1. Introduction

An ideal on a set X is a collection of subsets of X closed under the operations of taking finite unions and subsets of its elements. Ideals are a very useful notion in topology and set theory and have been studied for a long time. We present a survey of results about ideals on countable sets and include many open questions.

We have tried to include aspects that were not covered in the survey written by M. Hrušák ^[28]. We start by presenting two common forms to define ideals: based on submeasures or on collections of nowhere dense sets. A basic tool in the study of ideals are some orders to compare them: Katětov, Rudin-Keisler and Tukey order. We focus mostly on the Katětov order. The reader can consult ^[28], ^[51], ^[52], ^[53] for results on the Tukey order. One important ingredient of our presentation is that we deal mainly with definable ideals: Borel, analytic or co-analytic ideals. Another crucial aspect is the role played by combinatorial properties of ideals, a theme that has been very much studied and provides a common ground for the whole topic. Most of the work on ideals has been concentrated on tall ideals, nevertheless we include a section on Fréchet ideals (i.e., locally non tall ideals). Since the properties about ideals we are dealing with are, in one way or another, based on selection principles, we end the paper with a discussion of Borel selection principles for ideals, that is, the selection function is required to be Borel measurable.

We do not pretend to give a complete revision of this topic; in fact, the literature is vast and we have covered a small portion of it. Our purpose was to present some of the diverse ideas that have being used for studying ideals on countable sets and collect some open questions which were scattered in the literature.

2. Terminology

An ideal I on a set X is a collection of subsets of X such that:

(i) 0 ∈ I and X ∈ I.

(ii) If A, B ∈ I, then A U B ∈ I.

(iii) If A ⊆ B and B ∈ I, then A ∈ I.

Given an ideal I on X, the dual filter of I, denoted I*, is the collection of all sets X \ A with A ∈ I. We denote by I⁺ the collection of all subsets of X which do not belong to I. Two ideals I and J on X and Y respectively are isomorphic if there is a bijection f : X → Y such that E ∈ I if, and only if, f [E] ∈ J. Suppose X and Y are disjoint; then the free sum of I and J, denoted by I ⊕ J is defined on X U Y as follows: A ∈ I ⊕ J if A ∩ X ∈ I and A ∩ Y ∈ J.

We denote by 2^<w (respectively, ℕ^<w) the collection of all finite binary sequences (respectively, finite sequences of natural numbers). If x ∈ 2^ℕ , then x ↾ n is the sequence (x(0), … , x(n - 1)} for n ∈ ℕ.

Now we recall some combinatorial properties of ideals. We put A ⊆* B if A \ B is finite. An ideal I is a P-ideal, if for any family E_n ∈ I there is E ∈ I such that E_n ⊆* E for all n. This is one of the most studied class of ideals.

(p⁺) I is p⁺, if for every decreasing sequence (A_n)_n of sets in I⁺, there is A ∈ I⁺ such that A ⊆* A_n for all n ∈ ℕ. Following ^[31], we say that I is p^-, if for every decreasing sequence (A_n)_n of sets in I⁺ such that A_n \ A_n+1 ∈ I, there is B ∈ I⁺ such that B ⊆* A_n for all n.

The following notion was suggested by some results in ^[13], ^[20]. Let us call a scheme a collection {A_s: s ∈ 2^<w| such that and for all s ∈ 2^<w. An ideal is wp⁺, if for every scheme {A_s: s ∈ 2^<w} with A_∅ ∈ I⁺, there is B ∈ I⁺ and α ∈ 2^ℕ such that B ⊆* A _α↾_n for all n. (q⁺) I is q⁺, if for every A ∈ I⁺ and every partition (F _n ) _n of A into finite sets, there is S ∈ I⁺ such that S ⊆ A and S ∩ F _n has at most one element for each n. Such sets S are called (partial) selectors for the partition. If we allow partitions with pieces in I, we say that the ideal is weakly selective ws ^[31] (also called weakly Ramsey in ^[46]). Another natural variation is as follows: For every partition (Fn)n of a set A ∈ I⁺ with each piece F_n in I, there is S ∈ I⁺ such that S ⊆ A and S ∩ F_n is finite for all n. It is known that the last property is equivalent to p^- (see Theorem 8.2).

All spaces are assumed ∞ be regular and T₁. A collection B of non empty open subsets of X is a π-base, if every non empty open set contains a set belonging ∞ B. A point x of a topological space X is called a Fréchet point, if for every A with there is a sequence (x_n)_n in A converging ∞ x. It is well known that filters (or dually, ideals) are viewed as spaces with only one non isolated point. We recall this basic construction. Suppose Z = ℕ ∪ {∞} is a space such that ∞ is the only accumulation point. Then is the neighborhood filter of ∞. Conversely, given an ideal I over ℕ, we define a topology on ℕ ∪ {∞} by declaring that each n ∈ ℕ is isolated and I* is the neighborhood filter of ∞. We denote this space by Z(I). It is clear that the combinatorial properties of I and Z(I) are the same.

For n ∈ ℕ, we denote by X ^[n] the collection of n-elements subsets of X and X ^[∞] the collection of infinite subsets of X. The classical Ramsey theorem asserts that for every coloring c : ℕ^[2] → {0,1}, there is an infinite subset X of ℕ such that X is c-homogeneous, that is, c is constant in X^[2]. An ideal I is Ramsey at ℕ, when for any coloring c : ℕ^[2] → {0,1} there is a c-homogeneous set which is I-positive, we denoted it by If it is the case that for any coloring c and any X ∈ I⁺ there is a c-homogeneous set Y ∈ I⁺ contained in X, we shall write and call such ideal a Ramsey ideal. A collection A of subsets of a set X is tall, if for every infinite set A ⊆ X, there is an infinite set B ⊆ A with B ∈ A. Ramsey's theorem says that the collection of c-homogeneous sets is a tall family for any coloring c.

A general reference for all descriptive set theoretic notions used in this paper is ^[33]. A set is F_σ (also denoted ) if it is equal to the union of a countable collection of closed sets. Dually, a set is G_δ (also denoted ) if it is the intersection of a countable collection of open sets. The Borel hierarchy is the collection of classes and for α a countable ordinal. For instance, (which is also denoted by F_σδ) are the sets of the form where each F_n is an F_σ. A subset A of a Polish space is called analytic, if it is a continuous image of a Polish space. Equivalently, if there is a continuous function f : ℕ^ℕ - X with range A, where ℕ^ℕ is the space of irrationals. Every Borel subset of a Polish space is analytic. A subset of a Polish space is co-analytic if its complement is analytic. The class of analytic (resp., co-analytic) sets is denoted by (resp. ).

