Borel ideals Katˇ e tov 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 e tov 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 for A , B ∈ I , A ∪ B ∈ I , 1 for A , B ⊂ X , A ⊂ B and B ∈ I implies A ∈ I and 2 ∈ I . X / 3 Sonia Navarro Flores Ramsey spaces and the Katetov order
Borel ideals Katˇ e tov order Ellentuck space Topological Ramsey spaces Examples The ideal Fin, which consists of all non empty finite 1 subsets of ω. The ideal of meager subsets of R , 2 M = { M ⊂ R : M is meager } The eventually different ideal ED is such that A ∈ ED iff 3 ( ∃ m , n ∈ ω )( ∀ k > n )( |{ l : ( k , l ) ∈ A }| ≤ m ) ED fin = { A ∩ ∆ : A ∈ ED} where 4 ∆ = { ( n , m ) ∈ ω × ω : n ≤ m } . The ideal Fin × Fin is such that A ∈ Fin × Fin iff 5 { n : { m : ( n , m ) ∈ A } / ∈ Fin } ∈ Fin Sonia Navarro Flores Ramsey spaces and the Katetov order
Borel ideals Katˇ e tov order Ellentuck space Topological Ramsey spaces Definition Given an ideal I on X : We say that X is tall if for each Y ⊆ X infinite there 1 exists I ∈ I such that Y ∩ I is infinite. We denote by I + the family of I -positive sets, i.e. 2 subsets of X which are not in I . If Y ∈ I + , we denote by I ↾ Y the ideal { I ∩ Y : I ∈ I} 3 on 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 2 ω is given the product topology. We call an ideal I a Borel ideal on ω if I is a Borel subset of 2 ω . Sonia Navarro Flores Ramsey spaces and the Katetov order
Borel ideals Katˇ e tov order Ellentuck space Topological Ramsey spaces This order, called Kat ˇ etov order was introduced by M. Kat ˇ 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: I is Kat ˇ etov below J ( I ≤ K J ) if there is a function 1 f : ω → ω such that for all I ∈ I , f − 1 [ I ] ∈ J . I and J are Kat ˇ etov equivalent ( I ≃ K J ) if I ≤ K J 2 and J ≤ K I . Sonia Navarro Flores Ramsey spaces and the Katetov order
Borel ideals Katˇ e tov order Ellentuck space Topological Ramsey spaces Some basic properties of Katˇ e tov 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 e tov order Ellentuck space Topological Ramsey spaces Definition An ideal I is K -uniform if for every X ∈ I + , I ↾ X ≃ K I . In ”Katˇ e tov order on Borel ideals” M.Hrusak states the question: Question Is ED fin the only tall F σ ideal which is K -uniform? Sonia Navarro Flores Ramsey spaces and the Katetov order
Borel ideals Katˇ e tov 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 e tov 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 e tov 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 e tov order Ellentuck space Topological Ramsey spaces Authors defined the space R 1 motivated to a Tukey-classification problem. Definition (Dobrinen, Todorcevic) Let T denote the following infinite tree of height 2. � T = {��} ∪ {� n � : n < ω } ∪ {� n , i � : i ≤ n } . n <ω the n -th subtree of T is T ( n ) = {�� , � n � , � n , i � : i ≤ n } . The members X of R 1 are infinite subtrees of T which have the same structure as T . For n < ω , r n ( X ) denotes � i < n X ( i ). AR n = { r n ( X ) : X ∈ R 1 } , and AR = � n <ω AR n . Sonia Navarro Flores Ramsey spaces and the Katetov order
Borel ideals Katˇ e tov order Ellentuck space Topological Ramsey spaces Definition If X ∈ R 1 , s ∈ AR 1 , [ s , X ] = { Y ∈ R 1 : ( ∃ n ∈ ω )( s = r n ( X )) and Y ⊆ X } . Theorem(Dobrinen, Todotcevic) The set R 1 with the topology generated by basic sets [ s , X ] is a topological Ramsey space. Remark: ( ∀ X ∈ ED + fin )( ∃ Y ∈ R 1 )( Y ⊆ X ) . Sonia Navarro Flores Ramsey spaces and the Katetov order
Borel ideals Katˇ e tov order Ellentuck space Topological Ramsey spaces Hypercubes The space H 2 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 H 2 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 e tov order Ellentuck space Topological Ramsey spaces Definition For each k < ω and j ∈ 2, let A k , j be any linearly ordered set of size k + 1. Letting A k denote the product A k , 0 × A k , 1 . Let A = �� k , A k � : k < ω � . A is the maximal member of H 2 . We define B to be a member of H 2 if and only if B is a subset of A with the same structure. We use B ( k ) to denote � n k , B k � , the k -th block of B . The n -th approximation of B is r n ( B ) = � B (0) , ..., B ( n − 1) � . Let AH 2 n = { r n ( B ) : B ∈ H 2 } , the collection of all n -th approximations to members of H 2 . Let AH 2 = � n <ω AH 2 n . Sonia Navarro Flores Ramsey spaces and the Katetov order
Borel ideals Katˇ e tov order Ellentuck space Topological Ramsey spaces Definition If X ∈ H 2 , s ∈ AH 2 , [ s , X ] = { Y ∈ H 2 : ( ∃ n ∈ ω )( s = r n ( X )) and Y ⊆ X } . Theorem(Trujillo) The set H 2 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 e tov order Ellentuck space Topological Ramsey spaces Definition Let I H 2 be the ideal such that A / ∈ I H 2 iff ( ∀ n ∈ ω )( ∃ s , t ∈ [ ω ] n )( s × t ⊂ A ) Note that I + H 2 is G δ and then I H 2 is F σ . Also note that if X , Y ∈ H 2 , I H 2 ↾ X ≃ K I H 2 ↾ Y . Then for every X ∈ H 2 , I H 2 ↾ X is a K -uniform ideal. Now we want to prove that ED fin and I H 2 are not Katˇ e tov equivalent. Sonia Navarro Flores Ramsey spaces and the Katetov order
Borel ideals Katˇ e tov order Ellentuck space Topological Ramsey spaces For every ideal I on ω. Let r I be the least natural number k such that ( ∀ m )( ∀ c : [ ω ] 2 → m )( ∃ X ∈ I + )( | c ”[ X ] 2 | = k ) ( ω → ( I + ) 2 r I ). Proposition If I , J are ideals on ω such that I ≤ K J , then r I ≤ r J . So if I and J are Katˇ e tov equivalent, r I = r J . Sonia Navarro Flores Ramsey spaces and the Katetov order
Borel ideals Katˇ e tov 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 | ≤ r J ) . v X = f [ Y ] ∈ I + and | c ”[ X ] 2 | = | ϕ ′′ [ Y ] 2 | ≤ r J . Therefore r I ≤ r J . Sonia Navarro Flores Ramsey spaces and the Katetov order
Borel ideals Katˇ e tov order Ellentuck space Topological Ramsey spaces Proposition(Lafflame) If I = ED fin , r I = 3 . Proposition(Dobrinen,N) If I = I H 2 , r I = 5 . Sonia Navarro Flores Ramsey spaces and the Katetov order
Borel ideals Katˇ e tov order Ellentuck space Topological Ramsey spaces Proposition Fix X ∈ H 2 , then I H 2 ↾ X is a F σ and K -uniform ideal which is not Katˇ e tov equivalent to ED fin . Sonia Navarro Flores Ramsey spaces and the Katetov order
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