Homogeneous Spaces and Wadge Theory

Homogeneous spaces and Wadge theory

Sandra M¨uller

Universit¨atWien

January 2019

joint work with Rapha¨elCarroy and Andrea Medini

Arctic Set Theory Workshop 4

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 1 How I got interested in general topology

Our main tool: Wadge theory

The beauty of Hausdorﬀ operations

Putting everything together

Open questions and future goals

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 2 How I got interested in general topology

Our main tool: Wadge theory

The beauty of Hausdorﬀ operations

Putting everything together

Open questions and future goals

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 3 X X h x

ω Examples of homogeneous spaces: all discrete spaces, Q, 2 , ω ω ≈ R \ Q, all topological groups. We will focus on zero-dimensional homogeneous spaces, i.e. topological spaces which have a base of clopen sets. Fact X is a locally compact zero-dimensional homogeneous space iﬀ X is discrete, X ≈ 2ω, or X ≈ ω × 2ω. We will therefore focus on non-locally compact (equivalently, nowhere compact) zero-dimensional homogeneous spaces.

Homogeneous spaces

All our topological spaces will be separable and metrizable. A homeomorphism between two spaces X and Y is a bijective continuous function f such that the inverse f −1 is continuous as well. Deﬁnition A space X is homogeneous if for every x, y ∈ X there exists a homeomorphism h: X → X such that h(x) = y.

Sandra Müller (UniversitätWien) Homogeneous spaces and Wadge theory January 2019 4 ω Examples of homogeneous spaces: all discrete spaces, Q, 2 , ω ω ≈ R \ Q, all topological groups. We will focus on zero-dimensional homogeneous spaces, i.e. topological spaces which have a base of clopen sets. Fact X is a locally compact zero-dimensional homogeneous space iff X is discrete, X ≈ 2ω, or X ≈ ω × 2ω. We will therefore focus on non-locally compact (equivalently, nowhere compact) zero-dimensional homogeneous spaces.

Homogeneous spaces

X X h x

Sandra Müller (UniversitätWien) Homogeneous spaces and Wadge theory January 2019 4 We will focus on zero-dimensional homogeneous spaces, i.e. topological spaces which have a base of clopen sets. Fact X is a locally compact zero-dimensional homogeneous space iff X is discrete, X ≈ 2ω, or X ≈ ω × 2ω. We will therefore focus on non-locally compact (equivalently, nowhere compact) zero-dimensional homogeneous spaces.

Homogeneous spaces

X X h x

ω Examples of homogeneous spaces: all discrete spaces, Q, 2 , ω ω ≈ R \ Q, all topological groups.

Sandra Müller (UniversitätWien) Homogeneous spaces and Wadge theory January 2019 4 Fact X is a locally compact zero-dimensional homogeneous space iff X is discrete, X ≈ 2ω, or X ≈ ω × 2ω.

We will therefore focus on non-locally compact (equivalently, nowhere compact) zero-dimensional homogeneous spaces.

Homogeneous spaces

Deﬁnition A space X is homogeneous if for every x, y ∈ X there exists a homeomorphism h: X → X such that h(x) = y.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 4 Homogeneous spaces

Deﬁnition A space X is homogeneous if for every x, y ∈ X there exists a homeomorphism h: X → X such that h(x) = y.

Fact X is a locally compact zero-dimensional homogeneous space iﬀ X is discrete, X ≈ 2ω, or X ≈ ω × 2ω.

We will therefore focus on non-locally compact (equivalently, nowhere compact) zero-dimensional homogeneous spaces.

Sandra Müller (UniversitätWien) Homogeneous spaces and Wadge theory January 2019 4 Fact A zero-dimensional space X is h-homogeneous iff for all non-empty clopen proper subsets U, V of X there is a homeomorphism h: X → X such that h[U] = V .

X X Examples of h-homogeneous spaces: U h V ω ω Q, 2 , ω , any product of zero-dimensional h-homogeneous spaces (Medini, 2011)

h-homogeneity

Deﬁnition A space X is h-homogeneous if every non-empty clopen subset U of X (with the subspace topology) is homeomorphic to X.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 5 Examples of h-homogeneous spaces: ω ω Q, 2 , ω , any product of zero-dimensional h-homogeneous spaces (Medini, 2011)

h-homogeneity

Deﬁnition A space X is h-homogeneous if every non-empty clopen subset U of X (with the subspace topology) is homeomorphic to X.

Fact A zero-dimensional space X is h-homogeneous iﬀ for all non-empty clopen proper subsets U, V of X there is a homeomorphism h: X → X such that h[U] = V .

X X h U V

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 5 h-homogeneity

Deﬁnition A space X is h-homogeneous if every non-empty clopen subset U of X (with the subspace topology) is homeomorphic to X.