3. Some examples

In this section we present some examples of ideals. The interested reader can consult ^[28], ^[31], ^[43] where he can find many more interesting examples.

The simplest ideal is Fin, the collection of all finite subsets of ℕ. There are two natural ideals quite related to Fin which are defined on ℕ x ℕ.

In general, let I and J be ideals on X and Y respectively; its Fubini product I x J is an ideal on X x Y defined as follows: for A ⊆ X x Y, we let A _x = {y ∈ Y (x, y) ∈ A}.

By an abuse of notation, the ideals {∅} x Fin and Fin x {∅} are usually denoted ∅ x Fin and Fin x ∅, respectively. An ideal I on X is countably generated if there is a countable collection {A_n: n ∈ ℕ} of subsets of X such that E ∈ I if, and only if, there is n such that E ⊆ A₀ ∪ … ∪ A_n. The only countably generated ideals containing all finite sets are Fin and Fin x {∅} (see Proposition 1.2.8. in ^[12]).

Two very important ideals on ℚ are the ideal of nowhere dense subsets of ℚ (with its usual metric topology), denoted nwd(ℚ), and the ideal of null sets defined as follows:

In general, if X is a topological space, then nwd(X) denotes the ideal of nowhere dense subsets of X. Another very natural ideal associated to a space is defined as follows: For every point x ∈ X, let

In fact, every ideal on X is of the form J_x for some topology on X.

Two ideals on ℕ that have a very natural connection with number theory and real analysis are the following:

and

The ideal I_d consists of the asintotic density zero sets.

Let CL(2^ℕ) denote the collection of clopen subsests of 2^ℕ and λ the product measure on 2^ℕ. Notice that CL(2^ℕ) is countable. Let

Solecki's ideal S is the ideal on Ω generated by the following sets:

where x ∈ 2^ℕ. Solecki introduced S to characterize the ideals satisfying Fatou's lemma ^[50].

The following ideal is called the eventually different ideal:

The following restriction of also plays an important role in the study of combinatorial properties of ideals:

where Δ = {(n, m) ∈ ℕ x ℕ : m ≤ n}. Note that is (up to isomorphism) the unique ideal generated by the selectors of some partition of ℕ into finite sets {I _n: n ∈ ℕ} such that lim sup_n |1_n| = ∞. As we will see later, is critical for the q⁺-property.

Let conv be the ideal generated by the range of all convergent sequences of rationals numbers, where the convergence is in ℝ. In other words, conv is the collection of all subsets of ℚ such that the Cantor-Bendixon derivative of its closure in ℝ is finite.

Now we present a family of ideals defined by homogeneous sets for colorings. Let c : ℕ^[2] → {0,1} be a coloring. Recall that a set H ⊆ ℕ is c-homogeneous if c is constant in H^[2]. The collection hom(c) of all c-homogeneous sets is closed in 2^ℕ. Let I_hom(c) be the ideal generated by the c-homogeneous sets.

The infinite random graph on ℕ, also known as the Rado graph or the Erdös-Rényi graph (see, e.g., ^[5]) can be concisely described as follows. Recall that a family {X_n: n ∈ ℕ} of infinite subsets of ℕ is independent, if given two disjoint finite subsets F, E of ℕ the set is infinite. Let {X_n: n ∈ ℕ } be an independent family of subsets of N such that n ∈ X_m if, and only if, m ∈ X_n, for all m, n ∈ ℕ. The random graph is then (ℕ, E), where

The random graph is universal in the following sense. Given a graph there is a subset X ⊆ ℕ such that The random graph ideal R is the ideal on ℕ generated by cliques and free sets of the random graph or, equivalently, the homogeneous sets with respect to the random coloring c : [ℕ]² → 2 defined by c({m, n}) = 1 if, and only if, { m, n} ∈ E.

4. Complexity of ideals

We say that a collection A of subsets of a countable set X is analytic (resp. Borel), if A is analytic (resp. Borel) as a subset of the cantor cubet 2 ^X (identifying subsets of X with characteristic functions) ^[33]. The set ℕ^[∞] of infinite subsets of ℕ will be always considered with the subspace topology of 2^ℕ. We say that an ideal is analytic, if it is an analytic as a subset of 2^X. Since the collection of finite subsets of ℕ is a dense set in 2^ℕ, then there are no ideals containing Fin which are closed as subsets of 2^ℕ. On the other hand, if I is a G _δ ideal with Fin ⊆ I, then I* = {ℕ \ A : A ∈ I} is also dense G _δ (as the map A → ℕ \ A is an homeomorphism of 2^ℕ into itself). Therefore by the Baire category theorem I ∩ I* = ∅, which says that I = P(ℕ). So, the simplest Borel ideals have complexity F _σ. They have been quite investigated as we will see. Every analytic ideal is generated by a G _δ set, i.e., there is a G _δ G ⊆ I such that every A ∈ I is a subset of a finite union of elements of G ^[64] (see also [53, Theorem 8.1]).

Most of the theory of definable ideals has been concentrated on analytic ideals. There are a few results about co-analytic ideals. The following theorem provides a very general representation of analytic ideals on spaces of continuous functions. It is an instance of the ideal defined by (3).

Theorem 4.1 (Todorčević [57, Lemma 6.53]). Let I be an ideal over ℕ. The following are equivalent.

(i) I is analytic.

(ii) There are continuous functions f, f _n: ℕ^ℕ → ℝ for n ∈ ℕ such that f is an accumulation point of {f_n: n ∈ ℕ} respect to product topology on C_P(ℕ^ℕ) and

where the closure is taken in C_P(ℕ^ℕ).

There are some well known co-analytic ideals. Let WO(ℚ) be the ideal of well ordered subsets of ℚ. This is a typical complete co-analytic ideal. Let I _wf be the ideal on ℕ^<W generated by the well founded trees on ℕ, i.e., A ⊆ ℕ^<W belongs to I _wf, if there is a wellfounded tree T such that A ⊆ T. This is equivalent to say that the tree generated by A is well founded. Then I _wf is also a complete co-analytic ideal. In sections 6 and 9 we shall present another examples of co-analytic ideals (see also ^[16], ^[43]).

