A survey on the Borelness of the intersection operation

Transcription

1 A survey on the Borelness of the intersection operation The 2013 International Workshop on Logic Longyun Ding School of Mathematical Sciences Nankai University Mar 11th, 2013 Nanjin, China

2 Outline 1 Introduction 2 Intersection on F(X) 3 Intersection on Subs(X)

3 The Effros Borel spaces Let X be a Polish space, i.e., separable, completely metrizable topological space. Definition Let F(X) be the set of all closed subsets of X. The Effros Borel structure on F(X) is the σ-algebra generated by the sets {F F(X) : F U }, where U varies over open subsets of X.

4 The Effros Borel spaces Let X be a Polish space, i.e., separable, completely metrizable topological space. Definition Let F(X) be the set of all closed subsets of X. The Effros Borel structure on F(X) is the σ-algebra generated by the sets {F F(X) : F U }, where U varies over open subsets of X. Fact F(X) is a standard Borel space, i.e., there is a Polish topology τ on F(X) which induce the Effros Borel structure.

5 Facts of F(X) Theorem (Kuratowski Ryll-Nardzewski, The Selection Theorem) Let X be Polish. There is a sequence of Borel functions s n : F(X) X, such that for nonempty F F(X), {s n (F)} is dense in F.

6 Facts of F(X) Theorem (Kuratowski Ryll-Nardzewski, The Selection Theorem) Let X be Polish. There is a sequence of Borel functions s n : F(X) X, such that for nonempty F F(X), {s n (F)} is dense in F. Fact 1 The set of compact subsets of X, K(X), is a Borel set in F(X);

7 Facts of F(X) Theorem (Kuratowski Ryll-Nardzewski, The Selection Theorem) Let X be Polish. There is a sequence of Borel functions s n : F(X) X, such that for nonempty F F(X), {s n (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 F 1 F 2 is Borel in F(X) 2 ;

8 Facts of F(X) Theorem (Kuratowski Ryll-Nardzewski, The Selection Theorem) Let X be Polish. There is a sequence of Borel functions s n : F(X) X, such that for nonempty F F(X), {s n (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 F 1 F 2 is Borel in F(X) 2 ; 3 The function (F 1, F 2 ) F 1 F 2 is Borel from F(X) 2 to F(X);

9 Facts of F(X) Theorem (Kuratowski Ryll-Nardzewski, The Selection Theorem) Let X be Polish. There is a sequence of Borel functions s n : F(X) X, such that for nonempty F F(X), {s n (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 F 1 F 2 is Borel in F(X) 2 ; 3 The function (F 1, F 2 ) F 1 F 2 is Borel from F(X) 2 to F(X); 4 If f : X Y is continuous, then the map F f (F) is Borel from F(X) to F(Y ).

10 Facts of F(X) Theorem (Kuratowski Ryll-Nardzewski, The Selection Theorem) Let X be Polish. There is a sequence of Borel functions s n : F(X) X, such that for nonempty F F(X), {s n (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 F 1 F 2 is Borel in F(X) 2 ; 3 The function (F 1, F 2 ) F 1 F 2 is Borel from F(X) 2 to F(X); 4 If f : X Y is continuous, then the map F f (F) is Borel from F(X) to F(Y ). The function (F 1, F 2 ) F 1 F 2 is NOT necessarily Borel.

11 Borel G-spaces Definition Let G be a Polish group and X a Polish space. An action of G on X is a function : G X X, g (h x) = (gh) x and 1 G x = x. If is a Borel function, we say X is a Borel G-space.

12 Borel G-spaces Definition Let G be a Polish group and X a Polish space. An action of G on X is a function : G X X, g (h x) = (gh) x and 1 G x = x. If is a Borel function, we say X is a Borel G-space. We denote the orbit equivalence relation, E X G, as xe X G y g G(g x = y).

13 Becker-Kechris s theorem Theorem The following are equivalent: 1 E X G is Borel.

14 Becker-Kechris s theorem Theorem The following are equivalent: 1 E X G is Borel. 2 The map x G x = {g G : g x = x} is Borel from X to F(G).

15 Becker-Kechris s theorem Theorem The following are equivalent: 1 E X G is Borel. 2 The map x G x = {g G : g x = x} is Borel from X to F(G). 3 The map (x, u) G x,u = {g G : g x = u} is Borel from X 2 to F(G).

16 Closed subgroups of G Fact 1 Denote SG(G) the set of all closed subgroup of G. Then SG(G) is Borel in F(G).

17 Closed subgroups of G Fact 1 Denote SG(G) the set of all closed subgroup of G. Then SG(G) is Borel in F(G). 2 Denote S X (G) = {G x : x X }. Then S X (G) is Σ 1 1 and closed under conjugation, i.e., for g G and H S X (G) we have ghg 1 S X (G).

18 Diagonal actions Let G be a Polish group and X, Y be Borel G-spaces, then the diagonal action of G on X Y is: g (x, y) = (g x, g y).

19 Diagonal actions Let G be a Polish group and X, Y be Borel G-spaces, then the diagonal action of G on X Y is: g (x, y) = (g x, g y). Theorem Let S 1, S 2 be Σ 1 1 subsets of SG(G) both closed under conjugation. The following are equivalent: 1 The map (H, K) H K from S 1 S 2 to SG(G) is Borel.

20 Diagonal actions Let G be a Polish group and X, Y be Borel G-spaces, then the diagonal action of G on X Y is: g (x, y) = (g x, g y). Theorem Let S 1, S 2 be Σ 1 1 subsets of SG(G) both closed under conjugation. The following are equivalent: 1 The map (H, K) H K from S 1 S 2 to SG(G) is Borel. 2 There are Borel G-spaces X and Y such that S X (G) = S 1, S Y (G) = S 2 and EG X, E G Y X Y and EG are Borel.

21 Diagonal actions Let G be a Polish group and X, Y be Borel G-spaces, then the diagonal action of G on X Y is: g (x, y) = (g x, g y). Theorem Let S 1, S 2 be Σ 1 1 subsets of SG(G) both closed under conjugation. The following are equivalent: 1 The map (H, K) H K from S 1 S 2 to SG(G) is Borel. 2 There are Borel G-spaces X and Y such that S X (G) = S 1, S Y (G) = S 2 and EG X, E G Y X Y and EG are Borel. 3 There is α < ω 1 such that for any H S 1 and K S 2, HK Π 0 α.

22 Outline 1 Introduction 2 Intersection on F(X) 3 Intersection on Subs(X)

23 Borel Π 1 1-complete sets We say a Π 1 1 set A X is Borel Π1 1 -complete if for any Π1 1 set B Y, there is a Borel function f : Y X such that B = f 1 (A).

24 Borel Π 1 1-complete sets We say a Π 1 1 set A X is Borel Π1 1 -complete if for any Π1 1 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-founded trees in Tr. Then WF is Borel Π 1 1-complete in Tr.

25 Borel Π 1 1-complete sets We say a Π 1 1 set A X is Borel Π1 1 -complete if for any Π1 1 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-founded trees in Tr. Then WF is Borel Π 1 1-complete in Tr. Fact A Π 1 1 set A X is Borel Π1 1-complete iff there is a Borel function f : Tr X such that WF = f 1 (A).

26 Intersection on F(X) Theorem (Christensen) Let X be a Polish space and A X. Then the following are equivalent: 1 A A is K σ ; 2 The operation F F A is Borel; 3 The set {F F(X) : F A = } is Borel.

27 Intersection on F(X) Theorem (Christensen) Let X be a Polish space and A X. Then the following are equivalent: 1 A A is K σ ; 2 The operation F F A is Borel; 3 The set {F F(X) : F A = } is Borel. Theorem (Ding-Gao) Let A X be Σ 1 1. If A A is not K σ, then {F F(X) : F A = } is Borel Π 1 1 -complete.

28 Outline 1 Introduction 2 Intersection on F(X) 3 Intersection on Subs(X)

29 Notions on Banach spaces (1) 1 Let X be a Banach space, then the dual space X is the space of all bounded linear functional on X.

30 Notions on Banach spaces (1) 1 Let X be a Banach space, then the dual space X is the space of all bounded linear functional on X. 2 There is a natural map J : X X as J(x)(f ) = f (x) for all x X and f X. We identify x = J(x).

31 Notions on Banach spaces (1) 1 Let X be a Banach space, then the dual space X is the space of all bounded linear functional on X. 2 There is a natural map J : X X as J(x)(f ) = f (x) for all x X and f X. We identify x = J(x). 3 We say X is reflexive if X = X,

32 Notions on Banach spaces (1) 1 Let X be a Banach space, then the dual space X is the space of all bounded linear functional on X. 2 There is a natural map J : X X as J(x)(f ) = f (x) for all x X and f X. We identify x = J(x). 3 We say X is reflexive if X = X, and say X is quasi-reflexive if dim(x /X) <.

33 Borelness of intersection on Subs(X) Theorem (Ding-Gao) 1 If X is a quasi-reflexive separable Banach space, then the intersection operation (Y, Z) Y Z from Subs(X) 2 Subs(X) is Borel.

34 Borelness of intersection on Subs(X) Theorem (Ding-Gao) 1 If X is a quasi-reflexive separable Banach space, then the intersection operation (Y, Z) 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 V is quasi-reflexive, then the operation Y Y V from Subs(X) to Subs(X) is Borel.

35 Notions on Banach space (2) Definition A sequence (e n ) n=1 is called a basis of X if for every x X there is unique sequence of numbers (a n ) n=1 so that x = n=1 a ne n.

36 Notions on Banach space (2) Definition A sequence (e n ) n=1 is called a basis of X if for every x X there is unique sequence of numbers (a n ) n=1 so that x = n=1 a ne n. Definition We say a basis (e n ) n=1 is unconditional if, for every permutation π on N, (e π(n) ) n=1 is still a basis.

37 Notions on Banach space (2) Definition A sequence (e n ) n=1 is called a basis of X if for every x X there is unique sequence of numbers (a n ) n=1 so that x = n=1 a ne n. Definition We say a basis (e n ) n=1 is unconditional if, for every permutation π on N, (e π(n) ) n=1 is still a basis. Definition A sequence (X n ) n=1 of closed subspaces of X is called a Schauder decomposition of X if every x X has a unique representation of the form x = n=1 x n with x n X n for each n.

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

39 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 (V n ) n=1 with every V n non-reflexive, then the set {Y Subs(X) : Y V = {0}} is Borel Π 1 1 -complete. It follows that the intersection operation from Subs(X) 2 Subs(X) is not Borel.

40 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 1 1 -complete.

41 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 1 1 -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 1 1 -complete.

42 H.I. spaces and unconditional bases Definition A Banach space X is called hereditarily indecomposable (or H.I.) if any closed subspace of X cannot be written as Y Z with Y and Z infinite-dimensional.

43 H.I. spaces and unconditional bases Definition A Banach space X is called hereditarily indecomposable (or H.I.) if any closed subspace of X cannot be written as Y Z with Y and Z infinite-dimensional. Theorem (Gowers dichotomy theorem) Every Banach space X has a subspace which either has an unconditional basis or is H.I.

44 Main references 1 L. Ding, S. Gao, Diagonal actions and Borel equivalence relations, J. Symb. Logic 71 (2006), no.4, 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 sequence problem, J. Amer. Math. Soc. 6 (1993), W. T. Gowers, An infinite Ramsey theorem and some Banach-space dichotomies, Ann. Math. 156 (2002),

45 The end Thank you!