On equivalence relations generated by Cauchy sequences in countable - - PowerPoint PPT Presentation

▶
on equivalence relations generated by cauchy sequences in
SMART_READER_LITE
LIVE PREVIEW

On equivalence relations generated by Cauchy sequences in countable - - PowerPoint PPT Presentation

Oct 11, 2023 106 likes •794 views

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

slide-1
SLIDE 1

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

slide-2
SLIDE 2

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

slide-3
SLIDE 3

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

  • f X.

Definition A function f : X → Y is Borel function if f−1(U) is Borel for U

  • pen in Y .
  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-4
SLIDE 4

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Borel hierarchy

Σ0

1 = open,

Π0

1 = closed;

Σ0

2 = Fσ,

Π0

2 = Gδ;

for 1 ≤ α < ω1, Σ0

α = {

  • n∈ω

An : An ∈ Π0

αn, αn < α};

Π0

α = the complements of Σ0 α sets;

∆0

α = Σ0 α ∩ Π0 α.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-5
SLIDE 5

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

slide-6
SLIDE 6

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Σ1

1 sets and Π1 1 sets

Definition Let X be a Polish space. A subset A ⊆ X is analytic (or Σ1

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 Π1

1) if X \ A is Σ1 1.

Theorem (Suslin) Let A ⊆ X. Then A is Borel iff it is both Σ1

1 and Π1 1.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-7
SLIDE 7

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Σ1

1 sets and Π1 1 sets

Definition Let X be a Polish space. A subset A ⊆ X is analytic (or Σ1

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 Π1

1) if X \ A is Σ1 1.

Theorem (Suslin) Let A ⊆ X. Then A is Borel iff it is both Σ1

1 and Π1 1.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-8
SLIDE 8

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

1st dichotomy theorem

We say an equivalence relation E on X is Borel, Σ1

1, or Π1 1 if

{(x, y) ∈ X2 : xEy} is so in X2. Theorem (Silver, 1980) Let E be a Π1

1 equivalence relation. Then

E ≤B id(ω) or id(R) ≤B E.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-9
SLIDE 9

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

  • id(1)
  • id(2)

. . .

  • id(ω)
  • id(R)
  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-10
SLIDE 10

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

2nd dichotomy theorem

Definition E0 is the equivalence relation on {0, 1}ω defined by xE0y ⇐ ⇒ ∃m∀n ≥ m(x(n) = y(n)). Fact: E0 ∼B R/Q. Theorem (Harrington-Kechris-Louveau, 1990) Let E be a Borel equivalence relation. Then either E ≤B id(R) or E0 ≤B E.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-11
SLIDE 11

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

2nd dichotomy theorem

Definition E0 is the equivalence relation on {0, 1}ω defined by xE0y ⇐ ⇒ ∃m∀n ≥ m(x(n) = y(n)). Fact: E0 ∼B R/Q. Theorem (Harrington-Kechris-Louveau, 1990) Let E be a Borel equivalence relation. Then either E ≤B id(R) or E0 ≤B E.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-12
SLIDE 12

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

  • id(1)
  • id(2)

. . .

  • id(ω)
  • id(R)
  • E0
  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-13
SLIDE 13

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

3rd dichotomy theorem

Definition E1 is the equivalence relation on Rω defined by xE1y ⇐ ⇒ ∃m∀n ≥ m(x(n) = y(n)). Fact: E1 = Rω/c00, where c00 =

n Rn.

Theorem (Kechris-Louveau, 1997) If E ≤B E1, then E ≤B E0 or E ∼B E1.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-14
SLIDE 14

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

3rd dichotomy theorem

Definition E1 is the equivalence relation on Rω defined by xE1y ⇐ ⇒ ∃m∀n ≥ m(x(n) = y(n)). Fact: E1 = Rω/c00, where c00 =

n Rn.

Theorem (Kechris-Louveau, 1997) If E ≤B E1, then E ≤B E0 or E ∼B E1.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-15
SLIDE 15

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

  • id(1)
  • id(2)

. . .

  • id(ω)
  • id(R)
  • E0

❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵

E1 •

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-16
SLIDE 16

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 Eω on Xω defined by xEωy ⇐ ⇒ ∀n(x(n)Ey(n)). Fact: Eω

0 ∼B Rω/Qω.

Theorem (Hjorth-Kechris, 1997) If E ≤B Eω

0 , then E ≤B E0 or E ∼B Eω 0 .

