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 Answer (Vitali 1905) No! Under ZFC, there exists a non-measurable set.
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 Examples R, C, IN and separable Banach spaces are Polish.
The Baire space ωω and the Cantor space 2ω are also Polish.
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 The Baire space ωω and the Cantor space 2ω are also Polish.
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, IN 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, IN and separable Banach spaces are Polish.
The Baire space ωω and the Cantor space 2ω 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 < ω1,
0 Σ1(X) = U ⊆ X : U is open , ( ) [ Σ0 (X) = A{ : A ∈ Σ0 (X), β < α , α n n βn n n∈ω n o 0 { 0 Πα(X) = A : A ∈ Σα(X) , 0 0 0 ∆α(X) = Σα(X) ∩ Πα(X).
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern . α < ω1 . B
0 0 Σα Πα 0 ∆α
. .
0 0 Σ2 Π2 0 ∆2
0 0 Π1 ∆1
The Borel hierarchy
0 Σ1
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern . α < ω1 . B
0 0 Σα Πα 0 ∆α
. .
0 0 Σ2 Π2 0 ∆2
The Borel hierarchy
0 0 Σ1 0 Π1 ∆1
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern . α < ω1 . B
0 0 Σα Πα 0 ∆α
. .
0 Π2 0 ∆2
The Borel hierarchy
0 Σ2
0 0 Σ1 0 Π1 ∆1
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern . α < ω1 . B
0 0 Σα Πα 0 ∆α
. .
The Borel hierarchy
0 0 Σ2 Π2 0 ∆2
0 0 Σ1 0 Π1 ∆1
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern . α < ω1 . B
0 0 Σα Πα 0 ∆α
The Borel hierarchy
. .
0 0 Σ2 Π2 0 ∆2
0 0 Σ1 0 Π1 ∆1
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern . . B
0 Πα 0 ∆α
The Borel hierarchy
α < ω1
0 Σα
. .
0 0 Σ2 Π2 0 ∆2
0 0 Σ1 0 Π1 ∆1
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern . . B
The Borel hierarchy
α < ω1
0 0 Σα Πα 0 ∆α
. .
0 0 Σ2 Π2 0 ∆2
0 0 Σ1 0 Π1 ∆1
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern B
The Borel hierarchy
. α < ω1 .
0 0 Σα Πα 0 ∆α
. .
0 0 Σ2 Π2 0 ∆2
0 0 Σ1 0 Π1 ∆1
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern The Borel hierarchy
. α < ω1 . B
0 0 Σα Πα 0 ∆α
. .
0 0 Σ2 Π2 0 ∆2
0 0 Σ1 0 Π1 ∆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 I Wadge reducibility is topologically natural.
I It induces a quasi-order on P(X).
I It induces a partial order on the Wadge degrees called the Wadge order
P(X) ( / ≡w, ≤w).
Remarks
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 I It induces a quasi-order on P(X).
I It induces a partial order on the Wadge degrees called the Wadge order
P(X) ( / ≡w, ≤w).
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
I Wadge reducibility is topologically natural.
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern I It induces a partial order on the Wadge degrees called the Wadge order
P(X) ( / ≡w, ≤w).
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
I Wadge reducibility is topologically natural.
I 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
I Wadge reducibility is topologically natural.
I It induces a quasi-order on P(X).
I 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 I The Wadge order is a measure of topological com- plexity.
I The Wadge order is topologically meaningful:
The Wadge hierarchy is the ultimate analysis of P(ωω) in terms of topological complexity [...]. Andretta and Louveau.
Problem Understanding the shape of the Wadge order on differ- ent topological spaces.
The Wadge order
Remarks
I The Wadge order refines the Borel hierarchy.
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern I The Wadge order is topologically meaningful:
The Wadge hierarchy is the ultimate analysis of P(ωω) in terms of topological complexity [...]. Andretta and Louveau.
Problem Understanding the shape of the Wadge order on differ- ent topological spaces.
The Wadge order
Remarks
I The Wadge order refines the Borel hierarchy.
I 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 hierarchy is the ultimate analysis of P(ωω) in terms of topological complexity [...]. Andretta and Louveau.
Problem Understanding the shape of the Wadge order on differ- ent topological spaces.
The Wadge order
Remarks
I The Wadge order refines the Borel hierarchy.
I The Wadge order is a measure of topological com- plexity.
I The Wadge order is topologically meaningful:
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern Problem Understanding the shape of the Wadge order on differ- ent topological spaces.
The Wadge order
Remarks
I The Wadge order refines the Borel hierarchy.
I The Wadge order is a measure of topological com- plexity.