Fact A zero-dimensional space X is h-homogeneous iﬀ for all non-empty clopen proper subsets U, V of X there is a homeomorphism h: X → X such that h[U] = V .

X X Examples of h-homogeneous spaces: U h V ω ω Q, 2 , ω , any product of zero-dimensional h-homogeneous spaces (Medini, 2011)

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 5 Proof by picture. h0

x y

S −1 Now n∈ω(hn ∪ hn ) can be extended to a homeomorphism h: X → X such that h(x) = y and h−1(y) = x.

h-homogeneity versus homogeneity

Theorem (Folklore) Assume that X is a zero-dimensional space. If X is h-homogeneous, then X is homogeneous.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 6 h0

x y

S −1 Now n∈ω(hn ∪ hn ) can be extended to a homeomorphism h: X → X such that h(x) = y and h−1(y) = x.

h-homogeneity versus homogeneity

Theorem (Folklore) Assume that X is a zero-dimensional space. If X is h-homogeneous, then X is homogeneous.

Proof by picture.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 6 h0

x y

S −1 Now n∈ω(hn ∪ hn ) can be extended to a homeomorphism h: X → X such that h(x) = y and h−1(y) = x.

h-homogeneity versus homogeneity

Theorem (Folklore) Assume that X is a zero-dimensional space. If X is h-homogeneous, then X is homogeneous.

Proof by picture.

x y

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 6 h0

x y

S −1 Now n∈ω(hn ∪ hn ) can be extended to a homeomorphism h: X → X such that h(x) = y and h−1(y) = x.

h-homogeneity versus homogeneity

Theorem (Folklore) Assume that X is a zero-dimensional space. If X is h-homogeneous, then X is homogeneous.

Proof by picture.

x y

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 6 h0

x y

S −1 Now n∈ω(hn ∪ hn ) can be extended to a homeomorphism h: X → X such that h(x) = y and h−1(y) = x.

h-homogeneity versus homogeneity

Theorem (Folklore) Assume that X is a zero-dimensional space. If X is h-homogeneous, then X is homogeneous.

Proof by picture.

x y

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 6 h0

x y

S −1 Now n∈ω(hn ∪ hn ) can be extended to a homeomorphism h: X → X such that h(x) = y and h−1(y) = x.

h-homogeneity versus homogeneity

Theorem (Folklore) Assume that X is a zero-dimensional space. If X is h-homogeneous, then X is homogeneous.

Proof by picture.

x y

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 6 h1

x y

S −1 Now n∈ω(hn ∪ hn ) can be extended to a homeomorphism h: X → X such that h(x) = y and h−1(y) = x.

h-homogeneity versus homogeneity

Theorem (Folklore) Assume that X is a zero-dimensional space. If X is h-homogeneous, then X is homogeneous.

Proof by picture. h0

x y

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 6 S −1 Now n∈ω(hn ∪ hn ) can be extended to a homeomorphism h: X → X such that h(x) = y and h−1(y) = x.

h-homogeneity versus homogeneity

Theorem (Folklore) Assume that X is a zero-dimensional space. If X is h-homogeneous, then X is homogeneous.

Proof by picture. h0

x y

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 6 S −1 Now n∈ω(hn ∪ hn ) can be extended to a homeomorphism h: X → X such that h(x) = y and h−1(y) = x.

h-homogeneity versus homogeneity

Theorem (Folklore) Assume that X is a zero-dimensional space. If X is h-homogeneous, then X is homogeneous.

Proof by picture. h0

x y

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 6 h-homogeneity versus homogeneity

Theorem (Folklore) Assume that X is a zero-dimensional space. If X is h-homogeneous, then X is homogeneous.

x y

S −1 Now n∈ω(hn ∪ hn ) can be extended to a homeomorphism h: X → X such that h(x) = y and h−1(y) = x.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 6 Theorem (van Engelen, 1986) A Borel non-locally-compact subspace of 2ω is homogeneous if and only if it is h-homogeneous.

Question Can we say more under projective determinacy (PD) or AD?

h-homogeneity versus homogeneity

Theorem (Folklore) Assume that X is a zero-dimensional space. If X is h-homogeneous, then X is homogeneous.

But the converse does not hold in general. Theorem (van Mill, 1992) (AC) There exists a zero-dimensional homogeneous space that is not h-homogeneous.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 6 Question Can we say more under projective determinacy (PD) or AD?

h-homogeneity versus homogeneity

Theorem (Folklore) Assume that X is a zero-dimensional space. If X is h-homogeneous, then X is homogeneous.