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-17
SLIDE 17

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 Eω on Xω defined by xEωy ⇐ ⇒ ∀n(x(n)Ey(n)). Fact: Eω

0 ∼B Rω/Qω.

Theorem (Hjorth-Kechris, 1997) If E ≤B Eω

0 , then E ≤B E0 or E ∼B Eω 0 .

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-18
SLIDE 18

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

  • id(1)
  • id(2)

. . .

  • id(ω)
  • id(R)
  • E0

❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵

E1 •

✥✥✥✥✥✥✥✥✥✥

  • Eω
  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-19
SLIDE 19

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Sequence equivalence relations

Definition Let G be a Borel subgroup of Rω, then the Borel equivalence relation Rω/G is defined by x is equivalent to y ⇐ ⇒ x − y ∈ G. Fact: E1 = Rω/c00 = Rω/R<ω, Eω

0 ∼B Rω/Qω.

Denote c0 = {x ∈ Rω : lim

n→∞ |x(n)| = 0};

ℓp = {x ∈ Rω :

  • n

|x(n)|p < +∞}; ℓ∞ = {x ∈ Rω : sup

n |x(n)| < +∞}.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-20
SLIDE 20

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Sequence equivalence relations

Definition Let G be a Borel subgroup of Rω, then the Borel equivalence relation Rω/G is defined by x is equivalent to y ⇐ ⇒ x − y ∈ G. Fact: E1 = Rω/c00 = Rω/R<ω, Eω

0 ∼B Rω/Qω.

Denote c0 = {x ∈ Rω : lim

n→∞ |x(n)| = 0};

ℓp = {x ∈ Rω :

  • n

|x(n)|p < +∞}; ℓ∞ = {x ∈ Rω : sup

n |x(n)| < +∞}.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-21
SLIDE 21

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Below ℓ∞

Theorem (Dougherty-Hjorth, 1999) For p, q ∈ [1 + ∞), p ≤ q ⇐ ⇒ Rω/ℓp ≤B Rω/ℓq. Theorem (D. 2012) For p ∈ (0, 1], we have Rω/ℓp ∼B Rω/ℓ1. Theorem (Rosendal, 2005) Every Kσ equivalence relation on a Polish space is ≤B Rω/ℓ∞. Corollary E1 and Rω/ℓp are ≤B Rω/ℓ∞.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-22
SLIDE 22

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Below ℓ∞

Theorem (Dougherty-Hjorth, 1999) For p, q ∈ [1 + ∞), p ≤ q ⇐ ⇒ Rω/ℓp ≤B Rω/ℓq. Theorem (D. 2012) For p ∈ (0, 1], we have Rω/ℓp ∼B Rω/ℓ1. Theorem (Rosendal, 2005) Every Kσ equivalence relation on a Polish space is ≤B Rω/ℓ∞. Corollary E1 and Rω/ℓp are ≤B Rω/ℓ∞.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-23
SLIDE 23

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Below ℓ∞

Theorem (Dougherty-Hjorth, 1999) For p, q ∈ [1 + ∞), p ≤ q ⇐ ⇒ Rω/ℓp ≤B Rω/ℓq. Theorem (D. 2012) For p ∈ (0, 1], we have Rω/ℓp ∼B Rω/ℓ1. Theorem (Rosendal, 2005) Every Kσ equivalence relation on a Polish space is ≤B Rω/ℓ∞. Corollary E1 and Rω/ℓp are ≤B Rω/ℓ∞.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-24
SLIDE 24

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

  • id(1)
  • id(2)

. . .

  • id(ω)
  • id(R)
  • E0

❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵

E1 •

✥✥✥✥✥✥✥✥✥✥

  • Eω

❍ ❍ ❍ ❍

  • ℓ1
  • ℓp
  • ℓ∞
  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-25
SLIDE 25

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Rω/c0

Theorem (Hjorth, 2000) For p ∈ [1, +∞), Rω/ℓp and Rω/c0 are ≤B incomparable. Fact Eω

0 ≤B Rω/c0.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-26
SLIDE 26

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Rω/c0

Theorem (Hjorth, 2000) For p ∈ [1, +∞), Rω/ℓp and Rω/c0 are ≤B incomparable. Fact Eω

0 ≤B Rω/c0.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-27
SLIDE 27

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

  • id(1)
  • id(2)

. . .

