A survey on the Borelness of the intersectionoperation

2013 Workshop on Metamathematics and Metaphysics

Longyun Ding

School of Mathematical SciencesNankai University

June 15th, 2013School of Philosophy at Fudan University

Shanghai China

Introduction Intersection on F (X ) Intersection on Subs(X )

Outline

1 Introduction

2 Intersection on F (X )

3 Intersection on Subs(X )

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

The Effros Borel spaces

Let X be a Polish space, i.e., separable, completely metrizable topologicalspace.

Definition

Let F (X ) be the set of all closed subsets of X . The Effros Borelstructure on F (X ) is the σ-algebra generated by the sets

{F ∈ F (X ) : F ∩ U 6= ∅},

where U varies over open subsets of X .

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

The Effros Borel spaces

Let X be a Polish space, i.e., separable, completely metrizable topologicalspace.

Definition

Let F (X ) be the set of all closed subsets of X . The Effros Borelstructure on F (X ) is the σ-algebra generated by the sets

{F ∈ F (X ) : F ∩ U 6= ∅},

where U varies over open subsets of X .

Fact

F (X ) is a standard Borel space, i.e., there is a Polish topology τ onF (X ) which induce the Effros Borel structure.

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Facts of F (X )

Theorem (Kuratowski–Ryll-Nardzewski, The Selection Theorem)

Let X be Polish. There is a sequence of Borel functions sn : F (X ) → X,such that for nonempty F ∈ F (X ), {sn(F )} is dense in F .

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Facts of F (X )

Theorem (Kuratowski–Ryll-Nardzewski, The Selection Theorem)

Let X be Polish. There is a sequence of Borel functions sn : F (X ) → X,such that for nonempty F ∈ F (X ), {sn(F )} is dense in F .

Fact

1 The set of compact subsets of X , K (X ), is a Borel set in F (X );

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Facts of F (X )

Theorem (Kuratowski–Ryll-Nardzewski, The Selection Theorem)

Let X be Polish. There is a sequence of Borel functions sn : F (X ) → X,such that for nonempty F ∈ F (X ), {sn(F )} is dense in F .

Fact

1 The set of compact subsets of X , K (X ), is a Borel set in F (X );

2 The relation “x ∈ F” is Borel in X × F (X ), and “F1 ⊆ F2” is Borelin F (X )2;

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Facts of F (X )

Theorem (Kuratowski–Ryll-Nardzewski, The Selection Theorem)

Let X be Polish. There is a sequence of Borel functions sn : F (X ) → X,such that for nonempty F ∈ F (X ), {sn(F )} is dense in F .

Fact

1 The set of compact subsets of X , K (X ), is a Borel set in F (X );

2 The relation “x ∈ F” is Borel in X × F (X ), and “F1 ⊆ F2” is Borelin F (X )2;

3 The function (F1, F2) 7→ F1 ∪ F2 is Borel from F (X )2 to F (X );

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Facts of F (X )

Theorem (Kuratowski–Ryll-Nardzewski, The Selection Theorem)

Let X be Polish. There is a sequence of Borel functions sn : F (X ) → X,such that for nonempty F ∈ F (X ), {sn(F )} is dense in F .

Fact

1 The set of compact subsets of X , K (X ), is a Borel set in F (X );

2 The relation “x ∈ F” is Borel in X × F (X ), and “F1 ⊆ F2” is Borelin F (X )2;

3 The function (F1, F2) 7→ F1 ∪ F2 is Borel from F (X )2 to F (X );

4 If f : X → Y is continuous, then the map F 7→ f (F ) is Borel fromF (X ) to F (Y ).

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Facts of F (X )

Theorem (Kuratowski–Ryll-Nardzewski, The Selection Theorem)

Let X be Polish. There is a sequence of Borel functions sn : F (X ) → X,such that for nonempty F ∈ F (X ), {sn(F )} is dense in F .

