Ramsey spaces and the Katetov order Sonia Navarro Flores National - - PowerPoint PPT Presentation

Borel ideals Katˇ etov order Ellentuck space Topological Ramsey spaces

Ramsey spaces and the Katetov order

Sonia Navarro Flores

National University of Mexico BLAST 2018

August 8th, 2018

Sonia Navarro Flores Ramsey spaces and the Katetov order

Borel ideals Katˇ etov order Ellentuck space Topological Ramsey spaces

Ideals

Definition A family I ⊂ P(X) of subsets of a given set X is an ideal on X if

1

for A, B ∈ I, A ∪ B ∈ I,

2

for A, B ⊂ X, A ⊂ B and B ∈ I implies A ∈ I and

3

X / ∈ I.

Sonia Navarro Flores Ramsey spaces and the Katetov order

Borel ideals Katˇ etov order Ellentuck space Topological Ramsey spaces

Examples

1

The ideal Fin, which consists of all non empty finite subsets of ω.

2

The ideal of meager subsets of R, M = {M ⊂ R : M is meager}

3

The eventually different ideal ED is such that A ∈ ED iff (∃m, n ∈ ω)(∀k > n)(|{l : (k, l) ∈ A}| ≤ m)

4

EDfin = {A ∩ ∆ : A ∈ ED} where ∆ = {(n, m) ∈ ω × ω : n ≤ m}.

5

The ideal Fin × Fin is such that A ∈ Fin × Fin iff {n : {m : (n, m) ∈ A} / ∈ Fin} ∈ Fin

Sonia Navarro Flores Ramsey spaces and the Katetov order

Borel ideals Katˇ etov order Ellentuck space Topological Ramsey spaces

Definition Given an ideal I on X:

1

We say that X is tall if for each Y ⊆ X infinite there exists I ∈ I such that Y ∩ I is infinite.

2

We denote by I+ the family of I-positive sets, i.e. subsets of X which are not in I.

3

If Y ∈ I+, we denote by I ↾ Y the ideal {I ∩ Y : I ∈ I}

• n Y .

We will consider ideals on countable sets, so we pretend that they are, in fact, ideals on ω. We will identify P(ω) with 2ω is given the product topology. We call an ideal I a Borel ideal

• n ω if I is a Borel subset of 2ω.

Sonia Navarro Flores Ramsey spaces and the Katetov order

Borel ideals Katˇ etov order Ellentuck space Topological Ramsey spaces

This order, called Katˇ etov order was introduced by M. Katˇ etov in 1968 to study convergence in topological spaces. It has been used to clasify ultrafilters and ideals for mathematicians as Baumgartner, Solecky, Brendle and Hrusak. Definition Given two ideals I and J on ω we shall say that:

1

I is Katˇ etov below J (I ≤K J ) if there is a function f : ω → ω such that for all I ∈ I, f −1[I] ∈ J .

2

I and J are Katˇ etov equivalent (I ≃K J ) if I ≤K J and J ≤K I.

Sonia Navarro Flores Ramsey spaces and the Katetov order

Borel ideals Katˇ etov order Ellentuck space Topological Ramsey spaces

Some basic properties of Katˇ etov order are listed here. Let I and J be ideals on ω. I ≃K Fin if and only if I is not tall. If I ⊆ J then I ≤K J (f is the indentity function). If X ∈ I+ then I ≤K I ↾ ω (f is the inclusion map).

Sonia Navarro Flores Ramsey spaces and the Katetov order

Borel ideals Katˇ etov order Ellentuck space Topological Ramsey spaces

Definition An ideal I is K-uniform if for every X ∈ I+, I ↾ X ≃K I. In ”Katˇ etov order on Borel ideals” M.Hrusak states the question: Question Is EDfin the only tall Fσ ideal which is K-uniform?