  • id(ω)
  • id(R)
  • E0

❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵

E1 •

✥✥✥✥✥✥✥✥✥✥

  • Eω

❍ ❍ ❍ ❍

  • ℓ1
  • ℓp
  • ℓ∞
  • c0
  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-28
SLIDE 28

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

=+

Definition =+ is the equivalence relation on Rω defined by x =+ y ⇐ ⇒ {x(n) : n ∈ ω} = {y(n) : n ∈ ω}. Fact Eω

0 ≤B=+, while =+ and Rω/c0 are Borel incomparable.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-29
SLIDE 29

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

=+

Definition =+ is the equivalence relation on Rω defined by x =+ y ⇐ ⇒ {x(n) : n ∈ ω} = {y(n) : n ∈ ω}. Fact Eω

0 ≤B=+, while =+ and Rω/c0 are Borel incomparable.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-30
SLIDE 30

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

  • id(1)
  • id(2)

. . .

  • id(ω)
  • id(R)
  • E0

❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵

E1 •

✥✥✥✥✥✥✥✥✥✥

  • Eω

❍ ❍ ❍ ❍

  • ℓ1
  • ℓp
  • ℓ∞
  • c0

P P P P P P

=+ •

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-31
SLIDE 31

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Polish G-spaces and orbit equivalence relations

Definition Polish group: A topological group whose underlying space is Polish. G: Polish group, X: Polish space, a : G × X → X: continuous G-action on X. Definition Orbit equivalence relation: xEX

G y ⇐

⇒ ∃g ∈ G(g · x = y). Any EX

G is Σ1 1 equivalence relation.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-32
SLIDE 32

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Polish G-spaces and orbit equivalence relations

Definition Polish group: A topological group whose underlying space is Polish. G: Polish group, X: Polish space, a : G × X → X: continuous G-action on X. Definition Orbit equivalence relation: xEX

G y ⇐

⇒ ∃g ∈ G(g · x = y). Any EX

G is Σ1 1 equivalence relation.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-33
SLIDE 33

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Polish G-spaces and orbit equivalence relations

Definition Polish group: A topological group whose underlying space is Polish. G: Polish group, X: Polish space, a : G × X → X: continuous G-action on X. Definition Orbit equivalence relation: xEX

G y ⇐

⇒ ∃g ∈ G(g · x = y). Any EX

G is Σ1 1 equivalence relation.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-34
SLIDE 34

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Σ1

1 equivalence relations

Theorem (Kechris-Louveau, 1997) E1 ≤B EX

G for any Polish G-space X.

E0, E1, Rω/ℓp, Rω/ℓ∞: Fσ equivalence relations; Eω

0 , Rω/c0, =+: Π0 3 equivalence relations.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-35
SLIDE 35

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Σ1

1 equivalence relations

Theorem (Kechris-Louveau, 1997) E1 ≤B EX

G for any Polish G-space X.

E0, E1, Rω/ℓp, Rω/ℓ∞: Fσ equivalence relations; Eω

0 , Rω/c0, =+: Π0 3 equivalence relations.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-36
SLIDE 36

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

  • id(1)
  • id(2)

. . .

  • id(ω)
  • id(R)
  • E0

❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵

E1 •

✥✥✥✥✥✥✥✥✥✥

  • Eω

❍ ❍ ❍ ❍

  • ℓ1
  • ℓp
  • ℓ∞
  • c0

P P P P P P

=+ •

P P P P P P

  • EX

G

✦✦✦✦✦✦✦✦

  • Σ1

1

❅ ❅ ❅ ❅

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-37
SLIDE 37

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

slide-38
SLIDE 38

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Classification problems for Polish/compact metric spaces

Definition Polish metric space: separable complete metric space.

1 Iso/Isocpt: isometry among Polish/compact metric spaces 2 Hom/Homcpt: homeomorphism ... 3 Lip/Lipcpt: Lipschitz isomorphism ... 4 Uni/Unicpt: Uniform homeomorphism ...

Note: Unicpt = Homcpt.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-39
SLIDE 39

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Classification problems for Polish/compact metric spaces

Definition Polish metric space: separable complete metric space.

1 Iso/Isocpt: isometry among Polish/compact metric spaces 2 Hom/Homcpt: homeomorphism ... 3 Lip/Lipcpt: Lipschitz isomorphism ... 4 Uni/Unicpt: Uniform homeomorphism ...