I The Wadge order is topologically meaningful:
The Wadge hierarchy is the ultimate analysis of P(ωω) 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
I The Wadge order refines the Borel hierarchy.
I The Wadge order is a measure of topological com- plexity.
I The Wadge order is topologically meaningful:
The Wadge hierarchy is the ultimate analysis of P(ωω) 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 x0 x1 x2 ··· xn xn+1 ··· x = (xn)n∈ω
y0 s y1 ··· yn s ··· y = (yn)n∈ω
Definition Let S ∈ {2, ω} and A, B ⊆ Sω.
I II
The Wadge game Gw(A, B) where xn, yn ∈ S.
We say that II wins the game iff
x ∈ A ↔ y ∈ B.
The Wadge order
Good news If X = ωω or X = 2ω, we can use games!
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern x0 x1 x2 ··· xn xn+1 ··· x = (xn)n∈ω
y0 s y1 ··· yn s ··· y = (yn)n∈ω
I II
The Wadge game Gw(A, B) where xn, yn ∈ S.
We say that II wins the game iff
x ∈ A ↔ y ∈ B.
The Wadge order
Good news If X = ωω or X = 2ω, we can use games!
Definition Let S ∈ {2, ω} and A, B ⊆ Sω.
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern x0 x1 x2 ··· xn xn+1 ··· x = (xn)n∈ω
y0 s y1 ··· yn s ··· y = (yn)n∈ω
The Wadge game Gw(A, B) where xn, yn ∈ S.
We say that II wins the game iff
x ∈ A ↔ y ∈ B.
The Wadge order
Good news If X = ωω or X = 2ω, we can use games!
Definition Let S ∈ {2, ω} and A, B ⊆ Sω.
I II
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern x1 x2 ··· xn xn+1 ··· x = (xn)n∈ω
y0 s y1 ··· yn s ··· y = (yn)n∈ω
The Wadge game Gw(A, B) where xn, yn ∈ S.
We say that II wins the game iff
x ∈ A ↔ y ∈ B.
The Wadge order
Good news If X = ωω or X = 2ω, we can use games!
Definition Let S ∈ {2, ω} and A, B ⊆ Sω.
I x0 II
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern x1 x2 ··· xn xn+1 ··· x = (xn)n∈ω
s y1 ··· yn s ··· y = (yn)n∈ω
The Wadge game Gw(A, B) where xn, yn ∈ S.
We say that II wins the game iff
x ∈ A ↔ y ∈ B.
The Wadge order
Good news If X = ωω or X = 2ω, we can use games!
Definition Let S ∈ {2, ω} and A, B ⊆ Sω.
I x0
II y0
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern x2 ··· xn xn+1 ··· x = (xn)n∈ω
s y1 ··· yn s ··· y = (yn)n∈ω
The Wadge game Gw(A, B) where xn, yn ∈ S.
We say that II wins the game iff
x ∈ A ↔ y ∈ B.
The Wadge order
Good news If X = ωω or X = 2ω, we can use games!
Definition Let S ∈ {2, ω} and A, B ⊆ Sω.
I x0 x1
II y0
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern x2 ··· xn xn+1 ··· x = (xn)n∈ω
y1 ··· yn s ··· y = (yn)n∈ω
The Wadge game Gw(A, B) where xn, yn ∈ S.
We say that II wins the game iff
x ∈ A ↔ y ∈ B.
The Wadge order
Good news If X = ωω or X = 2ω, we can use games!
Definition Let S ∈ {2, ω} and A, B ⊆ Sω.
I x0 x1
II y0 s
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern ··· xn xn+1 ··· x = (xn)n∈ω
y1 ··· yn s ··· y = (yn)n∈ω
The Wadge game Gw(A, B) where xn, yn ∈ S.
We say that II wins the game iff
x ∈ A ↔ y ∈ B.
The Wadge order
Good news If X = ωω or X = 2ω, we can use games!
Definition Let S ∈ {2, ω} and A, B ⊆ Sω.
I x0 x1 x2
II y0 s
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern ··· xn xn+1 ··· x = (xn)n∈ω
··· yn s ··· y = (yn)n∈ω
The Wadge game Gw(A, B) where xn, yn ∈ S.
We say that II wins the game iff
x ∈ A ↔ y ∈ B.
The Wadge order
Good news If X = ωω or X = 2ω, we can use games!
Definition Let S ∈ {2, ω} and A, B ⊆ Sω.
I x0 x1 x2
II y0 s y1
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern xn xn+1 ··· x = (xn)n∈ω
yn s ··· y = (yn)n∈ω
The Wadge game Gw(A, B) where xn, yn ∈ S.