But the converse does not hold in general. Theorem (van Mill, 1992) (AC) There exists a zero-dimensional homogeneous space that is not h-homogeneous.

Theorem (van Engelen, 1986) A Borel non-locally-compact subspace of 2ω is homogeneous if and only if it is h-homogeneous.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 6 h-homogeneity versus homogeneity

Theorem (Folklore) Assume that X is a zero-dimensional space. If X is h-homogeneous, then X is homogeneous.

But the converse does not hold in general. Theorem (van Mill, 1992) (AC) There exists a zero-dimensional homogeneous space that is not h-homogeneous.

Theorem (van Engelen, 1986) A Borel non-locally-compact subspace of 2ω is homogeneous if and only if it is h-homogeneous.

Question Can we say more under projective determinacy (PD) or AD?

Sandra Müller (UniversitätWien) Homogeneous spaces and Wadge theory January 2019 6 Main ideas to extend van Engelen’s result beyond Borel spaces: The proof relies on an extremely refined classification of subsets of a Polish zero-dimensional space: the Wadge quasi-order. Need an understanding of the induced Wadge hierarchy in ωω beyond Borel classes. Need a method of transferring these results from ωω to 2ω. Want to apply a theorem of Steel in 2ω.

Yes!

Theorem (Carroy – Medini – M) (PD) A projective non-locally-compact subspace of 2ω is homogeneous if and only if it is h-homogeneous. (AD+DC) A non-locally-compact subspace of 2ω is homogeneous if and only if it is h-homogeneous.

Sandra Müller (UniversitätWien) Homogeneous spaces and Wadge theory January 2019 7 The proof relies on an extremely refined classification of subsets of a Polish zero-dimensional space: the Wadge quasi-order. Need an understanding of the induced Wadge hierarchy in ωω beyond Borel classes. Need a method of transferring these results from ωω to 2ω. Want to apply a theorem of Steel in 2ω.

Yes!

Main ideas to extend van Engelen’s result beyond Borel spaces:

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 7 Need an understanding of the induced Wadge hierarchy in ωω beyond Borel classes. Need a method of transferring these results from ωω to 2ω. Want to apply a theorem of Steel in 2ω.

Yes!

Main ideas to extend van Engelen’s result beyond Borel spaces: The proof relies on an extremely reﬁned classiﬁcation of subsets of a Polish zero-dimensional space: the Wadge quasi-order.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 7 Need a method of transferring these results from ωω to 2ω. Want to apply a theorem of Steel in 2ω.

Yes!

Main ideas to extend van Engelen’s result beyond Borel spaces: The proof relies on an extremely reﬁned classiﬁcation of subsets of a Polish zero-dimensional space: the Wadge quasi-order. Need an understanding of the induced Wadge hierarchy in ωω beyond Borel classes.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 7 Want to apply a theorem of Steel in 2ω.

Yes!

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 7 Yes!

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 7 How I got interested in general topology

Our main tool: Wadge theory

The beauty of Hausdorﬀ operations

Putting everything together

Open questions and future goals

Sandra Müller (UniversitätWien) Homogeneous spaces and Wadge theory January 2019 8 This is reflexive and transitive, it is a quasi-order. [A]W = {B | B ≤W A}. A pointclass Γ is a class of subsets closed under continuous preimages. c A is self-dual if A ≡W A , otherwise it is non-self-dual. This yields (under AD + DC) a very nice hierarchy of subsets of ωω. Theorem (Wadge, Martin – Monk)

Assuming AD and DC, ≤W satisﬁes the semi-well-ordering principle: ω ω (Wadge) For any A, B ⊆ ω either A ≤W B or B ≤W ω \ A.

(Martin – Monk) The quasi-order ≤W is well-founded.

Wadge reducibility

Deﬁnition For A ⊆ X and B ⊆ Y , we say that A Wadge (or continuously) reduces to B if there is a continuous function f : X → Y such that ω x ∈ A ⇔ f(x) ∈ B. For X = Y = ω , we write A ≤W B.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 9 [A]W = {B | B ≤W A}. A pointclass Γ is a class of subsets closed under continuous preimages. c A is self-dual if A ≡W A , otherwise it is non-self-dual. This yields (under AD + DC) a very nice hierarchy of subsets of ωω. Theorem (Wadge, Martin – Monk)

Assuming AD and DC, ≤W satisﬁes the semi-well-ordering principle: ω ω (Wadge) For any A, B ⊆ ω either A ≤W B or B ≤W ω \ A.

(Martin – Monk) The quasi-order ≤W is well-founded.

Wadge reducibility

This is reﬂexive and transitive, it is a quasi-order.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 9 A pointclass Γ is a class of subsets closed under continuous preimages. c A is self-dual if A ≡W A , otherwise it is non-self-dual. This yields (under AD + DC) a very nice hierarchy of subsets of ωω. Theorem (Wadge, Martin – Monk)

Assuming AD and DC, ≤W satisﬁes the semi-well-ordering principle: ω ω (Wadge) For any A, B ⊆ ω either A ≤W B or B ≤W ω \ A.

(Martin – Monk) The quasi-order ≤W is well-founded.

Wadge reducibility

This is reﬂexive and transitive, it is a quasi-order. [A]W = {B | B ≤W A}.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 9 c A is self-dual if A ≡W A , otherwise it is non-self-dual. This yields (under AD + DC) a very nice hierarchy of subsets of ωω. Theorem (Wadge, Martin – Monk)

Assuming AD and DC, ≤W satisﬁes the semi-well-ordering principle: ω ω (Wadge) For any A, B ⊆ ω either A ≤W B or B ≤W ω \ A.

(Martin – Monk) The quasi-order ≤W is well-founded.

Wadge reducibility

This is reﬂexive and transitive, it is a quasi-order. [A]W = {B | B ≤W A}. A pointclass Γ is a class of subsets closed under continuous preimages.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 9 This yields (under AD + DC) a very nice hierarchy of subsets of ωω. Theorem (Wadge, Martin – Monk)

Assuming AD and DC, ≤W satisﬁes the semi-well-ordering principle: ω ω (Wadge) For any A, B ⊆ ω either A ≤W B or B ≤W ω \ A.

(Martin – Monk) The quasi-order ≤W is well-founded.

Wadge reducibility

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 9 Wadge reducibility

This is reﬂexive and transitive, it is a quasi-order. [A]W = {B | B ≤W A}. A pointclass Γ is a class of subsets closed under continuous preimages. c A is self-dual if A ≡W A , otherwise it is non-self-dual. This yields (under AD + DC) a very nice hierarchy of subsets of ωω. Theorem (Wadge, Martin – Monk)

Assuming AD and DC, ≤W satisﬁes the semi-well-ordering principle: ω ω (Wadge) For any A, B ⊆ ω either A ≤W B or B ≤W ω \ A.

(Martin – Monk) The quasi-order ≤W is well-founded.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 9 Σ0 0 2 open sets Σ1 (for rank ω1)

0 clopen sets ∆1

0 Π2

0 closed sets Π1 cof = ω cof > ω

The length of the Wadge hierarchy of Borel sets is an ordinal of coﬁnality ω1 but strictly smaller than ω2. The length of the full Wadge hierarchy is Θ = sup{α | ∃f(f : R α)}.

Picture of the Wadge hierarchy

ω ω Recall: A ≤W B iﬀ there is a continuous function f : ω → ω such that x ∈ A ⇔ f(x) ∈ B. Write [A]W = {B | B ≤W A}.

ω ω [ω ]W = {ω }

[∅]W = {∅}

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 10 Σ0 0 2 open sets Σ1 (for rank ω1)

0 Π2

0 closed sets Π1 cof = ω cof > ω

The length of the Wadge hierarchy of Borel sets is an ordinal of coﬁnality ω1 but strictly smaller than ω2. The length of the full Wadge hierarchy is Θ = sup{α | ∃f(f : R α)}.

Picture of the Wadge hierarchy

ω ω Recall: A ≤W B iﬀ there is a continuous function f : ω → ω such that x ∈ A ⇔ f(x) ∈ B. Write [A]W = {B | B ≤W A}.

ω ω [ω ]W = {ω }

0 clopen sets ∆1

[∅]W = {∅}

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 10 0 Σ2 (for rank ω1)

0 Π2 cof = ω cof > ω

The length of the Wadge hierarchy of Borel sets is an ordinal of coﬁnality ω1 but strictly smaller than ω2. The length of the full Wadge hierarchy is Θ = sup{α | ∃f(f : R α)}.

Picture of the Wadge hierarchy

ω ω Recall: A ≤W B iﬀ there is a continuous function f : ω → ω such that x ∈ A ⇔ f(x) ∈ B. Write [A]W = {B | B ≤W A}.

0 open sets Σ1 ω ω [ω ]W = {ω }

0 clopen sets ∆1

[∅]W = {∅} 0 closed sets Π1

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 10 0 Σ2 (for rank ω1)

0 Π2 cof = ω cof > ω

The length of the Wadge hierarchy of Borel sets is an ordinal of coﬁnality ω1 but strictly smaller than ω2. The length of the full Wadge hierarchy is Θ = sup{α | ∃f(f : R α)}.

Picture of the Wadge hierarchy

ω ω Recall: A ≤W B iﬀ there is a continuous function f : ω → ω such that x ∈ A ⇔ f(x) ∈ B. Write [A]W = {B | B ≤W A}.

