The Wadge ordering over the Borel subsets of the Scott domain is - PowerPoint PPT Presentation

The Wadge ordering over the Borel subsets of the Scott domain is not wqo Workshop on Wadge Theory and Automata II, June 8th, 2018, Torino Louis Vuilleumier University of Lausanne and University Paris Diderot June 8th, 2018, Torino

Table of contents 1. Introduction 2. Quasi-Polish spaces, a good generalization 3. Wadge Theory on the Scott domain 4. New results On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Introduction On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Generalization of Polish Metrizable Non-Metrizable On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Generalization of Polish Metrizable Non-Metrizable Polish A Polish space is a separable completely metrizable space. On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Generalization of Polish Metrizable Non-Metrizable Polish DCPO A DCPO is a poset in which every directed subset has a supremum. On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Generalization of Polish Metrizable Non-Metrizable ω -cont. Polish domain DCPO An ω -continuous domain is a DCPO that has a countable domain theoretic basis. On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Generalization of Polish Metrizable Non-Metrizable ω -cont. Polish domain Quasi- Polish DCPO A quasi-Polish space is a countably based completely quasi-metrizable space. On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Quasi-Polish spaces, a good generalization On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

A generalization Examples of Polish spaces ω ω , 2 ω , R , C , R n , R ω , I ω , countable sets with the discrete topology, separable Banach spaces. On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

A generalization Examples of Polish spaces ω ω , 2 ω , R , C , R n , R ω , I ω , countable sets with the discrete topology, separable Banach spaces. Examples of quasi-Polish spaces 1. The Sierpinski space S = { 0, 1 } with the topology � � ∅ , S , { 1 } . On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

A generalization Examples of Polish spaces ω ω , 2 ω , R , C , R n , R ω , I ω , countable sets with the discrete topology, separable Banach spaces. Examples of quasi-Polish spaces 1. The Sierpinski space S = { 0, 1 } with the topology � � ∅ , S , { 1 } . 2. Consider P ( ω ) with the topology induced by the quasi-metric d ( x , y ) = sup { 2 − n : n ∈ x \ y } . This is the Scott domain , and it is a quasi-Polish space. On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

A generalization Examples of Polish spaces ω ω , 2 ω , R , C , R n , R ω , I ω , countable sets with the discrete topology, separable Banach spaces. Examples of quasi-Polish spaces 1. The Sierpinski space S = { 0, 1 } with the topology � � ∅ , S , { 1 } . 2. Consider P ( ω ) with the topology induced by the quasi-metric d ( x , y ) = sup { 2 − n : n ∈ x \ y } . This is the Scott domain , and it is a quasi-Polish space. Theorem (de Brecht) ◮ Every Polish space is quasi-Polish; ◮ Every ω -continuous domain is quasi-Polish; On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

