N_{2}) to dom (F) such that σ(A) ∈ F for all A ∈ N_{2}. If X is a Polish space and f : X → ℝ is of Baire class 2, there is a family (f_{p,q})_{(p,q)∈ω^{2}} of continuous functions : X → ℝ such that f = lim_{p→∞} lim_{q→∞} f_{p,q}. It follows that f = lim_{N_{2}} f_{p,q} hence that f = lim_{F} g_{n} where g_{n} = f_{p,q} with (p, q) = σ^{−1}(n).

And this shows that f ∈ C_{F}(X). □

Corollary 7.7. If F is a Σ_{1}^{1} filter then one and only one of the following two alternatives holds:

– either C_{F}(X) ⊂ B_{1}(X) for any Polish space X,

– or N_{2} ⊑ F.

Proof. By Theorem 2.6 the first alternative is equivalent to rk(F) ≤ 1, and by Theorem 7.5 the second one is is equivalent to rk(F) ≥ 2. □

Proposition 7.3 and Theorem 7.5 suggest the following conjecture for any countable ordinal ξ.

Conjecture 7.8. A Σ_{1}^{1} filter F is of rank ≥ ξ if and only if N_{ξ} ⊑ F.

8. Comparing the rank and the class of a Borel filter

Given a Borel set one is naturally interested in finding its “exact” complexity. For this the Wadge hierarchy provides a complete scale of comparison, but here we restrict this scale to the more classical Borel hierarchy by considering only the additive and multiplicative Borel classes. For such a class Γ we denote as usual by Γ̌ its dual class. We recall that by Wadge Theorem a set A is in Γ \ Γ̌ if and only if A is Γ-complete (see [7]). When we say that a Borel set A is of class ξ with no more precision we mean that A is in Σ_{ξ}^{0} or in Π_{ξ}^{0}.

The aim of this section is to discuss the relations between the rank of a Borel filter and its Borel class.

Proposition 8.1. For any ordinals ξ, η there exists a Borel filter of rank ξ and class ≥ η.

Proof. Without loss of generality we may assume that η ≥ ξ. Set D_{ξ} := dom (N_{ξ}), D_{η} := dom (N_{η}) and and consider on D := D_{ξ} + D_{η} the filter F of all subsets M such that M ∩ dom (N_{ξ}) ∈ N_{ξ} and M ∩ dom (N_{η}) ∈ N_{η}.

If G_{0} ⊂ P(dom (N_{ξ})) is any Σ_{1+ξ}^{0} set separating the filter N_{ξ} from its ideal then the set G = {M ⊂ D : M ∩ D_{ξ} ∈ G_{0}} is Σ_{1+ξ}^{0} too and G separates F from its ideal, hence rk(F) ≤ ξ.

To see that rk(F) = ξ it is enough to prove that any function f in B_{ξ}(2^{ω}) is limit along G of continuous functions on 2^{ω}. Since N_{ξ} generates B_{ξ} and N_{η} generates B_{η} ⊃ B_{ξ}, there are continuous functions f_{n} : 2^{ω} → ℝ for n ∈ D_{ξ} and f_{m} : 2^{ω} → ℝ for m ∈ D_{η} such that f = lim_{N_{ξ}} f_{n} = lim_{N_{η}} f_{m}. Then putting together these two families we get a family (f_{i})_{i∈D} such that f = lim_{G} f_{i}.

Finally consider the continuous mapping Ψ : M ↦ D_{ξ} + M from P(D_{η}) to P(D). One easily checks Ψ^{−1}(F) = N_{η}. Hence if F were of class < η then N_{η} would be of class < η too, which is impossible since N_{η} is of rank η. This proves that F is of class ≥ η . □

In fact for small ranks (0 or 1) we have even the following more precise result. Proposition 8.2. If i = 0 or 1 then for any Borel class Γ ⊃ Σ_{2}^{0} there exists in Γ \ Γ̌ a filter of rank i. Proof. Observe first that the case i = 0 follows from the case i = 1. To see this observe that if F is any free filter and a is any element of its domain then the non-free filter F(a) := {M ∈ F : a ∈ M} is of rank 0. Moreover F(a) is a Δ_{1}^{0} subset of F and F = Φ^{−1}(F(a)) where Φ : M ↦ M ∪ {a} is a continuous mapping. It follows from these observations that F is in some given class Γ if and only F(a) is in Γ.

So we now suppose that i = 1 and we treat separately the cases where Γ is additive or multiplicative. Observe that since Γ ⊃ Σ_{2}^{0}, then in the first case Γ = Σ_{ξ}^{0} with ξ ≥ 2, then Γ is at least of class 2.

We first treat the case where Γ is additive, and for this we shall make use of a family of filters considered by Lutzer, van Mill and Pol in [12]: For any x ∈ 2^{ω} set M_{x} = {s ∈ 2^{<ω} : s ≺ x}; then given any non-empty Borel subset A of 2^{ω} consider on the countable set 2^{<ω} the ideal ℐ_{A} generated by the set {M_{x} ∈ P(2^{<ω}) : x ∈ A} and let Φ_{A} = ℐ_{A}^{∗} be its dual filter. The exact complexity of these filters was computed by Calbrix in [2], and it turns out that for ξ ≥ 2, Φ_{A} is Σ_{ξ}^{0} if and only if A itself is Σ_{ξ}^{0}.

Observe that since Φ_{A} ⊂ Φ_{2^{ω}} then rk(Φ_{A}) ≤ rk(Φ_{2^{ω}}). Moreover, since ℐ_{2^{ω}} is generated by the compact set {M_{x} ∈ P(2^{<ω}) : x ∈ 2^{ω}} then ℐ_{2^{ω}} is Σ_{2}^{0} hence Φ_{2^{ω}} = ℐ_{A}^{∗} is Σ_{2}^{0} too, and it follows from Theorem 2.6 that rk(Φ_{2^{ω}}) ≤ 1. This proves that for all A, rk(Φ_{A}) ≤ 1 and since this filter is clearly free, then rk(Φ_{A}) = 1. Hence for all ξ ≥ 2 if we set set F_{ξ} := Φ_{A_{ξ}} where A_{ξ} is some Σ_{ξ}^{0}-complete set, it follows from the previous observations that F_{ξ} is Σ_{ξ}^{0}-complete and of rank 1.

For the case where Γ is multiplicative fix for all ξ ≥ 3 a sequence (ξ_{p})_{p∈ω} such that ξ = sup_{p}(ξ_{p} + 1) and let G_{ξ} be the complete filter sum of the family (F_{ξ_{p}})_{p∈ω}. Then by standard arguments one can prove that each G_{ξ} is Π_{ξ}^{0}-complete and it follows from Proposition 4.5 that G_{ξ} is of rank 1.

□

We consider now the reverse problem of seeking “simple” filters of some given rank ξ.

Notation 8.3. For all ξ > 0, if ξ = λ + n with λ limit and n < ω we set:

ξ^{∗} = \{ \begin{matrix}λ & if ξ = λ is limit , \\ 1 + λ + 2n − 1 & if ξ is successor . \end{matrix}

In particular if ξ = n is finite then ξ^{∗} = 2n.

Then summing up the conclusions of Corollary 2.10, Proposition 6.4 and Theorem 6.5 we can state:

Theorem 8.4. For all ξ > 0:

a) There is no Π_{1+ξ}^{0} filter of rank ≥ ξ.

b) There exists a Σ_{ξ^{∗}}^{0} filter of rank ξ.

Observe that when ξ^{∗} > 1 + ξ Theorem 8.4 does not give any information about the possible existence of some Π_{η}^{0} filter of rank ξ with η ≤ ξ^{∗}. In fact we conjecture that there is none:

Conjecture 8.5. There is no Π_{ξ^{∗}}^{0} filter of rank ≥ ξ.

Remark 8.6. If ξ = λ > 0 is limit then ξ^{∗} = λ = 1 + ξ. Similarly if ξ = λ + 1 with λ > 0 limit then ξ^{∗} = λ + 1 = 1 + ξ. Hence in both cases the conjecture is just a restatement of part a) of Theorem 8.4.

Thus the conjecture concerns only successor ranks. Observe also that if ξ = 1 then ξ^{∗} = 2 and the conjecture is also true since by Proposition 3.3 a filter of rank 1 is necessarily free, and it is a classical fact that such a filter is necessarily meager and cannot be a Π_{2}^{0} set.

9. Borel class of filters of rank ≥ 2

The aim of this section is to prove Conjecture 8.5 for ξ = 2 in which case ξ^{∗} = 4. Notice Theorem 8.4 ensures already that a Π_{3}^{0} filter is of rank ≤ 1.

Theorem 9.1. Any Π_{4}^{0} filter is of rank ≤ 1 .

In fact Theorem 9.1 will follow from some general combinatorial result on filters of rank ≥ 2. Before we state this result let us point out first a simple consequence of Theorem 2.6 which can also be derived from Theorem 1.4 of

In all the sequel when we speak about the intersection of some elements of a set G ⊂ P(ω) we shall always be implicitly refering to the intersection operation in P(ω).

Theorem 9.2. Let F be a Σ_{1}^{1} filter of rank ≥ 2. Then any Π_{3}^{0} set G ⊃ F contains two elements with empty intersection.

Proof. Observe that if G is a Π_{3}^{0} set then the set G ∩ G^{∗} is necessarily non empty for otherwise, by Lemma 1.4, F would be Δ_{3}^{0}-separated from F^{∗} and, by Theorem 2.6, F would then be of rank < 2. Now if M is any element of G ∩ G^{∗} then M and M^{c} are two disjoint elements of G. □

We now prove a Π_{4}^{0} version of this result from which Theorem 9.1 follows readily.

Theorem 9.3. Let F be a Σ_{1}^{1} filter of rank ≥ 2. Then any Π_{4}^{0} set G ⊃ F contains three elements with empty intersection.

Proof. By Theorem 7.5 we may and shall assume that F is the canonical filter N_{2} on ω^{2}. So let G be a Π_{4}^{0} set containing F = N_{2}.

Our goal is to construct three subsets M^{(0)}, M^{(1)}, M^{(2)} of ω^{2} in G such that ⋂_{i=0}^{i=2} M^{(i)} = ∅. In fact what we shall really and concretely construct are the characteristic functions x^{(i)} ∈ 2^{ω^{2}} of these sets, which will be defined under the form: x^{(i)} = ⋃_{n} c_{n}^{(i)} with c_{n}^{(i)} ∈ 2^{C_{n}} for some increasing sequence (C_{n}) of infinite subsets of ω^{2} which will be defined inductively. For these reasons it will be very important to avoid a systematic identification between P(ω^{2}) and 2^{ω^{2}}.

The construction is rather technical and necessitates the introduction of a number of preliminary notations.

We first set: F = {x ∈ 2^{ω^{2}} : [x = 1] ∈ F} and G = {x ∈ 2^{ω^{2}} : [x = 1] ∈ G}. Since G is a Π_{4}^{0} set we can fix a family (G_{(k,ℓ,m)})_{(k,ℓ,m)∈ω^{3}} of open subsets of 2^{ω^{2}} such that G = ⋂_{k} ⋃_{ℓ} ⋂_{m} G_{(k,ℓ,m)}. We also set for all k, G_{(k)} = ⋃_{ℓ} ⋂_{m} G_{(k,ℓ,m)} and for all ℓ, G_{(k,ℓ)} = ⋂_{m} G_{(k,ℓ,m)}; and we shall assume that the sequence (G_{(k)})_{k∈ω} is decreasing.

We denote by π = ω^{2} → ω the canonical projection on the first factor. We shall say that a subset M of ω^{2} is a cylinder if it is of the form M = M_{0} × ω with M_{0} = π(M) finite. We denote by C the set of all cylinders.

We denote by Ω the set of all partial mappings from ω^{2} to {0, 1}. If a ∈ Ω we denote as usual by dom (a) its domain. We consider on Ω the standard extension (inclusion) relation ⊂ .

We also consider on Ω the pointwise product operation: if a, b ∈ Ω then by definition dom (a.b) = dom (a)∩dom (b) and for any p = (p_{0}, p_{1}) ∈ dom (a.b), a.b(p) = a(p).b(p). We denote by 0 the constant mapping on ω^{2} with value 0; so a.b ⊂ 0 is equivalent to say that [a = 1] ∩ [b = 1] = ∅. Similarly we denote by 1 the constant mapping on ω^{2} with value 1. For any a ∈ Ω we set

N_{a} = {x ∈ 2^{ω^{2}} : a ⊂ x} .

Notice that if dom (a) is finite then N_{a} is a clopen subset of 2^{ω^{2}} but in general N_{a} is only closed.

Last we introduce a subset Σ of Ω × Ω which will play a crucial role in the proof and which is defined by:

(a, b) ∈ Σ ⟺ \{ \begin{matrix}dom (a) is finite, \\ dom (b) ∈ C, [b = 0] is finite, \\ dom (a) ∩ dom (b) = ∅. \end{matrix}

We consider on Σ the product ordering induced by (Ω, ⊂) and that we also denote for simplicity by ⊂ since there is no possible confusion. So:

(a, b) ⊂ (a′, b′) ⟺ a ⊆ a′ and b ⊆ b′ ⟹ a ∪ b ⊂ a′ ∪ b′ ⟹ b ⊂ b′

(the first equivalence is just the definition of a product ordering and the last implication follows from the conditions on the domains of a and b in the definition of Σ.)

Lemma 9.4. Let E be any subset of 2^{ω^{2}} and (a, b) in Σ. If for all (a′, b′) ⊃ (a, b) in Σ we have N_{a′∪ b′} ∩ E ≠ ∅ then N_{a∪ b} ∩ E is a dense subset of N_{a∪ b}.

Proof. Observe that if c is any finite extension of a ∪ b then c \ b is finite so (c\b, b) ∈ Σ and clearly (c\b, b) ⊃ (a, b), hence N_{(c\b)∪b}∩E = N_{c}∩E ≠ ∅. □

Lemma 9.5. Suppose that (F_{ℓ})_{l∈ω} is a countable covering of F. Then for all (a, b) in Σ there exist ℓ ∈ ω and (a′, b′) ⊃ (a, b) in Σ such that for all (a′′, b′′) ⊃ (a′, b′) in Σ, N_{a′′∪ b′′} ∩ F_{ℓ} ≠ ∅.

Proof. Fix (a, b) ∈ Σ; consider the cylinder A = π(dom (a)) × ω and let

Σ′ := {(a′, b′) ∈ Σ : (a′, b′) ⊃ (a, b) and dom (a′) ⊂ A} .

Observe that (a, b) ∈ Σ′.

\underline{Claim}: There exist ℓ ∈ ω and (a′, b′) ⊃ (a, b) in Σ′ such that

for all (a′′, b′′) ⊃ (a′, b′) in Σ′, N_{a′′∪ b′′} ∩ F_{ℓ} ≠ ∅. To prove the claim we proceed by contradiction: If not then starting from

(a, b) one constructs by a straightforward induction an increasing sequence (a_{ℓ}, b_{ℓ})_{ℓ∈ω} in Σ′ with the property that for ℓ, N_{a_{ℓ}∪ b_{ℓ}} ∩ F_{ℓ} = ∅. Let c = ⋃_{l}(a_{ℓ} ∪ b_{ℓ}) and consider x = c ∪ 1_{|ω^{2}\dom (c)} ∈ 2^{ω^{2}}.

Then [x = 0] ⊂ A ∪ ⋃_{ℓ} [b_{ℓ} = 0] and from the definition of Ω this latter set is in the ideal F^{∗} hence [x = 1] ∈ F and so x ∈ F. But for all ℓ, since x ⊃ a_{ℓ} ∪ b_{ℓ} then it follows from the construction that x ∉ F_{ℓ} ; hence x ∉ F which is a contradiction. This proves the claim from which we now derive the lemma.

So let ℓ and (a′, b′) be as in the claim: then clearly (a′, b′) ∈ Σ and (a′, b′) ⊃ (a, b). Consider any (a′′, b′′) in Σ such that (a′′, b′′) ⊃ (a′, b′). Set a′′′ = a′′_{|A} and B = dom (a′′) \ A; and observe that from the definition of Ω the set B is finite and disjoint from the cylinder dom (b′′) hence C = (π(B) × ω) \ B is disjoint from the cylinder dom (a′′) ∪ dom (b′′) and we can define b′′′ = b′′ ∪ a′′ ∪ 1_{|C} and since (a′′′, b′′′) ∈ Σ′ and (a′′′, b′′′) ⊃ (a′, b′) then by the claim N_{a′′′∪ b′′′} ∩ F_{ℓ} ≠ ∅, and since a′′ ∪ b′′ ⊂ a′′′ ∪ b′′′ then N_{a′′′∪ b′′′} ⊂ N_{a′′∪ b′′} and so N_{a′′∪ b′′} ∩ F_{ℓ} ≠ ∅. □

Before we state next lemma let us point out that our goal is to construct three sequences (c_{n}^{(i)})_{n∈ω} for i = 0, 1, 2 with similar properties. But for practical reasons we shall construct a unique sequence (c_{n})_{n∈ω} and then define c_{n}^{(i)} = c_{i+3n}.

Lemma 9.6. There exists a sequence ((a_{n}, b_{n}), C_{n}, ℓ_{n})_{n∈ω} in Σ × C × ω such that denoting by ≡ the equality modulo 3 on integers, and setting A_{n} = ⋃_{{k≢n \atop k<n}} C_{k} and B_{n} = ⋃_{{k≡n \atop k≤n}} C_{k} then for all n ≥ 0:

(i) C_{n} ∩ ⋃_{k<n} C_{k} = ∅ ,
(ii) (a_{n}, b_{n}) ⊃ (a_{n−3}, b_{n−3}) ,
(iii) dom (a_{n}) ⊂ A_{n} and dom (b_{n}) = B_{n} ,
(iv) dom (a_{n}) ⊃ A′_{n} := A_{n} ∩ (dom (a_{n−1}) ∪ dom (a_{n−2})) ,
(v) a_{n}.a_{n−1} ⊂ 0 and a_{n}.a_{n−2} ⊂ 0 ,
(vi) For all (a, b) ∈ Σ if (a, b) ⊃ (a_{n}, b_{n}) then N_{a∪b} ∩ G_{(n,ℓ_{n})} ≠ ∅ ,
(vii) N_{a_{n}∪ b_{n}} ⊂ ⋂_{{k≡n \atop k,m<n}} G_{(k,ℓ_{k},m)} ,

where by convention a_{k} = b_{k} = C_{k} = ∅ for k < 0.

Proof. Assume that ((a_{k}, b_{k}), C_{k}, ℓ_{k})_{k<n} is constructed, and observe that the sets A_{n} and A′_{n} are then already determined. Choose a finite set A such that A′_{n} ⊂ A ⊂ A_{n} and π(A_{n}) = π(A), then consider a = a_{n−3} ∪ 0_{|A}.

For all k < n with k ≡ n, (a, b_{n−3}) ∈ Σ and by (ii)_{k}

(a, b_{n−3}) ⊃ (a_{n−3}, b_{n−3}) ⊃ (a_{k}, b_{k}) .