0 open sets Σ1 ω ω [ω ]W = {ω }

0 clopen sets ∆1

[∅]W = {∅} 0 closed sets Π1

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 10 0 Σ2 (for rank ω1)

0 Π2 cof > ω

The length of the Wadge hierarchy of Borel sets is an ordinal of coﬁnality ω1 but strictly smaller than ω2. The length of the full Wadge hierarchy is Θ = sup{α | ∃f(f : R α)}.

Picture of the Wadge hierarchy

ω ω Recall: A ≤W B iﬀ there is a continuous function f : ω → ω such that x ∈ A ⇔ f(x) ∈ B. Write [A]W = {B | B ≤W A}.

0 open sets Σ1 ω ω [ω ]W = {ω }

0 clopen sets ∆1

[∅]W = {∅} 0 closed sets Π1 cof = ω

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 10 0 Σ2 (for rank ω1)

0 Π2

The length of the Wadge hierarchy of Borel sets is an ordinal of coﬁnality ω1 but strictly smaller than ω2. The length of the full Wadge hierarchy is Θ = sup{α | ∃f(f : R α)}.

Picture of the Wadge hierarchy

ω ω Recall: A ≤W B iﬀ there is a continuous function f : ω → ω such that x ∈ A ⇔ f(x) ∈ B. Write [A]W = {B | B ≤W A}.

0 open sets Σ1 ω ω [ω ]W = {ω }

0 clopen sets ∆1

[∅]W = {∅} 0 closed sets Π1 cof = ω cof > ω

Sandra Müller (UniversitätWien) Homogeneous spaces and Wadge theory January 2019 10 The length of the Wadge hierarchy of Borel sets is an ordinal of cofinality ω1 but strictly smaller than ω2. The length of the full Wadge hierarchy is Θ = sup{α | ∃f(f : R α)}.

Picture of the Wadge hierarchy

ω ω Recall: A ≤W B iﬀ there is a continuous function f : ω → ω such that x ∈ A ⇔ f(x) ∈ B. Write [A]W = {B | B ≤W A}.

Σ0 0 2 open sets Σ1 (for rank ω1) ω ω [ω ]W = {ω }

0 clopen sets ∆1

0 Π2 [∅]W = {∅} 0 closed sets Π1 cof = ω cof > ω

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 10 The length of the full Wadge hierarchy is Θ = sup{α | ∃f(f : R α)}.

Picture of the Wadge hierarchy

ω ω Recall: A ≤W B iﬀ there is a continuous function f : ω → ω such that x ∈ A ⇔ f(x) ∈ B. Write [A]W = {B | B ≤W A}.

Σ0 0 2 open sets Σ1 (for rank ω1) ω ω [ω ]W = {ω }

0 clopen sets ∆1

0 Π2 [∅]W = {∅} 0 closed sets Π1 cof = ω cof > ω

The length of the Wadge hierarchy of Borel sets is an ordinal of coﬁnality ω1 but strictly smaller than ω2.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 10 Picture of the Wadge hierarchy

ω ω Recall: A ≤W B iﬀ there is a continuous function f : ω → ω such that x ∈ A ⇔ f(x) ∈ B. Write [A]W = {B | B ≤W A}.

Σ0 0 2 open sets Σ1 (for rank ω1) ω ω [ω ]W = {ω }

0 clopen sets ∆1

0 Π2 [∅]W = {∅} 0 closed sets Π1 cof = ω cof > ω

The length of the Wadge hierarchy of Borel sets is an ordinal of coﬁnality ω1 but strictly smaller than ω2. The length of the full Wadge hierarchy is Θ = sup{α | ∃f(f : R α)}.

Sandra Müller (UniversitätWien) Homogeneous spaces and Wadge theory January 2019 10 S Call PUα(Γ) the class of all sets of the form An ∩ Bn, where An ∈ Γ, 0 and (Bn)n is a ∆1+α partition. Definition (Louveau – Saint-Raymond)

The level of a pointclass Γ is `(Γ) = sup{α < ω1 | Γ = PUα(Γ)}.

Given a pointclass Γ and a countable ordinal α, the α-expansion Γ(α) is 0 the class of all preimages of elements of Γ by Σ1+α-measurable functions. Theorem (Expansion Theorem, Saint-Raymond) (AD + DC) Let Γ be a non-self-dual Wadge pointclass and α a countable ordinal. Then the following are equivalent: 1 `(Γ) ≥ α, 2 Γ = Λ(α) for some non-self-dual Wadge class Λ.