Fact

1 The set of compact subsets of X , K (X ), is a Borel set in F (X );

2 The relation “x ∈ F” is Borel in X × F (X ), and “F1 ⊆ F2” is Borelin F (X )2;

3 The function (F1, F2) 7→ F1 ∪ F2 is Borel from F (X )2 to F (X );

4 If f : X → Y is continuous, then the map F 7→ f (F ) is Borel fromF (X ) to F (Y ).

The function (F1, F2) 7→ F1 ∩ F2 is NOT necessarily Borel.

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Borel G -spaces

Definition

Let G be a Polish group and X a Polish space. An action of G on X is afunction

· : G × X → X ,

g · (h · x) = (gh) · x and 1G · x = x .

If · is a Borel function, we say X is a Borel G-space.

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Borel G -spaces

Definition

Let G be a Polish group and X a Polish space. An action of G on X is afunction

· : G × X → X ,

g · (h · x) = (gh) · x and 1G · x = x .

If · is a Borel function, we say X is a Borel G-space.

We denote the orbit equivalence relation, EX

G, as

xEX

G y ⇐⇒ ∃g ∈ G(g · x = y).

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Becker-Kechris’s theorem

Theorem

The following are equivalent:

1 EXG

is Borel.

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Becker-Kechris’s theorem

Theorem

The following are equivalent:

1 EXG

is Borel.

2 The mapx 7→ Gx = {g ∈ G : g · x = x}

is Borel from X to F (G).

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Becker-Kechris’s theorem

Theorem

The following are equivalent:

1 EXG

is Borel.

2 The mapx 7→ Gx = {g ∈ G : g · x = x}

is Borel from X to F (G).

3 The map(x , u) 7→ Gx,u = {g ∈ G : g · x = u}

is Borel from X 2 to F (G).

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Closed subgroups of G

Fact

1 Denote SG(G) the set of all closed subgroup of G. Then SG(G) isBorel in F (G).

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Closed subgroups of G

Fact

1 Denote SG(G) the set of all closed subgroup of G. Then SG(G) isBorel in F (G).

2 Denote SX (G) = {Gx : x ∈ X}. Then SX (G) is Σ11 and closed

under conjugation, i.e., for g ∈ G and H ∈ SX (G) we havegHg−1 ∈ SX (G).

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Diagonal actions

Let G be a Polish group and X , Y be Borel G -spaces, then the diagonalaction of G on X × Y is: g · (x , y) = (g · x , g · y).

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Diagonal actions

Let G be a Polish group and X , Y be Borel G -spaces, then the diagonalaction of G on X × Y is: g · (x , y) = (g · x , g · y).

Theorem

Let S1, S2 be Σ11 subsets of SG(G) both closed under conjugation. The

following are equivalent:

1 The map (H , K ) 7→ H ∩ K from S1 × S2 to SG(G) is Borel.

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Diagonal actions

Let G be a Polish group and X , Y be Borel G -spaces, then the diagonalaction of G on X × Y is: g · (x , y) = (g · x , g · y).

Theorem

Let S1, S2 be Σ11 subsets of SG(G) both closed under conjugation. The

following are equivalent:

1 The map (H , K ) 7→ H ∩ K from S1 × S2 to SG(G) is Borel.

2 There are Borel G-spaces X and Y such that SX (G) = S1,SY (G) = S2 and EX

G, EY

Gand EX×Y

Gare Borel.

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Diagonal actions

Let G be a Polish group and X , Y be Borel G -spaces, then the diagonalaction of G on X × Y is: g · (x , y) = (g · x , g · y).

Theorem

Let S1, S2 be Σ11 subsets of SG(G) both closed under conjugation. The

following are equivalent:

1 The map (H , K ) 7→ H ∩ K from S1 × S2 to SG(G) is Borel.

2 There are Borel G-spaces X and Y such that SX (G) = S1,SY (G) = S2 and EX

G, EY

Gand EX×Y

Gare Borel.