It follows then from (vi) and Lemma 9.4 that each of the Π_{2}^{0} sets G_{(k,ℓ_{k})} is dense in N_{a∪b_{n−3}}. Hence ⋂_{{k≡n \atop k<n}} G_{(k,ℓ_{k})} is dense in N_{a∪b_{n−3}} and a fortiori the larger open set ⋂_{{k≡n \atop k,m<n}} G_{(k,ℓ_{k},m)}, so we can find a finite extension c of a∪b_{n−3} such that N_{c} ⊂ ⋂_{{k≡n \atop k,m<n}} G_{(k,ℓ_{k},m)}. Then applying Lemma 9.5 to (c\b_{n−3}, b_{n−3}) we get (a′, b′) ∈ Σ with (a′, b′) ⊃ (c \ b_{n−3}, b_{n−3}) ⊃ (a_{n−3}, b_{n−3}) such that for any (a_{n}, b_{n}) ∈ Σ with a_{n} ∪ b_{n} ⊃ c′ := a′ ∪ b′ condition (vi) is satisfied.

Observe that since dom (a) ∩ dom (b′) = ∅ then π(A_{n}) ∩ π(dom (b′)) = π(A ∩ dom (b′)) = ∅, consequently A_{n} ∩ dom (b′) = ∅.

Now set C′_{n} = (π(dom (c′))) × ω , C_{n} = C′_{n} \ ⋃_{k<n} C_{k}, B_{n} = C′_{n} \ A_{n} so that B_{n} = ⋃_{{k≡n \atop k≤n}} C_{k} and define a_{n} = c′_{|A_{n}} = a′_{|A_{n}} and b_{n} = c′_{|B_{n}} ∪ 1_{|B_{n}\dom (c′)}. Then one can easily check that (a_{n}, b_{n}) ∈ Σ and that conditions (i) - (vii) are all satisfied. □

To finish the proof of Theorem 9.3 set C = ω^{2} \⋃_{n∈ω} C_{n} and define for all i = 0, 1, 2, x^{(i)} := ⋃_{k} a_{i+3k}∪b_{i+3k}∪0_{|C}. It follows then from condition (vii) of Lemma 9.6 that each x^{(i)} ∈ G, and we now check that x^{(0)}. x^{(1)}. x^{(2)}(u) = 0 for all u ∈ ω^{2}. So fix such a u:

– If u ∈ C then by definition x^{(i)}(u) = 0 for all i.

– If not then by conditions (i) and (iii) there exists a unique pair (i, k) ∈ 3 × ω such that u ∈ dom (b_{i+3k}). Let m < n be the two first integers ≢ i such that u ∈ dom (a_{m}) ∩ dom (a_{n}). It follows from condition (iv) that we necessarily have: n = m + 1 or m + 2, hence m = n − 1 or n − 2, so by condition (v), a_{m}(u). a_{n}(u) = 0. But since m ≢ n then a_{m} ⊂ x^{(i′)}, a_{n} ⊂ x^{(i′′)} with {i, i′, i′′} = {0, 1, 2}, and it follows that x^{(0)}. x^{(1)}. x^{(2)}(u) = 0. □

Remarks 9.7. a) The reduction of the general case to the particular case F = N_{2} in the previous proof is not a matter of facility, and we do not know of any direct argument avoiding this reduction step. Observe in particular that the notion of cylinder, which played a crucial role in this proof, is meaningless in the general case, and the set C of all cylinders cannot be replaced by the ideal F^{∗}.

b) The conclusion of Theorem 9.3 cannot be strengthened to get two disjoint elements as in Theorem 9.2. Indeed if

G = {M ⊂ ω^{2} : ∃p, M(p) is cofinite and ∀q ≥ p, M(q) is infinite }

(where for all M ⊂ ω^{2} and p ∈ ω we set M(p) = {n ∈ ω : (p, n) ∈ M}) then G is clearly a Σ_{3}^{0} set containing the filter N_{2} and one easily checks that any two elements of G have non-empty intersection.

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Analyse Fonctionnelle, Institut de Mathématique de Jussieu, Boîte 186, 4, place Jussieu, 75252 Paris Cedex 05, FRANCE

  E-mail address: debs@math.jussieu.fr

Analyse Fonctionnelle, Institut de Mathématique de Jussieu, Boîte 186, 4 place Jussieu, F- 75252 Paris Cedex 05, FRANCE

  E-mail address: raymond@math.jussieu.fr