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Introduction Definitions Games on P( N ) Examples Results Towards a better Understanding of the Scott Domain Louis Vuilleumier joint work with Jacques Duparc Department of Information Systems University of Lausanne, Switzerland

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Introduction Definitions Games on P(N) Examples Results

Towards a better Understanding of the Scott Domain

Louis Vuilleumier

joint work with Jacques Duparc

Department of Information Systems University of Lausanne, Switzerland Jacques.Duparc@unil.ch Louis.Vuilleumier.1@unil.ch

September 7, 2016

Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 1 / 10

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Introduction Definitions Games on P(N) Examples Results

Classical Descriptive Set Theory

Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 2 / 10

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Introduction Definitions Games on P(N) Examples Results

Classical Descriptive Set Theory

Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 2 / 10

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Introduction Definitions Games on P(N) Examples Results

Classical Descriptive Set Theory

Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 2 / 10

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Introduction Definitions Games on P(N) Examples Results

Classical Descriptive Set Theory

Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 2 / 10

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Introduction Definitions Games on P(N) Examples Results

Classical Descriptive Set Theory

Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 2 / 10

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Introduction Definitions Games on P(N) Examples Results

Classical Descriptive Set Theory

Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 2 / 10

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Introduction Definitions Games on P(N) Examples Results

Classical Descriptive Set Theory

Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 2 / 10

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Introduction Definitions Games on P(N) Examples Results

Classical Descriptive Set Theory

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Introduction Definitions Games on P(N) Examples Results

Generalization

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Introduction Definitions Games on P(N) Examples Results

Generalization

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Introduction Definitions Games on P(N) Examples Results

Generalization

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Introduction Definitions Games on P(N) Examples Results

Generalization

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Introduction Definitions Games on P(N) Examples Results

Scott domain and continuous reduction

Definition Consider the set P(N) together with the topology — known as the Scott topology — generated by the basis B = {OF ∶ F finite subset of N} , where OF = {X ⊆ N ∶ F ⊆ X}. This space is the Scott domain.

Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 4 / 10

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Introduction Definitions Games on P(N) Examples Results

Scott domain and continuous reduction

Definition Consider the set P(N) together with the topology — known as the Scott topology — generated by the basis B = {OF ∶ F finite subset of N} , where OF = {X ⊆ N ∶ F ⊆ X}. This space is the Scott domain. Theorem (de Brecht, 2013) A space is quasi-Polish if and only if it is homeomorphic to some A ∈ Π0

2(P(N)).

Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 4 / 10

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Introduction Definitions Games on P(N) Examples Results

Scott domain and continuous reduction

Definition Consider the set P(N) together with the topology — known as the Scott topology — generated by the basis B = {OF ∶ F finite subset of N} , where OF = {X ⊆ N ∶ F ⊆ X}. This space is the Scott domain. Theorem (de Brecht, 2013) A space is quasi-Polish if and only if it is homeomorphic to some A ∈ Π0

2(P(N)).

Definition Let A, B ⊆ P(N). If there exists a continuous function f ∶ P(N) → P(N) such that f −1(B) = A, we say that A is continuously reducible or Wadge reducible to B, and we write A ≤W B.

Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 4 / 10

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Introduction Definitions Games on P(N) Examples Results

Scott domain and continuous reduction

Definition Consider the set P(N) together with the topology — known as the Scott topology — generated by the basis B = {OF ∶ F finite subset of N} , where OF = {X ⊆ N ∶ F ⊆ X}. This space is the Scott domain. Theorem (de Brecht, 2013) A space is quasi-Polish if and only if it is homeomorphic to some A ∈ Π0

2(P(N)).

Definition Let A, B ⊆ P(N). If there exists a continuous function f ∶ P(N) → P(N) such that f −1(B) = A, we say that A is continuously reducible or Wadge reducible to B, and we write A ≤W B. ≤W induces a quasi-order relation on the subsets of the Scott domain.

Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 4 / 10

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Introduction Definitions Games on P(N) Examples Results

Games on the Scott domain

Definition Let A, B ⊆ P(N). We define a game G∞(A, B). I II X0 Y0 X1 Y1 X2 Y2 ⋮ ⋮ X = ⋃n∈N Xn, Y = ⋃n∈N Yn Xn ∈ P<∞(N), Yn ∈ P(N) II wins if and only if (X ∈ A ↔ Y ∈ B). ⊆ ⊆ ⊆ ⊆ ⊆

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Introduction Definitions Games on P(N) Examples Results

Strategy for II

Definition An ultrapositional strategy for II in a game G∞(A, B) is an increasing function σ ∶ P<∞(N) → P(N), i.e. such that for all X0, X1 ∈ P<∞(N) with X0 ⊆ X1, we have σ(X0) ⊆ σ(X1).

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Strategy for II

Definition An ultrapositional strategy for II in a game G∞(A, B) is an increasing function σ ∶ P<∞(N) → P(N), i.e. such that for all X0, X1 ∈ P<∞(N) with X0 ⊆ X1, we have σ(X0) ⊆ σ(X1). Definition An ultrapositional strategy is winning for II if, following this ultrapositional strategy and whatever I plays, II wins the game G∞(A, B).