We do not know of any general theorem, as Theorem 4.1, for co-analytic ideals. So we state this question as follows.

Question 4.2. Is there a general representation theorem for co-analytic ideals?

5. Ideals based on submeasures

A natural and very important method for defining ideals is based on measures or, more generally, submeasures. In this section we present some of these ideas.

A function φ : P(ℕ) → [0, ∞] is a lower semicontinuous submeasure (lscsm) if

There are several ideals associated to a lscsm:

They satisfy the following relations:

The collection of ideals that can be represented by one these three forms have been extensively investigated. The work of Farah ^[12] and Solecki ^[49] are two of the most important early works for the study of the ideals associated to submeasures.

To each divergent series f : ℕ - [0, +∞) of possitive real numbers, we associate a measure on ℕ by

An ideal I is summable ^[42] if there is a divergent series f as above such that I = Fin(φ_f). Notice that Sum(φ_f) = Fin(φ_f). The usual notation for this ideal is I _f. A typical example is the following:

Another very natural way of defining lscsm is as follows. Let I _n be a partition of ℕ into finite sets. Let v _n be a measure on I _n (i.e., there is a function f_n: I _n → [0, +∞) such that . Let

y (A) = SUp Vn(A O In). n

Then φ is a lscsm and Exh(φ) is called a density ideal ^[12]. The prototype is the following

Example 5.1. Let φ_d: P(ℕ) - [0, +∞] given by

Then I_d = Exh(φ_d) is the ideal of asintotic density zero sets. We have

The Cantor set {0,1}^ℕ is a group with the product operation where {0,1} is the group ℤ₂; equivalently, viewing the elements of 2^ℕ as subsets of ℕ, then the algebraic operation is the symmetric difference. Then 2^ℕ is actually a Polish group. Every ideal on ℕ is a subgroup of 2^ℕ. Since there are no G _δ ideals (containing Fin), then none of these subgroups are Polish. However, the following weaker notion has been used to study subgroups of Polish groups. We say that a subgroup G of 2^ℕ is Polishable, if there is a Polish group topology on G such that the Borel structure of this topology is the same as the Borel structure G inherites from 2^ℕ.

The following representation of analytic P-ideals is the most fundamental result about them. It says that any P-ideal is in a sense similar to a density ideal.

Theorem 5.2 (S. Solecki ^[49]). Let I be an analytic ideal on ℕ. The following are equivalent:

(i) I is a P-ideal.

(ii) There is a lscsm φ such that I = Exh(φ).

(iii) I is Polishable.

In particular, every analytic P-ideal is F _σδ . Moreover, I is an F _σ P-ideal, if, and only if, there is a lscsm φ such that I = Exh(φ) = Fin(φ).

5.1. F _σ and F _σδ ideals

As we said, from the complexity point of view, F _a ideals are the simplest ones. In this section we present some results about them.

A set ⊆ P(ℕ) is hereditary if for every A ∈ K and B ⊆ A we have that B ∈ . A family of subsets of ℕ is said to be closed under finite changes if A△F ∈ for every A ∈ and a finite set F ⊆ ℕ. Given an hereditary collection , we denote by I_K the ideal generated by . That is to say

Theorem 5.3 (Mazur ^[42]). Let I be an ideal on ℕ. The following are equivalent.

(i) I is F_σ .

(ii) there is a hereditary closed collectionof subsets of ℕ such that I = I .

(iii) there is a lscsm φ such that I = Fin(φ).

An important example of F _σ ideals are I_hom(c), the ideal generated by the family hom(c) of homogeneous sets respect to a coloring c (see section 3). Notice that hom(c) is a closed hereditary collection of subsets of ℕ.

The following is part of the folklore (for a proof see, e.g., [31, Lemma 3.3]).

Theorem 5.4. Every F _σ ideal is p ⁺ .

An important question involving F _σ ideals is the following:

Question 5.5 (M. Hrušák ^[29]). Does every tall Borel ideal contain a tall F _σ ideal?

The previous question can be understood as asking whether an analog of the classical perfect set theorem holds for the collection of tall Borel ideals. However, the analogy is not complete, since there exists a ideal which does not contain any F _σ tall ideal (see [24, Theorem 4.24]).

We have seen in Theorem 5.2 that every analytic P-ideal is F _σδ. One could naturally ask whether such ideals are a countable intersection of F _σ ideals. Since this is not true in general, Farah ^[14] introduced a weaker property (we follow the presentation given in ^[30]). They called an ideal I Farah if there is a countable collection _n of closed hereditary families of subsets of ℕ such that

It is clear that every Farah ideal is F _σδ. In ^[14] it is shown that nwd(ℚ), null(ℚ) and every analytic P-ideal are Farah. However, there is no an F _σ ideal J such that nwd(ℚ) ⊆ J.

Theorem 5.6 (M. Hrušák and D. Meza-Alcántara ^[30]). Let I be an ideal on ℕ. The following are equivalent:

(i) I is Farah.

(ii) There is a sequence {F_n: n ∈ ℕ} of hereditary F _σ sets closed under finite changes such that

(iii) There is a sequence {F_n: n ∈ ℕ} of F _σ sets closed under finite changes such that

The previous result suggests a weaking of the notion of a Farah ideal. An ideal I is called weakly Farah ^[30] if there is a sequence {F_n: n G N} of hereditary F _a sets such that

I =Dn Fn.

Question 5.7.

• (i) (Farah ^[14]) Is every F _σδ ideal a Farah ideal?

• (ii) (M. Hrušák and D. Meza-Alcántara ^[30]) Is every F _σδ ideal weakly Farah? Is every weakly Farah ideal a Farah ideal?

5.2. Summable ideals on Banach spaces

The notion of a summable ideal has been extended to ideals where the sum is calculated in a Banach space or, more generally, in a Polish abelian group ^[2], ^[63], ^[11]. In this section we present some results and questions about this approach.

Let G be a Polish abelian group (with additive notation) or a Banach space. Let h : ℕ → G be a sequence. We say that the series is unconditional convergent in G if the net (where Fin is ordered by ⊆) is convergent in G. This is equivalent to require that is convergent in G for every permutation π of ℕ. Let h : ℕ → G be a sequence such that does not exist. The generalized summable ideal associated to G and h is the following ^[2]:

An ideal I is said to be G-representable, if there is h such that I = . Analogously, it is defined when an ideal is C-representable for C a class of abelian Polish groups.

We recall that a lscsm φ is non-pathological ^[12] if φ(A) is equal to the supremum of all v(A) for v a measure such that v ≤ φ. A P-ideal is non-pathological, if it is equal to Exh(φ) for some non-pathological lscsm φ.

Theorem 5.8 (Borodulin-Nadzieja, Farkas, Plebanek ^[2]).

• (i) An ideal is ℝ-representable if, and only if, it is summable.

• (ii) An ideal is Polish-representable if, and only if, it is an analytic P-ideal.

• (iii) An analytic P-ideal is Banach-representable if, and only if, it is non-pathological.

• (iv) A tall F _σ P-ideal is representable in c₀ if, and only if, it is summable.

• (v) There is an F _σ tall ideal representable in l ₁ which is not summable.

Question 5.9 (^[2]). How to characterize analytic P-ideals which are c₀-representable?

Question 5.10 (^[2]). How to characterize ideals which are l ₁-representable? Are they necessarily F _σ ?

6. Topological representations by nowhere dense sets

In this section we review some constructions of ideals motivated by the ideal of nowhere dense sets. We consider two different ways of presenting nwd(ℚ). For the first one, we see ℚ as a dense subset of R and we have the following representation:

On the other hand, if is a base for ℚ (of non empty open sets), we have

We will address each of these approaches in this section.

6.2. Topological representation

Suppose X is a Polish space and J is a σ-ideal of subsets of X. Let D ⊆ X be a countable dense set. An ideal I_J on D is defined as follows (see ^[44], ^[36] and the references therein). Let A ⊆ D; then,

An ideal I on ℕ has a topological representation ^[36] if there is Polish space X, a σ-ideal J on X and a countable dense set D ⊆ X such that I is isomorphic to I_J. Notice that, by definition, null(ℚ) has a topological representation in R. Both ideals nwd(Q) and null(Q) are tall and F _σδ. In ^[15] it is shown that nwd(ℚ) and null(ℚ) are not isomorphic and also that none of them is a P-ideal.

Topological representable ideals have the following interesting characterization. An ideal I is countably separated if there is a countable collection {X_n: n ∈ ℕ} such that for all A ∈ I and all B ∉ I, there is n such that A ∩ X_n = ∅ and B ∩ X_n ∉ I. This notion was motivated by the results in ^[54].

Theorem 6.6 ([36, Theorem 1.1]). Let I be an ideal on a countable set. The following are equivalent:

• (i) I has a topological representation.

• (ii) I has a topological representation on 2^ℕ with an ideal J generated by a collection of closed nowhere subsets of 2^ℕ.

• (iii) I is tall and countably separated.

Theorem 6.7 ([36, Corollary 1.5]). If a co-analytic ideal has a topological representation, then it is either -complete or F _σδ -complete.

Let us see some examples of ideals which are not topologically representable.

6.3. Marczewski-Burstin representations

Let F be a family of non empty subsets of X. The Marczewski ideal associated to F is defined as follows (see ^[38] and references therein):

If τ is a topology on X and F is a base for τ, then S⁰(F) is nwd(τ). If an ideal I is equal to S⁰(F) for some family of non empty subsets of X, then it is said that I is Marczewski-Burstin representable by F. When such F can be found countable, it is said that I is Marczewski-Burstin countably representable, which is denoted MBC.

Example 6.11 ([38, Theorem 4.12]). null(ℚ) is MBC.

It is clear that when F is an analytic collection of subsets of a countable set X, then S⁰(F) is at most Analogously to what happen with nwd(τ) (see Theorem 6.2), if F is an F _σ family, then S⁰(F) is

Theorem 6.12 ([38, Theorem 4.4]).

(i) Let I be an MBC ideal. Then I is F _σδ and countably separated.

(ii) If I is countably separated, then there is a MBC ideal J such that I ⊆ J.

There are two natural properties about F which imply that S⁰(F) is tall (see [38, Theorem 3.6]).

Question 6.13 (^[38]). Let f : ℕ → ℚ be a bijection. Let

Then J_c is an ideal. Is it MBC?

If (X, τ) is a countable topological space without isolated points and has a countable n-base, then nwd(τ) is isomorphic to ℚ and clearly is MBC. Thus we have the following.

Question 6.14. Let (X, τ) be a contable topological space without isolated points such that nwd(τ) is MBC. Is nwd(τ) isomorphic to nwd(ℚ)?

7. Ordering the collection of ideals

One of the main tools for the study of combinatorial properties of ideals are some orders (in fact, pre-order) defined on the collection of all ideals: Katětov order ≤_K, Rudin-Keisler order ≤_RK and Tukey order ≤_T.

Let I and J be two ideals on X and Y respectively. We say that I is Katětov below J, denoted I ≤_K J, if there is a function f : Y → X such that f^--1 [E] ∈ J for all E ∈ I. If f is finite-to-one, then we write I ≤_KB J and refer to the (pre)order ≤_KB as the Katětov-Blass order. We say that two ideals I and J are Katětov equivalent, denoted I ≈_K J, if I ≤_K J and J ≤_K I. Let Z(I) and Z(J) be the corresponding spaces defined in section 2. If f : Y → X is a function we will abuse the notation and consider f : Z(J) → Z(I) by letting f(∞) = ∞. If f is a Katětov reduction between I and J, then f : Z(J) → Z(I) is clearly continuous. Conversely, if there is f : Z(J) → Z(I) continuous with f^-1 (∞) = {∞}, then I ≤_K J.

Let (D, ≤) be a directed ordered set, i.e., for each x, y ∈ D, there is z ∈ D such that x, y ≤ z. A set A ⊆ D is bounded if there is x ∈ D such that y ≤ x for all y ∈ A. The dual notion to bounded set is that of cofinal set. A set A ⊆ D is cofinal, if for each x ∈ D, there is y ∈ A such that x ≤ y. Let D and E be two directed orders. A function f : D → E is called Tukey, if preimages under f of sets bounded in E are bounded in D. We write D ≤_T E if there is a Tukey function from D to E and we say that D is Tukey reducible to E.