We say that II wins the game iff
x ∈ A ↔ y ∈ B.
The Wadge order
Good news If X = ωω or X = 2ω, we can use games!
Definition Let S ∈ {2, ω} and A, B ⊆ Sω.
I x0 x1 x2 ···
II y0 s y1 ···
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern xn+1 ··· x = (xn)n∈ω
yn s ··· y = (yn)n∈ω
The Wadge game Gw(A, B) where xn, yn ∈ S.
We say that II wins the game iff
x ∈ A ↔ y ∈ B.
The Wadge order
Good news If X = ωω or X = 2ω, we can use games!
Definition Let S ∈ {2, ω} and A, B ⊆ Sω.
I x0 x1 x2 ··· xn
II y0 s y1 ···
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern xn+1 ··· x = (xn)n∈ω
s ··· y = (yn)n∈ω
The Wadge game Gw(A, B) where xn, yn ∈ S.
We say that II wins the game iff
x ∈ A ↔ y ∈ B.
The Wadge order
Good news If X = ωω or X = 2ω, we can use games!
Definition Let S ∈ {2, ω} and A, B ⊆ Sω.
I x0 x1 x2 ··· xn
II y0 s y1 ··· yn
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern ··· x = (xn)n∈ω
s ··· y = (yn)n∈ω
The Wadge game Gw(A, B) where xn, yn ∈ S.
We say that II wins the game iff
x ∈ A ↔ y ∈ B.
The Wadge order
Good news If X = ωω or X = 2ω, we can use games!
Definition Let S ∈ {2, ω} and A, B ⊆ Sω.
I x0 x1 x2 ··· xn xn+1
II y0 s y1 ··· yn
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern ··· x = (xn)n∈ω
··· y = (yn)n∈ω
The Wadge game Gw(A, B) where xn, yn ∈ S.
We say that II wins the game iff
x ∈ A ↔ y ∈ B.
The Wadge order
Good news If X = ωω or X = 2ω, we can use games!
Definition Let S ∈ {2, ω} and A, B ⊆ Sω.
I x0 x1 x2 ··· xn xn+1
II y0 s y1 ··· yn s
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern x = (xn)n∈ω
y = (yn)n∈ω
The Wadge game Gw(A, B) where xn, yn ∈ S.
We say that II wins the game iff
x ∈ A ↔ y ∈ B.
The Wadge order
Good news If X = ωω or X = 2ω, we can use games!
Definition Let S ∈ {2, ω} and A, B ⊆ Sω.
I x0 x1 x2 ··· xn xn+1 ···
II y0 s y1 ··· yn s ···
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern The Wadge game Gw(A, B) where xn, yn ∈ S.
We say that II wins the game iff
x ∈ A ↔ y ∈ B.
The Wadge order
Good news If X = ωω or X = 2ω, we can use games!
Definition Let S ∈ {2, ω} and A, B ⊆ Sω.
I x0 x1 x2 ··· xn xn+1 ··· x = (xn)n∈ω
II y0 s y1 ··· yn s ··· y = (yn)n∈ω
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern We say that II wins the game iff
x ∈ A ↔ y ∈ B.
The Wadge order
Good news If X = ωω or X = 2ω, we can use games!
Definition Let S ∈ {2, ω} and A, B ⊆ Sω.
I x0 x1 x2 ··· xn xn+1 ··· x = (xn)n∈ω
II y0 s y1 ··· yn s ··· y = (yn)n∈ω
The Wadge game Gw(A, B) where xn, yn ∈ S.
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern The Wadge order
Good news If X = ωω or X = 2ω, we can use games!
Definition Let S ∈ {2, ω} and A, B ⊆ Sω.
I x0 x1 x2 ··· xn xn+1 ··· x = (xn)n∈ω
II y0 s y1 ··· yn s ··· y = (yn)n∈ω
The Wadge game Gw(A, B) where xn, yn ∈ S.
We say that II wins the game iff
x ∈ A ↔ y ∈ B.
Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern A strategy σ is winning if it makes II win whatever I plays.
Theorem (Wadge) If X = ωω or X = 2ω, then
II has a w.s. ⇔ A ≤w B.