Levels and expansion

There is a method of jumping through the hierarchy.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 11 Given a pointclass Γ and a countable ordinal α, the α-expansion Γ(α) is 0 the class of all preimages of elements of Γ by Σ1+α-measurable functions. Theorem (Expansion Theorem, Saint-Raymond) (AD + DC) Let Γ be a non-self-dual Wadge pointclass and α a countable ordinal. Then the following are equivalent: 1 `(Γ) ≥ α, 2 Γ = Λ(α) for some non-self-dual Wadge class Λ.

Levels and expansion

There is a method of jumping through the hierarchy. S Call PUα(Γ) the class of all sets of the form An ∩ Bn, where An ∈ Γ, 0 and (Bn)n is a ∆1+α partition. Deﬁnition (Louveau – Saint-Raymond)

The level of a pointclass Γ is `(Γ) = sup{α < ω1 | Γ = PUα(Γ)}.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 11 Levels and expansion

The level of a pointclass Γ is `(Γ) = sup{α < ω1 | Γ = PUα(Γ)}.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 11 How I got interested in general topology

Our main tool: Wadge theory

The beauty of Hausdorﬀ operations

Putting everything together

Open questions and future goals

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 12 Some properties: ω ~ If D = {(1) }, then H{(1)ω}(A) is the countable intersection A0 ∩ A1 ∩ ... .

If Dn is the set of all s: ω → 2 such that s(n) = 1 (s(k) for k 6= n

arbitrary), then HDn (A0,A1,...) = An. T T i∈I HDi (A0,A1,...) = HD(A0,A1,...), where D = i∈I Di. S S i∈I HDi (A0,A1,...) = HD(A0,A1,...), where D = i∈I Di. X \HD(A0,A1,...) = H2ω\D(A0,A1,...). In particular, every combination of unions, intersections, and complements can be expressed as a Hausdorﬀ operation.

Hausdorﬀ operations

ω Given D ⊆ 2 and a sequence of sets A~ = A0,A1, ··· , deﬁne a set HD(A~) as follows: x ∈ HD(A~) ↔ {i ∈ ω | x ∈ Ai} ∈ D We call HD a Hausdorﬀ operation.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 13 If Dn is the set of all s: ω → 2 such that s(n) = 1 (s(k) for k 6= n

Hausdorﬀ operations

ω Given D ⊆ 2 and a sequence of sets A~ = A0,A1, ··· , deﬁne a set HD(A~) as follows: x ∈ HD(A~) ↔ {i ∈ ω | x ∈ Ai} ∈ D We call HD a Hausdorﬀ operation. Some properties: ω ~ If D = {(1) }, then H{(1)ω}(A) is the countable intersection A0 ∩ A1 ∩ ... .

Sandra Müller (UniversitätWien) Homogeneous spaces and Wadge theory January 2019 13 T T i∈I HDi (A0,A1,...) = HD(A0,A1,...), where D = i∈I Di. S S i∈I HDi (A0,A1,...) = HD(A0,A1,...), where D = i∈I Di. X \HD(A0,A1,...) = H2ω\D(A0,A1,...). In particular, every combination of unions, intersections, and complements can be expressed as a Hausdorff operation.

Hausdorﬀ operations

If Dn is the set of all s: ω → 2 such that s(n) = 1 (s(k) for k 6= n

arbitrary), then HDn (A0,A1,...) = An.

Sandra Müller (UniversitätWien) Homogeneous spaces and Wadge theory January 2019 13 S S i∈I HDi (A0,A1,...) = HD(A0,A1,...), where D = i∈I Di. X \HD(A0,A1,...) = H2ω\D(A0,A1,...). In particular, every combination of unions, intersections, and complements can be expressed as a Hausdorff operation.

Hausdorﬀ operations

If Dn is the set of all s: ω → 2 such that s(n) = 1 (s(k) for k 6= n

arbitrary), then HDn (A0,A1,...) = An. T T i∈I HDi (A0,A1,...) = HD(A0,A1,...), where D = i∈I Di.

Sandra Müller (UniversitätWien) Homogeneous spaces and Wadge theory January 2019 13 X \HD(A0,A1,...) = H2ω\D(A0,A1,...). In particular, every combination of unions, intersections, and complements can be expressed as a Hausdorff operation.

Hausdorﬀ operations

If Dn is the set of all s: ω → 2 such that s(n) = 1 (s(k) for k 6= n

arbitrary), then HDn (A0,A1,...) = An. T T i∈I HDi (A0,A1,...) = HD(A0,A1,...), where D = i∈I Di. S S i∈I HDi (A0,A1,...) = HD(A0,A1,...), where D = i∈I Di.

Sandra Müller (UniversitätWien) Homogeneous spaces and Wadge theory January 2019 13 In particular, every combination of unions, intersections, and complements can be expressed as a Hausdorff operation.