We shall focus only on the Katětov order as it is crucial for stating some important open questions. We shall follow the works of Hrušák ^[29] and Meza ^[43] (see also ^[31]) which are basic references on this topic. We refer the reader to ^[51], ^[52], ^[53] for results on Tukey order. The Rudin-Keisler order will be defined in section 9 to state some questions.

Theorem 7.1 (^[29]). Let I and J be two ideals on ℕ. Then,

For many combinatorial properties there are ideals (usually Borel ones of a low complexity) which are critical with respect to the given property, that is, they are maximal or minimal in the Katětov order ≤_K among all ideals satisfying the property. To illustrate this we present some examples (see ^[31] for many other similar results). A countable splitting family for an ideal I on ℕ is a countable collection X of infinite subsets of ℕ such that for every Y ∈ I⁺, there is such that

Theorem 7.2 (^[31]). Let I be a tall ideal on ℕ. Then,

The following theorems show some global properties of the Katětov order.

Theorem 7.3.

There is a result similar to part (ii) proved by Hrušák-Meza ^[32] showing that there is a universal analytic P-ideal.

Next results show two very interesting dichotomies. The ideals and were defined in section 3.

Theorem 7.4 (M. Hrušák ^[29]) (Category Dichotomy). Let I be a Borel ideal. Then either I ≤_K nwd(ℚ), or there is an I -positive set X such that ≤_K I ↾ X.

Theorem 7.5 (M. Hrušák ^[29]) (Measure Dichotomy). Let I be an analytic P-ideal. Then either I ≤_K Z, or there is an I-positive set X such that S ≤_K I ↾ X.

Question 7.6 (M. Hrušák ^[29]). Is R ≤_K S?

As we mentioned above there is no maximum among Borel ideals; however, we have the following.

Question 7.7 (H. Sakai ^[45]). Let 1 ≤ α ≤ w ₁. Is there a Borel ideal I such that J ≤_K I for all ideal J?

The following is a fundamental problem.

Question 7.8 (M. Hrušák ^[31]). If I is a Borel tall ideal, then either there is an I-positive set X such that I ↾ X ≥_K conv, or there is an F_σ-ideal J containing I.

See Theorem 8.9 for a partial answer to the previous question.

Question 7.9 (M. Hrušák ^[31]). Does every Borel ideal I satisfy that either I ≥_K Fin x Fin, or there is an F _σδ-ideal J such that I ⊆ J?

8. Ramsey and convergence properties

In this section we discuss some properties of ideals which have been motivated by properties of convergent sequences and series on ℝ ^[17], ^[18], ^[19], ^[20]: Bolzano-Weierstrass, Riemann's rearrangement Theorem and convergence in functional spaces. Those properties have a natural connection with Ramsey's theorem.

We have not included the game theoretic version of Ramsey properties which is indeed a very interesting approach. We refer the reader to the work of Laflamme ^[39], ^[40].

To each ideal there is an associated notion of convergence that we describe hereunder. Let X be a topological space and I an ideal on ℕ. A sequence (x_n)_n in X is I-convergent to x, if for every open set U of X with x ∈ U. Notice that Fin-convergence is the usual notion of convergence of sequences.

We recall that an ideal I is called Ramsey at ℕ when it satisfies and it is called Ramsey when (see section 2).¹

An ideal I has the Bolzano-Weierstrass property, denoted BW, if for any bounded sequence {x_n: n ∈ ℕ} of real numbers there is an I-positive set A such that {x_n: n ∈ A} is I-convergent. An ideal I has the finite Bolzano-Weierstrass property, denoted FinBW, if for any bounded sequence {x_n: n ∈ ℕ} of real numbers there is an I-positive set A such that {x_n: n ∈ A} is convergent. An ideal I is Mon (or monotone), if for any sequence {x_n: n ∈ ℕ} of real (equivalently rational) numbers there is an I-positive set X such that {x_n: n ∈ X} is monotone (possibly eventually constant). We say that I is hereditarely mononote, denoted h-Mon, if I ↾ A is Mon for all A ∉ I. Neither nwd(ℚ) nor I_d satisfy FinBW (see ^[19]).

Theorem 8.1 ([20, Theorem 3.16]). Let I be an ideal on ℕ. The following are equivalent:

The property (ii) above was denoted wp⁺in ^[3], and property (i) was denoted h-FinBW in ^[19], ^[20]. The following theorem summarizes several known results in the literature (see ^[3] for a proof and references).

Theorem 8.2. The following holds for ideals on a countable set.

The usual proof that Fin is a FinBW ideal shows in fact more: Any p⁺-ideal is FinBW.

Theorem 8.3 ([19, Theorem 3.4 and 4.1]). Every ideal that can be extended to an F_σ ideal satisfies FinBW.

Theorem 8.4 ([20, Fact 3.1 and Corollary 3.10]). If an ideal is Ramsey at ℕ, then it satisfies Mon, and if it is Mon, then FinBW holds. Moreover, any Mon analytic P-ideal is Ramsey at ℕ.

Example 8.5. I_1/n is FinBW but not Mon (see the remark after Corollary 3.10 in [20]).

FinBW is a Ramsey theoretic property as stated in the following theorems.

Theorem 8.6 ([20, Theorem 3.11]). Let I be a q ⁺ -ideal. Then the following are equivalent:

We have also a local version of the previous result.

Theorem 8.7 ([20, Theorem 3.16]). Let I be an ideal. Then the following are equivalent:

Perhaps one of the most intriguing question is the following.

Question 8.8 (Hrušák ^[31]). Is there a tall Ramsey Borel (or analytic) ideal?

A partial answer to Question 7.8 is the following.

Theorem 8.9 ([1, Proposition 6.5]). Let I be an analytic P-ideal. The following are equivalent.

We note that the equivalence of (i) and (ii) was proven in ^[43] (see section 5.1 in ^[31]), and that (ii) is equivalent to (iii) for analytic P-ideals was proven in [19, Theorem 4.2]. But the result was formally stated in [1, Proposition 6.5]). This motivates a reformulation of Question 7.8 as follows (see also Theorem 8.3).

Question 8.10 ([17, Problem 6.1]). Let I be a tall Borel FinBW ideal. Can I be extended to an F_σ ideal?