Sonia Navarro Flores Ramsey spaces and the Katetov order

Borel ideals Katˇ etov order Ellentuck space Topological Ramsey spaces

Ellentuck space

Let a ∈ [ω]<ω and A ∈ [ω]ω be such that max a < min A, let [a, A] = {B ∈ [ω]ω : a ⊏ B ⊆ a ∪ A}. The collection [ω]ω with the topology induced by basic sets [a, A] is the Ellentuck space.

Sonia Navarro Flores Ramsey spaces and the Katetov order

Borel ideals Katˇ etov order Ellentuck space Topological Ramsey spaces

Definition Let X be a subset of [ω]ω, X ⊆ [ω]ω is called Ramsey if for every basic set [a, A] there is B ∈ [A]ω with [a, B] ⊆ X or [a, B] ∩ X = ∅. Ellentuck’s Theorem Let X ⊆ [ω]ω. Then X is Ramsey if and only if X has the Baire Property in the Ellentuck topology.

Sonia Navarro Flores Ramsey spaces and the Katetov order

Borel ideals Katˇ etov order Ellentuck space Topological Ramsey spaces

Topological Ramsey spaces are an abstraction of the Ellentuck space that satisfy a general version of Ellentuck’s Theorem. In the book ”Introduction to Ramsey spaces”, Todorcevic defines axioms A.1 − A.4 by extracting properties of the Ellentuck space building on prior work of Carlson-Simpson. Topological Ramsey spaces has been studied by several mathematicians because of their applications to Tukey order theory and Bannach spaces.

Sonia Navarro Flores Ramsey spaces and the Katetov order

Borel ideals Katˇ etov order Ellentuck space Topological Ramsey spaces

Authors defined the space R1 motivated to a Tukey-classification problem. Definition (Dobrinen, Todorcevic) Let T denote the following infinite tree of height 2. T = {} ∪ {n : n < ω} ∪

• n<ω

{n, i : i ≤ n}. the n-th subtree of T is T(n) = {, n, n, i : i ≤ n}. The members X of R1 are infinite subtrees of T which have the same structure as T. For n < ω, rn(X) denotes

i<n X(i).

ARn = {rn(X) : X ∈ R1}, and AR =

n<ω ARn.

Sonia Navarro Flores Ramsey spaces and the Katetov order

Borel ideals Katˇ etov order Ellentuck space Topological Ramsey spaces

Definition If X ∈ R1, s ∈ AR1, [s, X] = {Y ∈ R1 : (∃n ∈ ω)(s = rn(X)) and Y ⊆ X}. Theorem(Dobrinen, Todotcevic) The set R1 with the topology generated by basic sets [s, X] is a topological Ramsey space. Remark: (∀X ∈ ED+

fin)(∃Y ∈ R1)(Y ⊆ X).

Sonia Navarro Flores Ramsey spaces and the Katetov order

Borel ideals Katˇ etov order Ellentuck space Topological Ramsey spaces

Hypercubes

The space H2 is a particular case of a class of topological Ramsey spaces constructed by using products of finite ordered relational structures from Fr¨ aiss´ e classes with the Ramsey

• property. The space H2 was constructed in order to force a

p-point which has initial Tukey structure exactly the Boolean algebra P(2).

Sonia Navarro Flores Ramsey spaces and the Katetov order

Borel ideals Katˇ etov order Ellentuck space Topological Ramsey spaces

Definition For each k < ω and j ∈ 2, let Ak,j be any linearly ordered set

• f size k + 1. Letting Ak denote the product Ak,0 × Ak,1. Let

A = k, Ak : k < ω. A is the maximal member of H2. We define B to be a member of H2 if and only if B is a subset of A with the same structure. We use B(k) to denote nk, Bk, the k-th block of B. The n-th approximation of B is rn(B) = B(0), ..., B(n − 1). Let AH2

n = {rn(B) : B ∈ H2},

the collection of all n-th approximations to members of H2. Let AH2 =

n<ω AH2 n.