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Introduction Definitions Games on P(N) Examples Results

Strategy for II

Definition An ultrapositional strategy for II in a game G∞(A, B) is an increasing function σ ∶ P<∞(N) → P(N), i.e. such that for all X0, X1 ∈ P<∞(N) with X0 ⊆ X1, we have σ(X0) ⊆ σ(X1). Definition An ultrapositional strategy is winning for II if, following this ultrapositional strategy and whatever I plays, II wins the game G∞(A, B). Proposition Let A, B ⊆ P(N). The following are equivalent.

1 A ≤W B, 2 II has a winning ultrapositional strategy in G∞(A, B). Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 6 / 10

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Introduction Definitions Games on P(N) Examples Results

Examples

Consider G∞ ({N}, P∞(N)) . σ ∶ P<∞(N) → P(N) X ↦ ⋃

n∈N {0,...,n}⊆X

{0, . . . , n}. It is a ultrapositional winning strategy. Hence, {N} ≤W P∞(N).

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Introduction Definitions Games on P(N) Examples Results

Examples

Consider G∞ ({N}, P∞(N)) . σ ∶ P<∞(N) → P(N) X ↦ ⋃

n∈N {0,...,n}⊆X

{0, . . . , n}. It is a ultrapositional winning strategy. Hence, {N} ≤W P∞(N). Consider G∞ ({{0}}, {{0}, {0, 1, 2}}) . σ ∶ P<∞(N) → P(N) ∅ ↦ ∅, {0} ↦ {0}, X ↦ {0, 1}

therwise.

It is a ultrapositional winning strategy. Hence, {{0}} ≤W {{0}, {0, 1, 2}}.

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Examples

Consider the game G∞({N}, {{0}}). I II ∅

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Introduction Definitions Games on P(N) Examples Results

Examples

Consider the game G∞({N}, {{0}}). I II ∅ ∅

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Introduction Definitions Games on P(N) Examples Results

Examples

Consider the game G∞({N}, {{0}}). I II ∅ ∅ {0}

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Introduction Definitions Games on P(N) Examples Results

Examples

Consider the game G∞({N}, {{0}}). I II ∅ ∅ {0} ∅

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Introduction Definitions Games on P(N) Examples Results

Examples

Consider the game G∞({N}, {{0}}). I II ∅ ∅ {0} ∅ {0, 1}

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Introduction Definitions Games on P(N) Examples Results

Examples

Consider the game G∞({N}, {{0}}). I II ∅ ∅ {0} ∅ {0, 1} ∅

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Introduction Definitions Games on P(N) Examples Results

Examples

Consider the game G∞({N}, {{0}}). I II ∅ ∅ {0} ∅ {0, 1} ∅ {0, 1, 2}

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Introduction Definitions Games on P(N) Examples Results

Examples

Consider the game G∞({N}, {{0}}). I II ∅ ∅ {0} ∅ {0, 1} ∅ {0, 1, 2} {0}

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Introduction Definitions Games on P(N) Examples Results

Examples

Consider the game G∞({N}, {{0}}). I II ∅ ∅ {0} ∅ {0, 1} ∅ {0, 1, 2} {0} {0, 1, 2}

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Introduction Definitions Games on P(N) Examples Results

Examples

Consider the game G∞({N}, {{0}}). I II ∅ ∅ {0} ∅ {0, 1} ∅ {0, 1, 2} {0} {0, 1, 2} {0}

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Introduction Definitions Games on P(N) Examples Results

Examples

Consider the game G∞({N}, {{0}}). I II ∅ ∅ {0} ∅ {0, 1} ∅ {0, 1, 2} {0} {0, 1, 2} {0} {0} {0, 1, 2} ☇☇☇ Hence, {N} ≰W {{0}}.

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Introduction Definitions Games on P(N) Examples Results

Examples

Consider the game G∞({N}, {{0}}). I II ∅ ∅ {0} ∅ {0, 1} ∅ {0, 1, 2} {0} {0, 1, 2} {0} {0} {0, 1, 2} ☇☇☇ Hence, {N} ≰W {{0}}. We also have : ({N}, {{0}}, {N}∁, {{0}}∁) is an antichain with respect to the quasi-order ≤W in the Scott domain.

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No self-dual set in the Scott domain

Proposition Let A ⊆ P(N), then A is non-self-dual.

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No self-dual set in the Scott domain

Proposition Let A ⊆ P(N), then A is non-self-dual. Proof. I II ∅ Y0 Y0 Y1 Y1 Y2 ⋮ ⋮ X = ⋃n∈N Yn, Y = ⋃n∈N Yn ⊆ ⊆ ⊆ ⊆ ⊆

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Antichains in the Scott domain

Proposition (V.) For every k ∈ N ∖ {0}, there exists a sequence (A1, . . . , Ak) such that, for all i, j ∈ {1, . . . , k}, i ≠ j, we have Ai ⊆ ∆0

3(P(N)) and Ai is not continuously reducible to

Aj.

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Antichains in the Scott domain

Proposition (V.) For every k ∈ N ∖ {0}, there exists a sequence (A1, . . . , Ak) such that, for all i, j ∈ {1, . . . , k}, i ≠ j, we have Ai ⊆ ∆0

3(P(N)) and Ai is not continuously reducible to

Aj. Merci !

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