Now we turn our attention ∞ another classical convergence property that can be reformulated in terms of ideals. A classical theorem of Riemann says that any conditional convergent series of reals numbers can be rearranged ∞ converge ∞ any given real number or ∞ diverge ∞ +∞ or -∞. In other words, if (a_n)_n is a conditional convergent series and r ∈ ℝ ∪ {+∞, -∞}, there is a permutation π : ℕ → ℕ such that In ^[18], ^(34] is considered a property of ideals motivated by Riemann's theorem. Let us say that an ideal I has the property R, if for any conditionally convergent series of real numbers and for any r ∈ ℝ ∪ {+∞, -∞}, there is a permutation π : ℕ → ℕ such that and

Similarly, I has property W, if for any conditionally convergent series of reals , there exists A ∈ I such that the restricted series is still conditionally convergent. In ^[34] it is studied similar properties but for series of vectors in ℝ².

Theorem 8.11 (Filipów-Szuca ^[18]). Let I be an ideal on ℕ. Then,

For instance, since I_d is not BW, then it has property R.

Theorem 8.12 (Filipów-Szuca [18, Theorem 3.3]). Let I be an ideal on N. The following statements are equivalent.

Question 8.13 (Klinga-Nowik ^[34]). Suppose that (i) I has the R property; (ii) is a conditionally convergent series of reals; (iii) is divergent and all b_n are positive reals. Does there exist W ∈ I such that is conditionally convergent and

Now we will look at some convergence properties on spaces of continuous functions. We start with the classical Arzelá-Ascoli's theorem characterizing compactness on the pointwise topology.

Theorem 8.14 ([17, Theorem 3.1]) (Ideal Version of Arzelà-Ascoli Theorem). Let I be an ideal on ℕ. The following conditions are equivalent.

Now we present an ideal version of the classical Helly's selection theorem in the space of monotone functions on the unit interval.

Theorem 8.15 ([17, Theorem 5.8]) (Ideal Version of Helly's Theorem). Let I be an ideal on ℕ. Suppose that I can be extended to an F_σ ideal. Then for every sequence (f_n)_n∈ℕ of uniformly bounded monotone real-valued functions defined on ℝ there is A ∈ I⁺ such that the subsequence (f_n)_n∈A is pointwise convergent.

We recall that, by Theorem 8.3, any ideal that can be extended to an F_σ ideal satisfies FinBW; thus, we have the following natural question.

Question 8.16 ([17, Problem 5.10]). Let I be an ideal on ℕ. Are the following conditions equivalent?

A summary of implications among some of the combinatorial properties studied is as follows. We abbreviate countably generated and countably separated by w-gen and w-sep, respectively. An ideal is Fréchet if it is locally non tall (they will be discussed in the next section).

9. Fréchet ideals

Many of the results presented so far were about tall ideals. In this section we study Fréchet ideals, a very important class of non tall ideals. This notion has a topological motivation but it can be expressed also as a combinatorial notion. Recall that to each ideal I on a set X is associate a topological space Z(I) on X ∪ {∞} (see section 2). We say that I is Fréchet if Z(I) is a Fréchet space. Notice that for E ⊆ X, we have

It is easy to verify that I is Fréchet if, and only if, for every A ∉ I there is an infinite B ⊆ A such that every infinite subset of B is not in I; that is to say, I ↾ A is not tall for every A ∉ I. In other words, an ideal is Fréchet if it is locally non tall.

Given a family A of infinite subsets of X, we define the orthogonal of A as follows ^[54]:

Notice that A^⊥ is an ideal. If A is an analytic family, then A^⊥ is co-analytic.

We denote by I(A) the ideal generated by A, that is to say,

Example 9.1. Let A _n = {n} x ℕ for n ∈ ℕ and A = {A_n: n ∈ ℕ }. Then I(A) = Fin x {∅}.

The following fact shows the importance of ⊥ to study Fréchet spaces.

Theorem 9.2. Let I be an ideal on X.

(i) An infinite set E ⊆ X is a convergent sequence to ∞ in Z(I) if, and only if, E ∈ I^⊥ .

(i) I is Fréchet if, and only if, I = (I^⊥)^⊥

Example 9.3. (Fin x {∅})^⊥ = {∅} x Fin and ({∅} x Fin)^⊥ = Fin x {∅}. In particular, Fin x {∅} and {∅} x Fin are Fréchet ideals.

Notice also that A^⊥ = ((A^⊥)^⊥)^⊥. In other words, A^⊥ is a Fréchet ideal for any family of sets A.

A family A of subsets of X is almost disjoint if A ∩ B is finite for all A, B ∈ A with A ≠ B. Typical examples of almost disjoint families are the following.

Example 9.4.

As we see next, almost disjoint families are tightly related to Fréchet ideals.

Theorem 9.5 (^[48]). Let I be an ideal on X. The following statements are equivalent.

Let us see some more examples of Fréchet ideals.

Example 9.6. Consider the ideal I _wf generated by the well founded trees on ℕ (see section 4). The orthogonal of I _wf is the ideal I_do generated by the finitely branching trees on ℕ, or equivalently, I_do consists of sets which are dominated by a branch:

The ideal I _wf is a complete co-analytic Fréchet ideal, while the ideal I_do is easily seen to be F _σδ (see [10, Example 2]).

9.1. Selective ideals

An ideal is selective if it is p⁺and q⁺. This is not the original definition given by Mathias [41] (who called them happy families) but it is a reformulation probably due to Kunen. The original first example of a selective ideal is the following:

Example 9.7 (Mathias ^[41]). Let A be an analytic almost disjoint family of infinite subsets of ℕ. Then I(A) is a selective ideal.

Next examples were found by Todorčević ^[55] in the realm of Banach spaces.

Example 9.8 ([57, Corollary 7.52]). Let f,f _n: X → ℝ be pointwise bounded continuous functions, and suppose that {f _n: n ∈ ℕ} accumulates to f. Let I _f be the ideal defined in Theorem 4.1, that is to say,

Then I_f is selective.

One of the reasons for being interested on selective ideals is due to the following.

Theorem 9.9 (Mathias ^[41]). Every selective ideal is Ramsey.

Selectivity is the combinatorial counterpart of the topological notion of bisequentiality (see [57, Theorem 7.53]) We only mention the following corollary of this fact which probably is due to Mathias ^[41].