Note: Unicpt = Homcpt.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-40
SLIDE 40

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Classification problems for Polish/compact metric spaces

Definition Polish metric space: separable complete metric space.

1 Iso/Isocpt: isometry among Polish/compact metric spaces 2 Hom/Homcpt: homeomorphism ... 3 Lip/Lipcpt: Lipschitz isomorphism ... 4 Uni/Unicpt: Uniform homeomorphism ...

Note: Unicpt = Homcpt.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-41
SLIDE 41

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Classification problems for Polish/compact metric spaces

Definition Polish metric space: separable complete metric space.

1 Iso/Isocpt: isometry among Polish/compact metric spaces 2 Hom/Homcpt: homeomorphism ... 3 Lip/Lipcpt: Lipschitz isomorphism ... 4 Uni/Unicpt: Uniform homeomorphism ...

Note: Unicpt = Homcpt.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-42
SLIDE 42

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Classification problems for Polish/compact metric spaces

Definition Polish metric space: separable complete metric space.

1 Iso/Isocpt: isometry among Polish/compact metric spaces 2 Hom/Homcpt: homeomorphism ... 3 Lip/Lipcpt: Lipschitz isomorphism ... 4 Uni/Unicpt: Uniform homeomorphism ...

Note: Unicpt = Homcpt.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-43
SLIDE 43

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Classification problems for Polish/compact metric spaces

Definition Polish metric space: separable complete metric space.

1 Iso/Isocpt: isometry among Polish/compact metric spaces 2 Hom/Homcpt: homeomorphism ... 3 Lip/Lipcpt: Lipschitz isomorphism ... 4 Uni/Unicpt: Uniform homeomorphism ...

Note: Unicpt = Homcpt.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-44
SLIDE 44

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Coding Polish metric spaces

Definition Let X ⊆ Rω×ω consisting of elements r = (ri,j) such that (1) ∀i, j ∈ ω (ri,j ≥ 0 ∧ (ri,j = 0 ⇐ ⇒ i = j)); (2) ∀i, j ∈ ω (ri,j = rj,i); (3) ∀i, j, k ∈ ω (ri,j ≤ ri,k + rj,k). X is a Polish subspace of Rω×ω. Denote Xr the completion of (ω, r). Definition Xcpt = {r ∈ X : Xr is compact}.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-45
SLIDE 45

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Coding Polish metric spaces

Definition Let X ⊆ Rω×ω consisting of elements r = (ri,j) such that (1) ∀i, j ∈ ω (ri,j ≥ 0 ∧ (ri,j = 0 ⇐ ⇒ i = j)); (2) ∀i, j ∈ ω (ri,j = rj,i); (3) ∀i, j, k ∈ ω (ri,j ≤ ri,k + rj,k). X is a Polish subspace of Rω×ω. Denote Xr the completion of (ω, r). Definition Xcpt = {r ∈ X : Xr is compact}.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-46
SLIDE 46

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Coding Polish metric spaces

Definition Let X ⊆ Rω×ω consisting of elements r = (ri,j) such that (1) ∀i, j ∈ ω (ri,j ≥ 0 ∧ (ri,j = 0 ⇐ ⇒ i = j)); (2) ∀i, j ∈ ω (ri,j = rj,i); (3) ∀i, j, k ∈ ω (ri,j ≤ ri,k + rj,k). X is a Polish subspace of Rω×ω. Denote Xr the completion of (ω, r). Definition Xcpt = {r ∈ X : Xr is compact}.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-47
SLIDE 47

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Isometry and Homeomorpism

Theorem (Gromov) Isocpt ∼B id(R). Theorem (Gao-Kechris, 2003) Iso is a universal orbit equivalence relation. Theorem (Zielinski, 2016) Iso ∼B Homcpt. Fact Hom is an Σ1

2 equivalence relation on X.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-48
SLIDE 48

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Isometry and Homeomorpism

Theorem (Gromov) Isocpt ∼B id(R). Theorem (Gao-Kechris, 2003) Iso is a universal orbit equivalence relation. Theorem (Zielinski, 2016) Iso ∼B Homcpt. Fact Hom is an Σ1

2 equivalence relation on X.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-49
SLIDE 49

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Isometry and Homeomorpism

Theorem (Gromov) Isocpt ∼B id(R). Theorem (Gao-Kechris, 2003) Iso is a universal orbit equivalence relation. Theorem (Zielinski, 2016) Iso ∼B Homcpt. Fact Hom is an Σ1

2 equivalence relation on X.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-50
SLIDE 50

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

  • id(1)
  • id(2)

. . .

  • id(ω)
  • id(R)∼B Isocpt
  • E0

❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵

E1 •

✥✥✥✥✥✥✥✥✥✥

  • Eω

❍ ❍ ❍ ❍

  • ℓ1
  • ℓp
  • ℓ∞
  • c0

P P P P P P

=+ •

P P P P P P

  • EX

G ∼B Iso ∼B Homcpt

✦✦✦✦✦✦✦✦

  • Σ1

1

❅ ❅ ❅ ❅

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-51
SLIDE 51

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Lipschitz isomorphism and uniform homeomorphsim

Theorem (Rosendal, 2005) Lipcpt ∼B Rω/ℓ∞. Theorem (Ferenczi-Louveau-Rosendal, 2009) Lip ∼B Uni are uinversal Σ1

1 equivalence relations.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-52
SLIDE 52

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Lipschitz isomorphism and uniform homeomorphsim

Theorem (Rosendal, 2005) Lipcpt ∼B Rω/ℓ∞. Theorem (Ferenczi-Louveau-Rosendal, 2009) Lip ∼B Uni are uinversal Σ1

1 equivalence relations.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-53
SLIDE 53

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

  • id(1)
  • id(2)

. . .

  • id(ω)
  • id(R)∼B Isocpt
  • E0

❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵

E1 •

✥✥✥✥✥✥✥✥✥✥

  • Eω

❍ ❍ ❍ ❍

  • ℓ1
  • ℓp
  • ℓ∞∼B Lipcpt
  • c0

P P P P P P

=+ •

P P P P P P

  • EX

G ∼B Iso ∼B Unicpt

✦✦✦✦✦✦✦✦

  • Σ1

1∼B Lip ∼B Uni

❅ ❅ ❅ ❅

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-54
SLIDE 54

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

slide-55
SLIDE 55

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Cauchy sequence equivalence relation

Fact Let r, s ∈ X. Then the following are equivalent: (a) (ω, r) and (ω, s) have the same set of Cauchy sequences; (b) there exists a homeomorphism ϕ : Xr → Xs with ϕ ↾ ω = id(ω). Definition Cauchy sequence equivalence relation: For r, s ∈ X, rEcss iff (ω, r) and (ω, s) have the same set of Cauchy sequences. Theorem (D.-Gu, 2018) Ecs is a Π1

1-complete equivalence relation. So Ecs and Lip (or

Uni) are Borel incomparable.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-56
SLIDE 56

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Cauchy sequence equivalence relation

Fact Let r, s ∈ X. Then the following are equivalent: (a) (ω, r) and (ω, s) have the same set of Cauchy sequences; (b) there exists a homeomorphism ϕ : Xr → Xs with ϕ ↾ ω = id(ω). Definition Cauchy sequence equivalence relation: For r, s ∈ X, rEcss iff (ω, r) and (ω, s) have the same set of Cauchy sequences. Theorem (D.-Gu, 2018) Ecs is a Π1

1-complete equivalence relation. So Ecs and Lip (or

Uni) are Borel incomparable.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-57
SLIDE 57

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Cauchy sequence equivalence relation

Fact Let r, s ∈ X. Then the following are equivalent: (a) (ω, r) and (ω, s) have the same set of Cauchy sequences; (b) there exists a homeomorphism ϕ : Xr → Xs with ϕ ↾ ω = id(ω). Definition Cauchy sequence equivalence relation: For r, s ∈ X, rEcss iff (ω, r) and (ω, s) have the same set of Cauchy sequences. Theorem (D.-Gu, 2018) Ecs is a Π1

1-complete equivalence relation. So Ecs and Lip (or

Uni) are Borel incomparable.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-58
SLIDE 58

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Restriction on compact metric spaces

Denote Ecsc = Ecs ↾ Xcpt. Theorem (D.-Gu, 2018)

1 Ecsc is Π0

3-equivalence relation;

2 Ecsc ∼ EX

G for some Polish group G and Polish G-space X;

3 Rω/c0 ≤B Ecsc; 4 =+≤B Ecsc.

Question: Does Rω/ℓ1 ≤B Ecsc?

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-59
SLIDE 59

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Restriction on compact metric spaces

Denote Ecsc = Ecs ↾ Xcpt. Theorem (D.-Gu, 2018)

1 Ecsc is Π0

3-equivalence relation;

2 Ecsc ∼ EX

G for some Polish group G and Polish G-space X;

3 Rω/c0 ≤B Ecsc; 4 =+≤B Ecsc.

Question: Does Rω/ℓ1 ≤B Ecsc?

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-60
SLIDE 60

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Restriction on compact metric spaces

Denote Ecsc = Ecs ↾ Xcpt. Theorem (D.-Gu, 2018)

1 Ecsc is Π0

3-equivalence relation;

2 Ecsc ∼ EX

G for some Polish group G and Polish G-space X;

3 Rω/c0 ≤B Ecsc; 4 =+≤B Ecsc.

Question: Does Rω/ℓ1 ≤B Ecsc?

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-61
SLIDE 61

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Restriction on compact metric spaces

Denote Ecsc = Ecs ↾ Xcpt. Theorem (D.-Gu, 2018)

1 Ecsc is Π0

3-equivalence relation;

2 Ecsc ∼ EX

G for some Polish group G and Polish G-space X;

3 Rω/c0 ≤B Ecsc; 4 =+≤B Ecsc.

Question: Does Rω/ℓ1 ≤B Ecsc?

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-62
SLIDE 62

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Restriction on compact metric spaces

Denote Ecsc = Ecs ↾ Xcpt. Theorem (D.-Gu, 2018)

1 Ecsc is Π0

3-equivalence relation;

2 Ecsc ∼ EX

G for some Polish group G and Polish G-space X;

3 Rω/c0 ≤B Ecsc; 4 =+≤B Ecsc.

Question: Does Rω/ℓ1 ≤B Ecsc?

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-63
SLIDE 63

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Restriction on compact metric spaces

Denote Ecsc = Ecs ↾ Xcpt. Theorem (D.-Gu, 2018)

1 Ecsc is Π0

3-equivalence relation;

2 Ecsc ∼ EX

G for some Polish group G and Polish G-space X;

3 Rω/c0 ≤B Ecsc; 4 =+≤B Ecsc.

Question: Does Rω/ℓ1 ≤B Ecsc?

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-64
SLIDE 64

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

  • id(1)
  • id(2)

. . .

  • id(ω)
  • id(R)
  • E0

❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵

E1 •

✥✥✥✥✥✥✥✥✥✥

  • Eω

❍ ❍ ❍ ❍

  • ℓ1
  • ℓp
  • ℓ∞
  • c0

P P P P P P

=+ •

P P P P P P

  • Iso ∼B Homcpt = Unicpt

✦✦✦✦✦✦✦✦

  • Lip ∼B Uni

❅ ❅ ❅ ❅

  • Ecs

✁ ✁ ✁ ❵ ❵ ❵ ❵ ❵ ❅ ❅

  • Ecsc
  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-65
SLIDE 65

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

Some invariant subsets of Ecsc

Xn = {r ∈ Xcpt : card(X

′ r) = n},

Xω = {r ∈ Xcpt : card(X

′′ r) = 1}.

Fact r ∈ Xn ⇐ ⇒ Xr ∼ = ω · n + 1, r ∈ Xω ⇐ ⇒ Xr ∼ = ω2 + 1. Y = {r ∈ Xω : Xr = ω}.

  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-66
SLIDE 66

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

  • id(1)
  • id(2)

. . .

  • id(ω) ∼B Ecsc ↾ X1
  • id(R)
  • E0 ∼B Ecsc ↾ Xn (n ≥ 2)

❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵

E1 •

✥✥✥✥✥✥✥✥✥✥

  • Eω

0 ∼B Ecsc ↾ Y

❍ ❍ ❍ ❍

  • ℓ1
  • ℓp
  • ℓ∞
  • c0

P P P P P P

=+ • ∼B Ecsc ↾ Xω

P P P P P P

  • Iso

✦✦✦✦✦✦✦✦

  • Lip

❅ ❅ ❅ ❅

  • Ecs

✁ ✁ ✁ ❵ ❵ ❵ ❵ ❵ ❅ ❅

  • Ecsc
  • L. Ding

On equivalence relations generated by Cauchy sequences

slide-67
SLIDE 67

Borel reduction Classifying Polish metric spaces Cauchy sequence equivalence relation

The end

Thank you!

  • L. Ding

On equivalence relations generated by Cauchy sequences