Hausdorﬀ operations

If Dn is the set of all s: ω → 2 such that s(n) = 1 (s(k) for k 6= n

Sandra Müller (UniversitätWien) Homogeneous spaces and Wadge theory January 2019 13 Hausdorff operations

If Dn is the set of all s: ω → 2 such that s(n) = 1 (s(k) for k 6= n

Sandra Müller (UniversitätWien) Homogeneous spaces and Wadge theory January 2019 13 Theorem (Addison, van Wesep) ω ω (AD + DC) Γ is a non-self-dual Wadge class in 2 iff Γ = ΓD(2 ) for some D ⊆ 2ω.

This in fact works for all Polish zero-dimensional spaces X instead of 2ω.

From ωω to 2ω

ω For D ⊆ 2 , deﬁne ΓD(X) as the collection of all subsets of X that are the result of applying HD on open sets of X. Lemma (Relativization Lemma) Given two spaces X and Y , and D ⊆ 2ω. −1 If f : X → Y is continuous and A ∈ ΓD(Y ) then f [A] ∈ ΓD(X).

Assume Y ⊆ X, then A ∈ ΓD(Y ) if and only if there is A˜ ∈ ΓD(X) such that A = A˜ ∩ Y .

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 14 From ωω to 2ω

Assume Y ⊆ X, then A ∈ ΓD(Y ) if and only if there is A˜ ∈ ΓD(X) such that A = A˜ ∩ Y .

Theorem (Addison, van Wesep) ω ω (AD + DC) Γ is a non-self-dual Wadge class in 2 iﬀ Γ = ΓD(2 ) for some D ⊆ 2ω.

This in fact works for all Polish zero-dimensional spaces X instead of 2ω.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 14 How I got interested in general topology

Our main tool: Wadge theory

The beauty of Hausdorﬀ operations

Putting everything together

Open questions and future goals

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 15 A is everywhere properly Γ if for all s ∈ 2<ω, A ∩ [s] ∈ Γ \ Γˇ, where Γˇ = {2ω \ X | X ∈ Γ}.

Theorem (Steel, 1980) (AD + DC) Let Γ be a reasonably closed Wadge class of subsets of 2ω. Take X,Y ⊆ 2ω such that both X and Y are everywhere properly Γ, and either they are both meager, or both Baire. Then there is a homeomorphism h: 2ω → 2ω such that h[X] = Y .

Steel’s theorem ...

Let A ⊆ 2ω and let Γ be a pointclass. For s ∈ 2<ω, let [s] = {x ∈ 2ω | s ⊆ x}. Say that 0 0 Γ is reasonably closed if it is closed under ∩Π2 and ∪Σ2.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 16 Theorem (Steel, 1980) (AD + DC) Let Γ be a reasonably closed Wadge class of subsets of 2ω. Take X,Y ⊆ 2ω such that both X and Y are everywhere properly Γ, and either they are both meager, or both Baire. Then there is a homeomorphism h: 2ω → 2ω such that h[X] = Y .

Steel’s theorem ...

Let A ⊆ 2ω and let Γ be a pointclass. For s ∈ 2<ω, let [s] = {x ∈ 2ω | s ⊆ x}. Say that 0 0 Γ is reasonably closed if it is closed under ∩Π2 and ∪Σ2. A is everywhere properly Γ if for all s ∈ 2<ω, A ∩ [s] ∈ Γ \ Γˇ, where Γˇ = {2ω \ X | X ∈ Γ}.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 16 Steel’s theorem ...

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 16 Every homogeneous space is either meager or Baire ω So if X ⊆ 2 is homogeneous, Γ = [X]W is reasonably closed and X is everywhere properly Γ, then for any X ∩ [s] for s ∈ 2<ω we can apply Steel’s theorem. I.e. X and X ∩ [s] are homeomorphic. A result of Terada (1993) yields that X is h-homogeneous.

... and how we are going to use it

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 17 ω So if X ⊆ 2 is homogeneous, Γ = [X]W is reasonably closed and X is everywhere properly Γ, then for any X ∩ [s] for s ∈ 2<ω we can apply Steel’s theorem. I.e. X and X ∩ [s] are homeomorphic. A result of Terada (1993) yields that X is h-homogeneous.

... and how we are going to use it

Every homogeneous space is either meager or Baire

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 17 I.e. X and X ∩ [s] are homeomorphic. A result of Terada (1993) yields that X is h-homogeneous.