A generalization Examples of Polish spaces ω ω , 2 ω , R , C , R n , R ω , I ω , countable sets with the discrete topology, separable Banach spaces. Examples of quasi-Polish spaces 1. The Sierpinski space S = { 0, 1 } with the topology � � ∅ , S , { 1 } . 2. Consider P ( ω ) with the topology induced by the quasi-metric d ( x , y ) = sup { 2 − n : n ∈ x \ y } . This is the Scott domain , and it is a quasi-Polish space. Theorem (de Brecht) ◮ Every Polish space is quasi-Polish; ◮ Every ω -continuous domain is quasi-Polish; ◮ A metrizable space is quasi-Polish iff it is Polish. On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Borel hierarchy in quasi-Polish spaces Problem The Borel classes on quasi-Polish spaces do not give a well behaved hierarchy, i.e. a nested sequence of collection of subsets. On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Borel hierarchy in quasi-Polish spaces Problem The Borel classes on quasi-Polish spaces do not give a well behaved hierarchy, i.e. a nested sequence of collection of subsets. Example In the Sierpinski space S , the open set { 1 } is not Σ 0 2 . On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Borel hierarchy in quasi-Polish spaces Problem The Borel classes on quasi-Polish spaces do not give a well behaved hierarchy, i.e. a nested sequence of collection of subsets. Example In the Sierpinski space S , the open set { 1 } is not Σ 0 2 . Solution (Selivanov) Slight modification of the definition of the Borel hierar- chy for non-metrizable sets. Let 2 � α < ω 1 , � � � Σ 0 n ∈ Σ 0 ( A n \ A ′ n ) : A n , A ′ α ( X ) = β n , β n < α . n ∈ ω On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Borel hierarchy in all topological spaces On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Hierarchies in quasi-Polish spaces Theorem Let X be an uncountable quasi-Polish space, then the Borel hierarchy on X does not collapse. On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Hierarchies in quasi-Polish spaces Theorem Let X be an uncountable quasi-Polish space, then the Borel hierarchy on X does not collapse. On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Hierarchies in quasi-Polish spaces Theorem Let X be an uncountable quasi-Polish space, then the Borel hierarchy on X does not collapse. Theorem (Hausdorff-Kuratowski) If X is a quasi-Polish space and 1 � θ < ω 1 , then ∆ 0 � D α ( Σ 0 θ + 1 ( X ) = θ ( X )) . 1 � α<ω 1 On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Hierarchies in quasi-Polish spaces Theorem Let X be an uncountable quasi-Polish space, then the Borel hierarchy on X does not collapse. Theorem (Hausdorff-Kuratowski) If X is a quasi-Polish space and 1 � θ < ω 1 , then ∆ 0 � D α ( Σ 0 θ + 1 ( X ) = θ ( X )) . 1 � α<ω 1 On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Subspaces of quasi-Polish spaces Theorem (Kuratowski) Let X be a quasi-metrizable space, Y be completely quasi-metrizable, X , Y be countably based, A ⊆ X with f : A → Y continuous. Then there exists G ∈ Π 0 2 ( X ) with A ⊆ G and a contin- uous extension g : G → Y of f . On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Subspaces of quasi-Polish spaces Theorem (Kuratowski) Let X be a quasi-metrizable space, Y be completely quasi-metrizable, X , Y be countably based, A ⊆ X with f : A → Y continuous. Then there exists G ∈ Π 0 2 ( X ) with A ⊆ G and a contin- uous extension g : G → Y of f . On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Subspaces of quasi-Polish spaces Theorem (Kuratowski) Let X be a quasi-metrizable space, Y be completely quasi-metrizable, X , Y be countably based, A ⊆ X with f : A → Y continuous. Then there exists G ∈ Π 0 2 ( X ) with A ⊆ G and a contin- uous extension g : G → Y of f . Theorem A subspace Y ⊆ X of a quasi-Polish space is quasi- Polish if and only if Y ∈ Π 0 2 ( X ) . On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Subspaces of quasi-Polish spaces Theorem (Kuratowski) Let X be a quasi-metrizable space, Y be completely quasi-metrizable, X , Y be countably based, A ⊆ X with f : A → Y continuous. Then there exists G ∈ Π 0 2 ( X ) with A ⊆ G and a contin- uous extension g : G → Y of f . Theorem A subspace Y ⊆ X of a quasi-Polish space is quasi- Polish if and only if Y ∈ Π 0 2 ( X ) . On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Subspaces of quasi-Polish spaces Theorem (Kuratowski) Let X be a quasi-metrizable space, Y be completely quasi-metrizable, X , Y be countably based, A ⊆ X with f : A → Y continuous. Then there exists G ∈ Π 0 2 ( X ) with A ⊆ G and a contin- uous extension g : G → Y of f . Theorem A subspace Y ⊆ X of a quasi-Polish space is quasi- Polish if and only if Y ∈ Π 0 2 ( X ) . Theorem (de Brecht) A space X is a quasi-Polish space if and only if X ∈ Π 0 2 ( P ( ω )) . On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Further extensions of classical results As in the Polish case, there exists (de Brecht): On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Further extensions of classical results As in the Polish case, there exists (de Brecht): ◮ A game theoretical characterization of quasi-Polish spaces; On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino