On equivalence relations generated by Cauchy sequences in countable metric spaces CTFM 2019, Wuhan University of Technology Longyun Ding School of Mathematical Sciences Nankai University 23 March 2019
Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation Outline 1 Borel reduction 2 Classifying Polish metric spaces 3 Cauchy sequence equivalence relation L. Ding On equivalence relations generated by Cauchy sequences
Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation Borel sets and Borel functions Definition Polish space : a separable, completely metrizable topological space. Let X, Y be two Polish spaces. Definition B ( X ) : Borel sets of X is the σ -algebra generated by the open sets of X . Definition A function f : X → Y is Borel function if f − 1 ( U ) is Borel for U open in Y . L. Ding On equivalence relations generated by Cauchy sequences
Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation Borel hierarchy Σ 0 Π 0 1 = open , 1 = closed ; Σ 0 Π 0 2 = F σ , 2 = G δ ; for 1 ≤ α < ω 1 , � Σ 0 A n : A n ∈ Π 0 α = { α n , α n < α } ; n ∈ ω Π 0 α = the complements of Σ 0 α sets; ∆ 0 α = Σ 0 α ∩ Π 0 α . L. Ding On equivalence relations generated by Cauchy sequences
Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation Borel reducibility between equivalence relations Let X, Y be Polish spaces and E, F equivalence relations on X, Y respectively. Definition E ≤ B F : There is a Borel function θ : X → Y such that, for all x, y ∈ X , xEy ⇐ ⇒ θ ( x ) Fθ ( y ) . E ∼ B F : E ≤ B F and F ≤ B E ; E < B F : E ≤ B F but not F ≤ B E . L. Ding On equivalence relations generated by Cauchy sequences
Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation Σ 1 1 sets and Π 1 1 sets Definition Let X be a Polish space. A subset A ⊆ X is analytic (or Σ 1 1 ) if there is a Polish space Y and a closed subset C ⊆ X × Y such that x ∈ A ⇐ ⇒ ∃ y ∈ Y (( x, y ) ∈ C ) . A subset A ⊆ X is co-analytic (or Π 1 1 ) if X \ A is Σ 1 1 . Theorem (Suslin) Let A ⊆ X . Then A is Borel iff it is both Σ 1 1 and Π 1 1 . L. Ding On equivalence relations generated by Cauchy sequences
Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation Σ 1 1 sets and Π 1 1 sets Definition Let X be a Polish space. A subset A ⊆ X is analytic (or Σ 1 1 ) if there is a Polish space Y and a closed subset C ⊆ X × Y such that x ∈ A ⇐ ⇒ ∃ y ∈ Y (( x, y ) ∈ C ) . A subset A ⊆ X is co-analytic (or Π 1 1 ) if X \ A is Σ 1 1 . Theorem (Suslin) Let A ⊆ X . Then A is Borel iff it is both Σ 1 1 and Π 1 1 . L. Ding On equivalence relations generated by Cauchy sequences
Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation 1st dichotomy theorem We say an equivalence relation E on X is Borel, Σ 1 1 , or Π 1 1 if { ( x, y ) ∈ X 2 : xEy } is so in X 2 . Theorem (Silver, 1980) Let E be a Π 1 1 equivalence relation. Then E ≤ B id( ω ) or id( R ) ≤ B E. L. Ding On equivalence relations generated by Cauchy sequences
Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation • id( R ) • id( ω ) . . . • id(2) • id(1) L. Ding On equivalence relations generated by Cauchy sequences
Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation 2nd dichotomy theorem Definition E 0 is the equivalence relation on { 0 , 1 } ω defined by xE 0 y ⇐ ⇒ ∃ m ∀ n ≥ m ( x ( n ) = y ( n )) . Fact: E 0 ∼ B R / Q . Theorem (Harrington-Kechris-Louveau, 1990) Let E be a Borel equivalence relation. Then either E ≤ B id( R ) or E 0 ≤ B E . L. Ding On equivalence relations generated by Cauchy sequences
Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation 2nd dichotomy theorem Definition E 0 is the equivalence relation on { 0 , 1 } ω defined by xE 0 y ⇐ ⇒ ∃ m ∀ n ≥ m ( x ( n ) = y ( n )) . Fact: E 0 ∼ B R / Q . Theorem (Harrington-Kechris-Louveau, 1990) Let E be a Borel equivalence relation. Then either E ≤ B id( R ) or E 0 ≤ B E . L. Ding On equivalence relations generated by Cauchy sequences
Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation • E 0 • id( R ) • id( ω ) . . . • id(2) • id(1) L. Ding On equivalence relations generated by Cauchy sequences
Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation 3rd dichotomy theorem Definition E 1 is the equivalence relation on R ω defined by xE 1 y ⇐ ⇒ ∃ m ∀ n ≥ m ( x ( n ) = y ( n )) . Fact: E 1 = R ω /c 00 , where c 00 = � n R n . Theorem (Kechris-Louveau, 1997) If E ≤ B E 1 , then E ≤ B E 0 or E ∼ B E 1 . L. Ding On equivalence relations generated by Cauchy sequences
Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation 3rd dichotomy theorem Definition E 1 is the equivalence relation on R ω defined by xE 1 y ⇐ ⇒ ∃ m ∀ n ≥ m ( x ( n ) = y ( n )) . Fact: E 1 = R ω /c 00 , where c 00 = � n R n . Theorem (Kechris-Louveau, 1997) If E ≤ B E 1 , then E ≤ B E 0 or E ∼ B E 1 . L. Ding On equivalence relations generated by Cauchy sequences
Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation ❵ E 1 • ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ • E 0 • id( R ) • id( ω ) . . . • id(2) • id(1) L. Ding On equivalence relations generated by Cauchy sequences
Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation 4th dichotomy theorem Definition Let E be an equivalence relation on X . The equivalence relation E ω on X ω defined by xE ω y ⇐ ⇒ ∀ n ( x ( n ) Ey ( n )) . Fact: E ω 0 ∼ B R ω / Q ω . Theorem (Hjorth-Kechris, 1997) If E ≤ B E ω 0 , then E ≤ B E 0 or E ∼ B E ω 0 . L. Ding On equivalence relations generated by Cauchy sequences
Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation 4th dichotomy theorem Definition Let E be an equivalence relation on X . The equivalence relation E ω on X ω defined by xE ω y ⇐ ⇒ ∀ n ( x ( n ) Ey ( n )) . Fact: E ω 0 ∼ B R ω / Q ω . Theorem (Hjorth-Kechris, 1997) If E ≤ B E ω 0 , then E ≤ B E 0 or E ∼ B E ω 0 . L. Ding On equivalence relations generated by Cauchy sequences
Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation ❵ ✥✥✥✥✥✥✥✥✥✥ • E ω E 1 • ❵ ❵ ❵ ❵ 0 ❵ ❵ ❵ ❵ ❵ • E 0 • id( R ) • id( ω ) . . . • id(2) • id(1) L. Ding On equivalence relations generated by Cauchy sequences
Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation Sequence equivalence relations Definition Let G be a Borel subgroup of R ω , then the Borel equivalence relation R ω /G is defined by x is equivalent to y ⇐ ⇒ x − y ∈ G. Fact: E 1 = R ω /c 00 = R ω / R <ω , E ω 0 ∼ B R ω / Q ω . Denote c 0 = { x ∈ R ω : lim n →∞ | x ( n ) | = 0 } ; ℓ p = { x ∈ R ω : | x ( n ) | p < + ∞} ; � n ℓ ∞ = { x ∈ R ω : sup n | x ( n ) | < + ∞} . L. Ding On equivalence relations generated by Cauchy sequences
Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation Sequence equivalence relations Definition Let G be a Borel subgroup of R ω , then the Borel equivalence relation R ω /G is defined by x is equivalent to y ⇐ ⇒ x − y ∈ G. Fact: E 1 = R ω /c 00 = R ω / R <ω , E ω 0 ∼ B R ω / Q ω . Denote c 0 = { x ∈ R ω : lim n →∞ | x ( n ) | = 0 } ; ℓ p = { x ∈ R ω : | x ( n ) | p < + ∞} ; � n ℓ ∞ = { x ∈ R ω : sup n | x ( n ) | < + ∞} . L. Ding On equivalence relations generated by Cauchy sequences
Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation Below ℓ ∞ Theorem (Dougherty-Hjorth, 1999) ⇒ R ω /ℓ p ≤ B R ω /ℓ q . For p, q ∈ [1 + ∞ ) , p ≤ q ⇐ Theorem (D. 2012) For p ∈ (0 , 1] , we have R ω /ℓ p ∼ B R ω /ℓ 1 . Theorem (Rosendal, 2005) Every K σ equivalence relation on a Polish space is ≤ B R ω /ℓ ∞ . Corollary E 1 and R ω /ℓ p are ≤ B R ω /ℓ ∞ . L. Ding On equivalence relations generated by Cauchy sequences
Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation Below ℓ ∞ Theorem (Dougherty-Hjorth, 1999) ⇒ R ω /ℓ p ≤ B R ω /ℓ q . For p, q ∈ [1 + ∞ ) , p ≤ q ⇐ Theorem (D. 2012) For p ∈ (0 , 1] , we have R ω /ℓ p ∼ B R ω /ℓ 1 . Theorem (Rosendal, 2005) Every K σ equivalence relation on a Polish space is ≤ B R ω /ℓ ∞ . Corollary E 1 and R ω /ℓ p are ≤ B R ω /ℓ ∞ . L. Ding On equivalence relations generated by Cauchy sequences
Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation Below ℓ ∞ Theorem (Dougherty-Hjorth, 1999) ⇒ R ω /ℓ p ≤ B R ω /ℓ q . For p, q ∈ [1 + ∞ ) , p ≤ q ⇐ Theorem (D. 2012) For p ∈ (0 , 1] , we have R ω /ℓ p ∼ B R ω /ℓ 1 . Theorem (Rosendal, 2005) Every K σ equivalence relation on a Polish space is ≤ B R ω /ℓ ∞ . Corollary E 1 and R ω /ℓ p are ≤ B R ω /ℓ ∞ . L. Ding On equivalence relations generated by Cauchy sequences
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