... and how we are going to use it

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 17 ... and how we are going to use it

Every homogeneous space is either meager or Baire ω So if X ⊆ 2 is homogeneous, Γ = [X]W is reasonably closed and X is everywhere properly Γ, then for any X ∩ [s] for s ∈ 2<ω we can apply Steel’s theorem. I.e. X and X ∩ [s] are homeomorphic. A result of Terada (1993) yields that X is h-homogeneous.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 17 ... and how we are going to use it

Corollary (AD + DC) If X ⊆ 2ω is homogeneous, generates a reasonably closed Wadge class Γ and X is everywhere properly Γ, then X is h-homogeneous.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 17 Proposition A good class is reasonably closed.

Theorem ω 0 Let X ⊆ 2 . If X/∈ ∆Dω(Σ2) is homogeneous, then [X]W is good.

0 (Recall: The case X ∈ ∆Dω(Σ2) was analyzed by van Engelen already.) Theorem (AD+DC) A non-locally-compact subspace of 2ω is homogeneous if and only if it is h-homogeneous.

Good Wadge classes are reasonably closed...

A pointclass Γ in 2ω is good if 0 ∆Dω(Σ2) ⊆ Γ, Γ is non-self-dual, and `(Γ) ≥ 1.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 18 Theorem ω 0 Let X ⊆ 2 . If X/∈ ∆Dω(Σ2) is homogeneous, then [X]W is good.

0 (Recall: The case X ∈ ∆Dω(Σ2) was analyzed by van Engelen already.) Theorem (AD+DC) A non-locally-compact subspace of 2ω is homogeneous if and only if it is h-homogeneous.

Good Wadge classes are reasonably closed...

A pointclass Γ in 2ω is good if 0 ∆Dω(Σ2) ⊆ Γ, Γ is non-self-dual, and `(Γ) ≥ 1. Proposition A good class is reasonably closed.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 18 Theorem (AD+DC) A non-locally-compact subspace of 2ω is homogeneous if and only if it is h-homogeneous.

...and almost all homogeneous spaces are good

A pointclass Γ in 2ω is good if 0 ∆Dω(Σ2) ⊆ Γ, Γ is non-self-dual, and `(Γ) ≥ 1. Proposition A good class is reasonably closed.

Theorem ω 0 Let X ⊆ 2 . If X/∈ ∆Dω(Σ2) is homogeneous, then [X]W is good.

0 (Recall: The case X ∈ ∆Dω(Σ2) was analyzed by van Engelen already.)

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 18 ...and almost all homogeneous spaces are good

A pointclass Γ in 2ω is good if 0 ∆Dω(Σ2) ⊆ Γ, Γ is non-self-dual, and `(Γ) ≥ 1. Proposition A good class is reasonably closed.

Theorem ω 0 Let X ⊆ 2 . If X/∈ ∆Dω(Σ2) is homogeneous, then [X]W is good.

0 (Recall: The case X ∈ ∆Dω(Σ2) was analyzed by van Engelen already.) Theorem (AD+DC) A non-locally-compact subspace of 2ω is homogeneous if and only if it is h-homogeneous.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 18 How I got interested in general topology

Our main tool: Wadge theory

The beauty of Hausdorﬀ operations

Putting everything together

Open questions and future goals

Sandra Müller (UniversitätWien) Homogeneous spaces and Wadge theory January 2019 19 Theorem (van Engelen, 1994) Let X be a zero-dimensional Borel space. Then the following are equivalent X is homeomorphic to a filter. X is homogeneous, meager, homeomorphic to its square, and not locally compact.

Question Can this be generalized to all zero-dimensional projective spaces (under PD), or all zero-dimensional spaces (under AD + DC)?

Van Engelen’s characterization of Borel ﬁlters

As topological groups, all ﬁlters are homogeneous, but there is a characterization for Borel spaces.

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 20 Question Can this be generalized to all zero-dimensional projective spaces (under PD), or all zero-dimensional spaces (under AD + DC)?

Van Engelen’s characterization of Borel ﬁlters

As topological groups, all ﬁlters are homogeneous, but there is a characterization for Borel spaces. Theorem (van Engelen, 1994) Let X be a zero-dimensional Borel space. Then the following are equivalent X is homeomorphic to a ﬁlter. X is homogeneous, meager, homeomorphic to its square, and not locally compact.

Sandra Müller (UniversitätWien) Homogeneous spaces and Wadge theory January 2019 20 Van Engelen’s characterization of Borel filters

Question Can this be generalized to all zero-dimensional projective spaces (under PD), or all zero-dimensional spaces (under AD + DC)?

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 20 “The Wadge hierarchy is the ultimate analysis of P(ωω) in terms of topological complexity [...]”

(Andretta and Louveau in the Introduction to Cabal Part III)

Thank you for your attention!

Sandra M¨uller (Universit¨atWien) Homogeneous spaces and Wadge theory January 2019 21