Theorem 9.10. Every selective analytic ideal is Fréchet.

As we already said, if I is analytic, then I^⊥ is co-analytic. Motivated by the study of Rosenthal compacta Krawczyk ^[35] and Todorčević ^[56], ^[57] have shown the following (see also ^[10]):

Theorem 9.11. If I is a selective analytic ideal not countably generated, then I^⊥ is a complete co-analytic set.

The following examples illustrate the previous result.

Example 9.12. Let A be the almost disjoint family given in Example 9.4(ii), and let I be I (A). Then I is selective (see Example 9.7) and it is analytic (actually it is F _a ), but it is not countable generated. Hence, I^⊥ is (see [10, Example 1]).

Example 9.13. In Example 9.6 we presented the ideal I _wf generated by the well-founded trees on N (see section 4). The orthogonal of I _wf is the ideal I_do consists of sets which are dominated by a branch. The ideal I _wf is a complete co-analytic set, while the ideal I _do is easily seen to be F _σδ , it is not countably generated and it is not selective (see [10, Example 2] ).

9.2. Orthogonal Borel families

Two families A and B of infinite subsets of N are called orthogonal, if A ∩ B is finite for all A ∈ A and B ∈ B ^[54]. In this section we are interested in pairs of orthogonal families which are both Borel. An example is A = ∅ x Fin and B = Fin x ∅. The next theorem says this is the only possible such pair (I, I^⊥) when one of them is a P-ideal.

Theorem 9.14 (Todorčević, [54, Theorem 7]). Let I be an analytic P-ideal. Then I^⊥ is countably generated if, and only if, I^⊥ is Borel.

In ^[27] was constructed a family non isomorphic Fréchet ideals such that both I and I^⊥ are Borel. In fact, every ideal in is F _σδ. Let us recall its definition.

Let {K _n: n ∈ ℕ} be a partition of X. For n ∈ ℕ, let I_n be an ideal on K _n . The direct sum, denoted by is defined by

For instance, if each I_n is isomorphic to Fin, then ⊕_nI_n is isomorphic to {∅} x Fin. If each I_n is Fréchet, then ⊕_nI_n is also Fréchet.

The family is the smallest collection of ideals on ℕ containing Fin and closed under countable direct sums and the operation of taking orthogonal. The family has some interesting properties.

Theorem 9.15 (Guevara-Uzcátegui ^[27]). Let I be an analytic selective ideal on ℕ and A ⊆ ℕ. The following statements are equivalent:

• (i) I ↾ A is countably generated.

• (ii) I^⊥ ↾ A ∈ .

• (iii) I^⊥ ↾ A is Borel.

• (iv)

Theorem 9.16 (Guevara-Uzcátegui ^[27]). For every A ⊆ ℕ^<w , the following statements are equivalent:

• (i) I _wf ↾ A belongs to B.

• (ii) I _wf ↾ A is Borel.

• (iii)

Another interesting co-analytic ideal is WO(ℚ), the collection of well founded subsets of WO(ℚ). For simplicity, we will write WO instead of WO(ℚ). We first observe that WO^⊥ is the ideal of well founded subsets of (ℚ, <*) where <* is the reversed order of ℚ. In fact, the map x → -x from ℚ onto ℚ is an isomorphism between WO and WO ^⊥ . In particular, WO is a Fréchet ideal. A linear order (L, <) is said to be scattered, if it does not contain a order-isomorphic copy of ℚ.

Theorem 9.17 (Guevara-Uzcátegui ^[27]). For every A ⊆ ℚ, the following statements are equivalent:

• (i) A is scattered (with the order inherited from ℚ).

• (ii) WO ↾ A belongs to.

• (iii) WO ↾ A is Borel.

• (iv) WO WO \ A.

It is known that every F _σ tall ideal is not Ramsey, and also that there is a co-analytic tall Ramsey ideal ^[31]. We have already stated the basic question of whether there is a Ramsey tall Borel ideal (see Question 8.8). A seemingly weaker question is

Question 9.18. Is there a non Fréchet Ramsey Borel (or analytic) ideal?

The only Borel Fréchet pairs (I, I^⊥) we are aware of are given by the ideals in . So the natural question is:

Question 9.19. Is there a Borel Fréchet ideal with Borel orthogonal not isomorphic to an ideal in B?

A related question is the following

Question 9.20. Are there others Fréchet ideals satisfying the conclusion of theorem 9.16?

Since every Fréchet ideal is Katětov equivalent to Fin, then Katětov order is trivial among Fréchet ideals. But the Rudin-Keisler order is not trivial on Fréchet ideals ^[22], ^[21]. We say I ≤ _RK J if there is a function f : ℕ → ℕ such that f^{- 1} [E] ∈ J if, and only if, E ∈ I.

Theorem 9.21 (García-Ortiz ^[21]).

• (i) There are strictly increasing ≤_RK -chains of Fréchet idelas of size c⁺ . Such chains can be constructed ≤_RK -above every Fréchet ideal.

• (ii) For every infinite cardinal K ≤, there is a ≤_RK -antichain of size K. It is natural then to ask:

Question 9.22. How are the ideals in ordered according to ≤_RK?

F. Guevara ^[26] has classified the ideals in according to the Tukey order: Except for the countable generated, every ideal in is Tukey equivalent to ℕ^ℕ.

Obviously a Fréchet ideal cannot be topologically representable as it is not tall (see Theorem 6.6). It is easy to check that any Fréchet ideal is weakly selective. Thus the following question is appropriate.

Question 9.23. When is a Fréchet ideal countably separated?

F. Guevara ^[26] has shown that all ideals in are countably separated.

10. Uniform selection properties

As we have seen, most of the combinatorial properties for ideals are in fact selection properties. In this section we analyze the issue of whether the selector can be found Borel measurable. This question can be regarded as one instance of the classical uniformization problem in descriptive set theory: Let B ⊆ X x Y be a Borel set where X and Y are Polish spaces. A Borel uniformization for B is a Borel function F : X → Y such that (x, F(x)) ∈ B for all x ∈ proyx (B). It is well known that, in general, such Borel function does not exist (see section 18 of ^[33]).