3 There is α < ω1 such that for any H ∈ S1 and K ∈ S2, HK ∈ Π0α.

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Outline

1 Introduction

2 Intersection on F (X )

3 Intersection on Subs(X )

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Borel Π11-complete sets

We say a Π11 set A ⊆ X is Borel Π

11-complete if for any Π

11 set B ⊆ Y ,

there is a Borel function f : Y → X such that B = f −1(A).

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Borel Π11-complete sets

We say a Π11 set A ⊆ X is Borel Π

11-complete if for any Π

11 set B ⊆ Y ,

there is a Borel function f : Y → X such that B = f −1(A).

Fact

Let Tr be the set of all trees on ω, and WF the set of all well-foundedtrees in Tr. Then WF is Borel Π

11-complete in Tr.

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Borel Π11-complete sets

We say a Π11 set A ⊆ X is Borel Π

11-complete if for any Π

11 set B ⊆ Y ,

there is a Borel function f : Y → X such that B = f −1(A).

Fact

Let Tr be the set of all trees on ω, and WF the set of all well-foundedtrees in Tr. Then WF is Borel Π

11-complete in Tr.

Fact

A Π11 set A ⊆ X is Borel Π

11-complete iff there is a Borel function

f : Tr → X such that WF = f −1(A).

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Intersection on F (X )

Theorem (Christensen)

Let X be a Polish space and A ⊆ X. Then the following are equivalent:

1 A ∩ ∂A is Kσ;

2 The operation F 7→ F ∩ A is Borel;

3 The set {F ∈ F (X ) : F ∩ A = ∅} is Borel.

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Intersection on F (X )

Theorem (Christensen)

Let X be a Polish space and A ⊆ X. Then the following are equivalent:

1 A ∩ ∂A is Kσ;

2 The operation F 7→ F ∩ A is Borel;

3 The set {F ∈ F (X ) : F ∩ A = ∅} is Borel.

Theorem (Ding-Gao)

Let A ⊆ X be Σ11. If A ∩ ∂A is not Kσ, then {F ∈ F (X ) : F ∩ A = ∅} is

Borel Π11-complete.

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Outline

1 Introduction

2 Intersection on F (X )

3 Intersection on Subs(X )

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Notions on Banach spaces (1)

1 Let X be a Banach space, then the dual space X ∗ is the space of allbounded linear functional on X .

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Notions on Banach spaces (1)

1 Let X be a Banach space, then the dual space X ∗ is the space of allbounded linear functional on X .

2 There is a natural map J : X → X ∗∗ as J(x)(f ) = f (x) for all x ∈ Xand f ∈ X ∗. We identify x = J(x).

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Notions on Banach spaces (1)

1 Let X be a Banach space, then the dual space X ∗ is the space of allbounded linear functional on X .

2 There is a natural map J : X → X ∗∗ as J(x)(f ) = f (x) for all x ∈ Xand f ∈ X ∗. We identify x = J(x).

3 We say X is reflexive if X = X ∗∗,

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Notions on Banach spaces (1)

1 Let X be a Banach space, then the dual space X ∗ is the space of allbounded linear functional on X .

2 There is a natural map J : X → X ∗∗ as J(x)(f ) = f (x) for all x ∈ Xand f ∈ X ∗. We identify x = J(x).

3 We say X is reflexive if X = X ∗∗, and say X is quasi-reflexive ifdim(X ∗∗/X ) < ∞.

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Borelness of intersection on Subs(X )

Theorem (Ding-Gao)

1 If X is a quasi-reflexive separable Banach space, then the intersectionoperation (Y , Z ) 7→ Y ∩ Z from Subs(X )2 → Subs(X ) is Borel.

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Borelness of intersection on Subs(X )

Theorem (Ding-Gao)

1 If X is a quasi-reflexive separable Banach space, then the intersectionoperation (Y , Z ) 7→ Y ∩ Z from Subs(X )2 → Subs(X ) is Borel.

2 Let X be a separable Banach space, V a closed subspace of X . If Vis quasi-reflexive, then the operation Y 7→ Y ∩ V from Subs(X ) toSubs(X ) is Borel.