Sonia Navarro Flores Ramsey spaces and the Katetov order

Borel ideals Katˇ etov order Ellentuck space Topological Ramsey spaces

Definition If X ∈ H2, s ∈ AH2, [s, X] = {Y ∈ H2 : (∃n ∈ ω)(s = rn(X)) and Y ⊆ X}. Theorem(Trujillo) The set H2 with the topology generated by basic sets [s, X] is a topological Ramsey space.

Sonia Navarro Flores Ramsey spaces and the Katetov order

Borel ideals Katˇ etov order Ellentuck space Topological Ramsey spaces

Definition Let IH2 be the ideal such that A / ∈ IH2 iff (∀n ∈ ω)(∃s, t ∈ [ω]n)(s × t ⊂ A) Note that I+

H2 is Gδ and then IH2 is Fσ. Also note that if

X, Y ∈ H2, IH2 ↾ X ≃K IH2 ↾ Y . Then for every X ∈ H2, IH2 ↾ X is a K-uniform ideal. Now we want to prove that EDfin and IH2 are not Katˇ etov equivalent.

Sonia Navarro Flores Ramsey spaces and the Katetov order

Borel ideals Katˇ etov order Ellentuck space Topological Ramsey spaces

For every ideal I on ω. Let rI be the least natural number k such that (∀m)(∀c : [ω]2 → m)(∃X ∈ I+)(|c”[X]2| = k) (ω → (I+)2

rI).

Proposition If I, J are ideals on ω such that I ≤K J , then rI ≤ rJ . So if I and J are Katˇ etov equivalent, rI = rJ .

Sonia Navarro Flores Ramsey spaces and the Katetov order

Borel ideals Katˇ etov order Ellentuck space Topological Ramsey spaces

i Let m be a natural number and c : [ω]2 → m be a coloring. ii (∃f : ω → ω)(∀I ∈ I)(f −1[I] ∈ J ). iii Let ϕ : [ω]2 → m be such that ϕ({m, n}) = c({f (m), f (n)}). iv (∃Y ∈ J +)(|ϕ”[Y ]2| ≤ rJ ). v X = f [Y ] ∈ I+ and |c”[X]2| = |ϕ′′[Y ]2| ≤ rJ . Therefore rI ≤ rJ .

Sonia Navarro Flores Ramsey spaces and the Katetov order

Borel ideals Katˇ etov order Ellentuck space Topological Ramsey spaces

Proposition(Lafflame) If I = EDfin, rI = 3. Proposition(Dobrinen,N) If I = IH2, rI = 5.

Sonia Navarro Flores Ramsey spaces and the Katetov order

Borel ideals Katˇ etov order Ellentuck space Topological Ramsey spaces

Proposition Fix X ∈ H2, then IH2 ↾ X is a Fσ and K-uniform ideal which is not Katˇ etov equivalent to EDfin.

Sonia Navarro Flores Ramsey spaces and the Katetov order

Borel ideals Katˇ etov order Ellentuck space Topological Ramsey spaces

References

1

Dobrinen, N. and Todorcevic, S., A new class of Ramsey-classification Theorems and their Applications in the Tukey theory of Ultrafilters, Part 1,Transactions of the American Mathematical Society, 2014, Vol.366.

2

Dobrinen, N., Mijares, Jos´ e and Trujillo, T., Topological Ramsey spaces from Fra¨ ıss´ e classes, Ramsey-classification theorems, and initial structures in the Tukey types of p-points, 34 pp, Archive for Mathematical Logic, special issue in honor of James E. Baumgartner.

3

Hrusak, Michael, Katˇ etov order on Borel ideals, Accepted to Archive for Mathematical Logic.

4

Kechris, Alexander S., Classical Descriptive Set Theory, Springer-Verlag, 1994.

5

Trujillo,T., Topological Ramsey spaces, associated ultrafilters, and their applications to the Tukey theory of

Sonia Navarro Flores Ramsey spaces and the Katetov order