As an illustration of the problem we are interested, let us consider the notion of tallness. Let C be a tall Borel (analytic, co-analytic) family of infinite subsets of ℕ. A very natural question is whether there is a Borel function F : 2^ℕ → 2^ℕ such that for all A ⊆ ℕ infinite, F (A) is an infinite subset of A and F (A) ∈ C. That is to say, F witness in a Borel way that C is tall. In this case we can say that C is uniformly tall or that C has a Borel selector. This problem was studied in ^[24] and, in particular, they showed that there is a tall F_σ ideal which is not uniformly tall.

10.2. Uniformly tall ideals

From Theorem 10.4, using the front ℕ^[2], we obtain that hom(c) is a uniformly tall collection, and thus I_hom(c) is a uniformly tall ideal for any coloring c of pairs of natural numbers. It should be clear that if a collection C contains hom(c) for some coloring c, then C is also uniformly tall. In fact, most of the examples we know of uniformly tall families are of that type. This could be regarded as a method for showing that a given family is uniformly tall (see example 10.6 below).

In particular, the random graph ideal R (see section 3) is uniformly tall. Thus, from the universal property of the random graph, we have that R ≤_k I if there is a ⊆ [ℕ]² such that hom( ) ⊆ I. Therefore, if R ≤_k I, then I has a Borel selector. That is the case with all examples studied in ^[29], ^[31]. Even Solecki's ideal S has a Borel selector ^[23], even though it is not known whether it is Katětov above R (see Question 7.6).

Example 10.6 (^[24]). The families of sets listed below are all uniformly tall. This is proved by finding a coloring c : ℕ^[2] → {0,1} such that hom(c) is a subset of the given family. The coloring used is the Sierpiński's coloring: Let X = {x_n: n ∈ ℕ} be a countable set and a total order on X. Define c : X^[2] → {0,1} by c({x_n,x_m}) = 0 if, and only if, n < m and x_n x_m . The c-homogeneous sets are the -monotone sequences in X:

• (i) nwd(τ), where (X, τ) is a Hausdorff countable space without isolated points.

• (ii) Let X be a compact metric space and (x_n)_n be a sequence in X. Consider C(x_n)_n = {A ⊆ ℕ : (x_n)_n∈A is convergent}.

• (iii) Let WO(ℚ) be the collection of all well-ordered subsets of ℚ respect the usual order. Let WO(ℚ)* the collection of well ordered subsets of (ℚ, <*), where <* is the reversed order of the usual order of ℚ. Then, C = WO(ℚ) U WO(ℚ)* is a tall family. Notice that C is

It is not true that Galvin's theorem 10.3 holds uniformly. In fact, there is ⊆ Fin such that hom( ) is not uniformly tall (see [24, Theorem 4.21]). Moreover, there is an Fj tall ideal which is not uniformly tall (see [24, Theorem 4.18]). Since the proof of this fact is not constructive, we naturally have the following:

Question 10.7 ([²⁴]). Find a concrete example of an Fσ tall ideal without a Borel selector.

Tall Fσ ideals are not q+ (otherwise they would be selective and thus Fréchet, see Theorems 5.4 and 9.10). This suggests the following:

Question 10.8. Is there a weakly selective (or q+) tall Borel ideal without a Borel selector?

Property q+ might be relevant as the next result suggests.

Theorem 10.9. Let I be an analytic P-ideal. The following assertions are equivalent:

• (i) I is tall.

• (ii) I has a continuous selector.

• (iii) I is not q+ at ℕ.

Since the generalized summable ideals (see section 5.2) are somewhat similar to P-ideals, the previous result naturally suggests the following.

Question 10.10. Let be a generalized summable ideal. Suppose is tall. Is it uniformly tall?

The following result characterizes tall ideals with continuous selectors.

Theorem 10.11 (J. Grebík and M. Hrušák [23, Proposition 25]). Let I be a Borel tall ideal. Then I has a continuous selector if, and only if, for every family {Xn : n ∈ ℕ} of infinite subsets of ℕ there is an A ∈ I such that A ∩ Xn ≠ ∅ for all n ∈ ℕ.

These are the only results concerning the complexity of the selector functions. So we naturally wonder if there is a bound in the Borel complexity of the selector for Borel tall ideals.

Ideals admitting a topological representation (as defined in section 6.2) are tall and countably separated. So we have the following question (a negative answer of it will solve Question 10.8, as countably separated ideals are weakly selective [37, Proposition 4.3]).

Question 10.12. Suppose I is a co-analytic ideal with a topological representation. Is I uniformly tall?

Another question we could ask is whether there is a "simple basis" for the collection of all tall families. More precisely we have the following question:

Question 10.13. Let C be a tall family of infinite subsets of ℕ. Suppose that C is analytic or co-analytic. Is there ⊆ Fin such that hom( ) ⊆ C?

The restriction on the complexity is necessary as there is a tall ideal I such that hom( ) ⊈ I for all ⊆ Fin. In particular, I does not contain any closed hereditary tall set (see [24, Theorem 4.24]).

Some test families for the previous question are the following:

Example 10.14.

• (a) Let C1 and C2 be two tall hereditary families with Borel selector. It is easy to verify that C1 ∩ C2 is also uniformly tall. Let B1 and B2 two fronts on ℕ, and Fi ⊆ Bi, for i = 0,1; is there a front B3 and F3 ⊆ B3 such that hom(F3) ⊆ hom(F1) ∩ hom(F2)? Or more generally, given Fi ⊆ Fin, for i = 0,1, is there F3 ⊆ Fin such that hom(F3) ⊆ hom(F1) ∩ hom(F2) ?

• (b) Let A be an almost disjoint analytic family of infinite subsets of ℕ. Let C(A) be I(A) ∪ (I(A))⊥. Then C(A) is a tall family. The question would be for which families A there is ⊆ Fin such that hom() ⊆ C(A).

• (c) Consider the following generalization of Example 10.6 (ii). Let K be a sequentially compact space, and (x_n)_n be a sequence on K. Let

Then C(x_n)_n is tall.

A particular interesting example is for K a separable Rosenthal compacta. By Debs' theorem ^[7], ^[8] (see also ^[9]), in every Rosenthal compacta, C(x_n)_n is uniformly tall. When K is not first countable C(x_n)_n, is a complete co-analytic subset of ℕ^[∞]. We do not know if there is ⊆ Fin such that hom( ) ⊆ C(x_n)_n.