Strategies
Definition A strategy for II in the Wadge game Gw(A, B) is a func- tion:
σ : S Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern Theorem (Wadge) If X = ωω or X = 2ω, then II has a w.s. ⇔ A ≤w B. Strategies Definition A strategy for II in the Wadge game Gw(A, B) is a func- tion: σ : S A strategy σ is winning if it makes II win whatever I plays. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern Strategies Definition A strategy for II in the Wadge game Gw(A, B) is a func- tion: σ : S A strategy σ is winning if it makes II win whatever I plays. Theorem (Wadge) If X = ωω or X = 2ω, then II has a w.s. ⇔ A ≤w B. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern Theorem (Wadge’s Lemma, Wadge) ω If A, B are Borel sets and A w B, then B ≤w S \ A. ω In particular, A w B and B w A implies A ≡w S \ B. Theorem (Martin) The quasi-order ≤w is well-founded on the Borel sub- sets of Sω. The Wadge order Theorem (Borel determinacy, Martin) ω If A, B are Borel subsets of S , then Gw(A, B) is deter- mined, i.e., one player has a winning strategy. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern Theorem (Martin) The quasi-order ≤w is well-founded on the Borel sub- sets of Sω. The Wadge order Theorem (Borel determinacy, Martin) ω If A, B are Borel subsets of S , then Gw(A, B) is deter- mined, i.e., one player has a winning strategy. Theorem (Wadge’s Lemma, Wadge) ω If A, B are Borel sets and A w B, then B ≤w S \ A. ω In particular, A w B and B w A implies A ≡w S \ B. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern The Wadge order Theorem (Borel determinacy, Martin) ω If A, B are Borel subsets of S , then Gw(A, B) is deter- mined, i.e., one player has a winning strategy. Theorem (Wadge’s Lemma, Wadge) ω If A, B are Borel sets and A w B, then B ≤w S \ A. ω In particular, A w B and B w A implies A ≡w S \ B. Theorem (Martin) The quasi-order ≤w is well-founded on the Borel sub- sets of Sω. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern [∅]W ··· ··· ··· ω [ω ]W cof (λ) = ω cof (λ) > ω The Wadge order on the Cantor space and the Baire space [∅]W limit levels ··· ··· ··· ω [2 ]W Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern The Wadge order on the Cantor space and the Baire space [∅]W limit levels ··· ··· ··· ω [2 ]W [∅]W ··· ··· ··· ω [ω ]W cof (λ) = ω cof (λ) > ω Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern Theorem (Schlicht) B(X) Let X be a Polish space. Then ( / ≡w, ≤w) is a well- quasi-order if and only if X is zero-dimensional. The Wadge order on Polish spaces Definition Let X be a Polish space. Then X is zero-dimensional if it admits a basis consisting of clopen sets. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern The Wadge order on Polish spaces Definition Let X be a Polish space. Then X is zero-dimensional if it admits a basis consisting of clopen sets. Theorem (Schlicht) B(X) Let X be a Polish space. Then ( / ≡w, ≤w) is a well- quasi-order if and only if X is zero-dimensional. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern The Scott domain Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern Definition The Scott domain Pω is the set Πn∈N2 equipped with the product of the Sierpinski´ topology TS on 2 = {0, 1}, i.e., TS = ∅, {1}, {0, 1} . Remark The topology of the Cantor space 2ω is the topology of positive and negative information, whereas the topol- ogy of the Scott domain Pω is the topology of positive information only. The Scott domain Recall ω The Cantor space 2 is the set Πn∈N2 equipped with the product of the discrete topology on 2 = {0, 1}. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern Remark The topology of the Cantor space 2ω is the topology of positive and negative information, whereas the topol- ogy of the Scott domain Pω is the topology of positive information only. The Scott domain Recall ω The Cantor space 2 is the set Πn∈N2 equipped with the product of the discrete topology on 2 = {0, 1}. Definition The Scott domain Pω is the set Πn∈N2 equipped with the product of the Sierpinski´ topology TS on 2 = {0, 1}, i.e., TS = ∅, {1}, {0, 1} . Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern The Scott domain Recall ω The Cantor space 2 is the set Πn∈N2 equipped with the product of the discrete topology on 2 = {0, 1}. Definition The Scott domain Pω is the set Πn∈N2 equipped with the product of the Sierpinski´ topology TS on 2 = {0, 1}, i.e., TS = ∅, {1}, {0, 1} . Remark The topology of the Cantor space 2ω is the topology of positive and negative information, whereas the topol- ogy of the Scott domain Pω is the topology of positive information only. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern The majority of this paper will be dedicated to showing the naturalness of extending the descriptive set theory of Polish spaces to the class of quasi-Polish spaces. M. de Brecht Remark The Scott domain Pω was the first denotational seman- tic of the λ-calculus. The Scott domain Pω is universal among quasi-Polish spaces. The Scott domain and quasi-Polish spaces Definition A quasi-Polish space is second countable completely quasi-metrizable space, where a quasi-metric is a met- ric whose symmetry condition has been dropped. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern Remark The Scott domain Pω was the first denotational seman- tic of the λ-calculus. The Scott domain Pω is universal among quasi-Polish spaces. The Scott domain and quasi-Polish spaces Definition A quasi-Polish space is second countable completely quasi-metrizable space, where a quasi-metric is a met- ric whose symmetry condition has been dropped. The majority of this paper will be dedicated to showing the naturalness of extending the descriptive set theory of Polish spaces to the class of quasi-Polish spaces. M. de Brecht Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern The Scott domain and quasi-Polish spaces Definition A quasi-Polish space is second countable completely quasi-metrizable space, where a quasi-metric is a met- ric whose symmetry condition has been dropped. The majority of this paper will be dedicated to showing the naturalness of extending the descriptive set theory of Polish spaces to the class of quasi-Polish spaces. M. de Brecht Remark The Scott domain Pω was the first denotational seman- tic of the λ-calculus. The Scott domain Pω is universal among quasi-Polish spaces. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern Theorem (V.) 0 The Wadge order on the ∆2-subsets of Pω has an infi- nite strictly decreasing sequence and an infinite set of pairwise incomparable elements. The Scott domain Problem Understand the shape of the Wadge order on the Scott domain Pω. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern The Scott domain Problem Understand the shape of the Wadge order on the Scott domain Pω. Theorem (V.) 0 The Wadge order on the ∆2-subsets of Pω has an infi- nite strictly decreasing sequence and an infinite set of pairwise incomparable elements. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern In this case, if A, B ⊆ X, A is F-reducible to B, written A ≤F B, whenever there exists a function f ∈ F such that f −1(B) = A. Other reducibility notions Definition If X is a topological space, then F ⊆ {f : X → X} is a reducibility notion if it contains the identity and is closed under composition. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern Other reducibility notions Definition If X is a topological space, then F ⊆ {f : X → X} is a reducibility notion if it contains the identity and is closed under composition. In this case, if A, B ⊆ X, A is F-reducible to B, written A ≤F B, whenever there exists a function f ∈ F such that f −1(B) = A. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern Theorem (V.) There exists a reducibility notion F ⊆ F2 such that the quasi-order ≤F on the Borel subsets of the Scott do- main is well-founded and admits maximal antichains of size 2. In particular, if F 0 is a reducibility notion such that 0 F2 ⊆ F , then the quasi-order ≤F 0 on the Borel subsets of the Scott domain is well-founded and admits maxi- mal antichains of size 2. Other reducibilities n −1 0 0o F2 = f : P(N) → P(N): f (U) ∈ Σ2 for all U ∈ Σ1 . Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern In particular, if F 0 is a reducibility notion such that 0 F2 ⊆ F , then the quasi-order ≤F 0 on the Borel subsets of the Scott domain is well-founded and admits maxi- mal antichains of size 2. Other reducibilities n −1 0 0o F2 = f : P(N) → P(N): f (U) ∈ Σ2 for all U ∈ Σ1 . Theorem (V.) There exists a reducibility notion F ⊆ F2 such that the quasi-order ≤F on the Borel subsets of the Scott do- main is well-founded and admits maximal antichains of size 2. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern Other reducibilities n −1 0 0o F2 = f : P(N) → P(N): f (U) ∈ Σ2 for all U ∈ Σ1 . Theorem (V.) There exists a reducibility notion F ⊆ F2 such that the quasi-order ≤F on the Borel subsets of the Scott do- main is well-founded and admits maximal antichains of size 2. In particular, if F 0 is a reducibility notion such that 0 F2 ⊆ F , then the quasi-order ≤F 0 on the Borel subsets of the Scott domain is well-founded and admits maxi- mal antichains of size 2. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern 2. Investigate other reducibilities; 3. Generalize to other quasi-Polish spaces. Thank you. Further Directions ∆0(Pω) 1. Give a complete description of ( 2 / ≡w, ≤w); Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern 3. Generalize to other quasi-Polish spaces. Thank you. Further Directions ∆0(Pω) 1. Give a complete description of ( 2 / ≡w, ≤w); 2. Investigate other reducibilities; Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern Thank you. Further Directions ∆0(Pω) 1. Give a complete description of ( 2 / ≡w, ≤w); 2. Investigate other reducibilities; 3. Generalize to other quasi-Polish spaces. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern Further Directions ∆0(Pω) 1. Give a complete description of ( 2 / ≡w, ≤w); 2. Investigate other reducibilities; 3. Generalize to other quasi-Polish spaces. Thank you. Descriptive Set Theory and the Wadge order Unil, Paris-VII April 25th, 2019, Bern