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Notions on Banach space (2)

Definition

A sequence (en)∞

n=1 is called a basis of X if for every x ∈ X there isunique sequence of numbers (an)

∞

n=1 so that x =∑

∞

n=1 anen.

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Notions on Banach space (2)

Definition

A sequence (en)∞

n=1 is called a basis of X if for every x ∈ X there isunique sequence of numbers (an)

∞

n=1 so that x =∑

∞

n=1 anen.

Definition

We say a basis (en)∞

n=1 is unconditional if, for every permutation π on N,(eπ(n))

∞

n=1 is still a basis.

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Notions on Banach space (2)

Definition

A sequence (en)∞

n=1 is called a basis of X if for every x ∈ X there isunique sequence of numbers (an)

∞

n=1 so that x =∑

∞

n=1 anen.

Definition

We say a basis (en)∞

n=1 is unconditional if, for every permutation π on N,(eπ(n))

∞

n=1 is still a basis.

Definition

A sequence (Xn)∞

n=1 of closed subspaces of X is called a Schauderdecomposition of X if every x ∈ X has a unique representation of theform x =

∑∞

n=1 xn with xn ∈ Xn for each n.

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Non-Borelness of intersection on Subs(X )

Theorem (Ding-Gao)

Let X = V ⊕H where V , H are two infinite-dimensional closed subspaces.If V has a Schauder decomposition (Vn)

∞

n=1 with every Vn non-reflexive,then the set {Y ∈ Subs(X ) : Y ∩ V = {0}} is Borel Π

11-complete.

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Non-Borelness of intersection on Subs(X )

Theorem (Ding-Gao)

Let X = V ⊕H where V , H are two infinite-dimensional closed subspaces.If V has a Schauder decomposition (Vn)

∞

n=1 with every Vn non-reflexive,then the set {Y ∈ Subs(X ) : Y ∩ V = {0}} is Borel Π

11-complete.

It follows that the intersection operation from Subs(X )2 → Subs(X ) isnot Borel.

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Intersection on Subs(X ) and SG(X )

Theorem (Ding-Gao)

Let X = V ⊕H where V , H are two infinite-dimensional closed subspaces,then the set {Y ∈ SG(X ) : Y ∩ V = {0}} is Borel Π

11-complete.

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Intersection on Subs(X ) and SG(X )

Theorem (Ding-Gao)

Let X = V ⊕H where V , H are two infinite-dimensional closed subspaces,then the set {Y ∈ SG(X ) : Y ∩ V = {0}} is Borel Π

11-complete.

Theorem (D.)

Let X = V ⊕ R where V has an unconditional basis, then the set{Y ∈ SG(X ) : Y ∩ V = {0}} is Borel Π

11-complete.

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

H.I. spaces and unconditional bases

Definition

A Banach space X is called hereditarily indecomposable (or H.I.) if anyclosed subspace of X cannot be written as Y ⊕ Z with Y and Zinfinite-dimensional.

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

H.I. spaces and unconditional bases

Definition

A Banach space X is called hereditarily indecomposable (or H.I.) if anyclosed subspace of X cannot be written as Y ⊕ Z with Y and Zinfinite-dimensional.

Theorem (Gowers’ dichotomy theorem)

Every Banach space X has a subspace which either has an unconditionalbasis or is H.I.

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

Main references

1 L. Ding, S. Gao, Diagonal actions and Borel equivalence relations, J.Symb. Logic 71 (2006), no.4, 1081–1096.

2 L. Ding, S. Gao, On the Borelness of the intersection operation,Israel J. Math. to appear.

3 W. T. Gowers, B. Maurey, The unconditional basic sequenceproblem, J. Amer. Math. Soc. 6 (1993), 851–874.

4 W. T. Gowers, An infinite Ramsey theorem and some Banach-spacedichotomies, Ann. Math. 156 (2002), 797–833.

L. Ding SM-NNK

A survey on the Borelness of the intersection operation

Introduction Intersection on F (X ) Intersection on Subs(X )

The end

Thank you!

L. Ding SM-NNK

A survey on the Borelness of the intersection operation