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Descriptive Set Theory and the Wadge order Reducibility notions for the Scott domain PhDs in Logic XI, April 25th, 2019, Bern, Switzerland Louis Vuilleumier University of Lausanne and University Paris Diderot April 25th, 2019, Bern

Descriptive Set Theory Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Measure Problem Question (Lebesgue 1902) Does there exist a function m which associates to each bounded set of real numbers X a non-negative real number m ( X ) such that the following conditions hold? i) m is not always 0; ii) m is invariant under translation; iii) m is σ -additive. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Measure Problem Question (Lebesgue 1902) Does there exist a function m which associates to each bounded set of real numbers X a non-negative real number m ( X ) such that the following conditions hold? i) m is not always 0; ii) m is invariant under translation; iii) m is σ -additive. Answer (Vitali 1905) No! Under ZFC, there exists a non-measurable set. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

Descriptive set theory is the study of “definable sets” in Polish (i.e. separable completely metrizable) spaces. In this theory, sets are classified in hierarchies, accord- ing to the complexity of their definitions [...]. A. Kechris Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

Descriptive set theory is the study of “definable sets” in Polish (i.e. separable completely metrizable) spaces. In this theory, sets are classified in hierarchies, accord- ing to the complexity of their definitions [...]. A. Kechris Examples R , C , I N and separable Banach spaces are Polish. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

Descriptive set theory is the study of “definable sets” in Polish (i.e. separable completely metrizable) spaces. In this theory, sets are classified in hierarchies, accord- ing to the complexity of their definitions [...]. A. Kechris Examples R , C , I N and separable Banach spaces are Polish. The Baire space ω ω and the Cantor space 2 ω are also Polish. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Cantor Space Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

Descriptive Set Theory The Borel hierarchy for metrizable spaces Let X be a metrizable space, we set for 2 ≤ α < ω 1 , Σ 0 � � 1 ( X ) = U ⊆ X : U is open , � � � A ∁ Σ 0 n : A n ∈ Σ 0 α ( X ) = β n ( X ) , β n < α , n ∈ ω A : A ∁ ∈ Σ 0 � � Π 0 α ( X ) = α ( X ) , ∆ 0 α ( X ) = Σ 0 α ( X ) ∩ Π 0 α ( X ) . Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Borel hierarchy Σ 0 1 Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Borel hierarchy Σ 0 Π 0 1 ∆ 0 1 1 Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Borel hierarchy Σ 0 2 Σ 0 Π 0 1 ∆ 0 1 1 Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Borel hierarchy Σ 0 Π 0 2 2 ∆ 0 2 Σ 0 Π 0 1 ∆ 0 1 1 Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Borel hierarchy . . . Σ 0 Π 0 2 2 ∆ 0 2 Σ 0 Π 0 1 ∆ 0 1 1 Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Borel hierarchy α < ω 1 Σ 0 α . . . Σ 0 Π 0 2 2 ∆ 0 2 Σ 0 Π 0 1 ∆ 0 1 1 Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Borel hierarchy α < ω 1 Σ 0 Π 0 α α ∆ 0 α . . . Σ 0 Π 0 2 2 ∆ 0 2 Σ 0 Π 0 1 ∆ 0 1 1 Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Borel hierarchy . . α < ω 1 . Σ 0 Π 0 α α ∆ 0 α . . . Σ 0 Π 0 2 2 ∆ 0 2 Σ 0 Π 0 1 ∆ 0 1 1 Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Borel hierarchy . . α < ω 1 . B Σ 0 Π 0 α α ∆ 0 α . . . Σ 0 Π 0 2 2 ∆ 0 2 Σ 0 Π 0 1 ∆ 0 1 1 Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Definition Let X be a topological space, and A , B ⊆ X . The set A is Wadge reducible to B , denoted by A ≤ w B , if there exists a continuous function f : X → X such that f − 1 ( B ) = A . Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Definition Let X be a topological space, and A , B ⊆ X . The set A is Wadge reducible to B , denoted by A ≤ w B , if there exists a continuous function f : X → X such that f − 1 ( B ) = A . Remarks ◮ Wadge reducibility is topologically natural. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Definition Let X be a topological space, and A , B ⊆ X . The set A is Wadge reducible to B , denoted by A ≤ w B , if there exists a continuous function f : X → X such that f − 1 ( B ) = A . Remarks ◮ Wadge reducibility is topologically natural. ◮ It induces a quasi-order on P ( X ) . Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Definition Let X be a topological space, and A , B ⊆ X . The set A is Wadge reducible to B , denoted by A ≤ w B , if there exists a continuous function f : X → X such that f − 1 ( B ) = A . Remarks ◮ Wadge reducibility is topologically natural. ◮ It induces a quasi-order on P ( X ) . ◮ It induces a partial order on the Wadge degrees called the Wadge order ( P ( X ) / ≡ w , ≤ w ) . Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Remarks ◮ The Wadge order refines the Borel hierarchy. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Remarks ◮ The Wadge order refines the Borel hierarchy. ◮ The Wadge order is a measure of topological com- plexity. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Remarks ◮ The Wadge order refines the Borel hierarchy. ◮ The Wadge order is a measure of topological com- plexity. ◮ The Wadge order is topologically meaningful: Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Remarks ◮ The Wadge order refines the Borel hierarchy. ◮ The Wadge order is a measure of topological com- plexity. ◮ The Wadge order is topologically meaningful: The Wadge hierarchy is the ultimate analysis of P ( ω ω ) in terms of topological complexity [...]. Andretta and Louveau. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Remarks ◮ The Wadge order refines the Borel hierarchy. ◮ The Wadge order is a measure of topological com- plexity. ◮ The Wadge order is topologically meaningful: The Wadge hierarchy is the ultimate analysis of P ( ω ω ) in terms of topological complexity [...]. Andretta and Louveau. Problem Understanding the shape of the Wadge order on differ- ent topological spaces. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Good news If X = ω ω or X = 2 ω , we can use games! Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Good news If X = ω ω or X = 2 ω , we can use games! Definition Let S ∈ { 2 , ω } and A , B ⊆ S ω . Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Good news If X = ω ω or X = 2 ω , we can use games! Definition Let S ∈ { 2 , ω } and A , B ⊆ S ω . I II Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Good news If X = ω ω or X = 2 ω , we can use games! Definition Let S ∈ { 2 , ω } and A , B ⊆ S ω . x 0 I II Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Good news If X = ω ω or X = 2 ω , we can use games! Definition Let S ∈ { 2 , ω } and A , B ⊆ S ω . x 0 I y 0 II Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Good news If X = ω ω or X = 2 ω , we can use games! Definition Let S ∈ { 2 , ω } and A , B ⊆ S ω . x 0 x 1 I y 0 II Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Good news If X = ω ω or X = 2 ω , we can use games! Definition Let S ∈ { 2 , ω } and A , B ⊆ S ω . x 0 x 1 I y 0 s II Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern

The Wadge order Good news If X = ω ω or X = 2 ω , we can use games! Definition Let S ∈ { 2 , ω } and A , B ⊆ S ω . x 0 x 1 x 2 I y 0 s II Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern