[PPT] - Introduction to Descriptive Set Theory (MATH40350) Dr Richard Smith PowerPoint Presentation

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Introduction to Descriptive Set Theory (MATH40350)

Dr Richard Smith (http://maths.ucd.ie/~rsmith)

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 1 / 31

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Metric spaces Polish spaces

Separability and 2nd countability

Theorem 1.2.10 (TOP ×)

Let X be a metric space. Then X is 2nd countable if and only if it is separable.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 2 / 31

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Metric spaces Polish spaces

Separability and 2nd countability

Theorem 1.2.10 (TOP ×)

Let X be a metric space. Then X is 2nd countable if and only if it is separable.

Corollary 1.2.11 (TOP ×)

Let X be a separable metric space. Then every subspace of X is also separa- ble.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 2 / 31

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Metric spaces Polish spaces

Cantor’s intersection theorem

Theorem 1.2.17

Let X be completely metrisable, with compatible metric d. Suppose that (Fn) is a sequence of closed non-empty subsets of X satisfying

1

Fn+1 ⊆ Fn (i.e. (Fn) is decreasing)

2

diam (Fn) → 0 as n → ∞. Then the intersection F = ∞

n=1 Fn is a singleton.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 3 / 31

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Metric spaces Building new Polish spaces from old

Examples of Gδ sets

Example 1.3.8

1

Any open subset of a metric space is a Gδ.

2

R \ Q is clearly not open, but it is a Gδ. Enumerate Q as (qn) and put Un = R \ {qn}, then each Un is open and R \ Q = ∞

n=0 Un.

3

(TOP ×) Any closed subset F of a metric space is a Gδ. Define Un =

x ∈ X : d(x, y) < 2−n for some y ∈ F
.

Each Un is open and F = ∞

n=0 Un.

4

The set of points of continuity of an arbitrary function f : R − → R is a Gδ (Question 2, Exercise Sheet 1).

5

Q is not a Gδ in R (same question).

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 4 / 31

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Metric spaces Building new Polish spaces from old

Polish subspaces

Proposition 1.3.1

Let X be completely metrisable and take closed Y ⊆ X. Then Y is completely

metrisable. In particular, any closed subspace of a Polish space is Polish, by

Corollary 1.2.11.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 5 / 31

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Metric spaces Building new Polish spaces from old

Polish subspaces

Proposition 1.3.1

Let X be completely metrisable and take closed Y ⊆ X. Then Y is completely

metrisable. In particular, any closed subspace of a Polish space is Polish, by

Corollary 1.2.11.

Proposition 1.3.2

Let X be completely metrisable and let U ⊆ X be open. Then U is completely

metrisable. In particular, any open subset of a Polish space is Polish, by Corol-

lary 1.2.11.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 5 / 31

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Metric spaces Building new Polish spaces from old

Polish subspaces

Proposition 1.3.1

Let X be completely metrisable and take closed Y ⊆ X. Then Y is completely

metrisable. In particular, any closed subspace of a Polish space is Polish, by

Corollary 1.2.11.

Proposition 1.3.2

Let X be completely metrisable and let U ⊆ X be open. Then U is completely

metrisable. In particular, any open subset of a Polish space is Polish, by Corol-

lary 1.2.11.

Theorem 1.3.9

Let X be a completely metrisable space and let G ⊆ X be a Gδ in X. Then G is also completely metrisable. In particular, if X is Polish then so is G, by Corollary 1.2.11.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 5 / 31

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Metric spaces Building new Polish spaces from old

Products of Polish spaces

Proposition 1.3.11

Let (Xn)∞

n=0 be a sequence of Polish spaces, with corresponding compatible

metrics dn ≤ 2−n. Then the product X = ∞

n=0 Xn, with metric defined by

d(x, y) =

∞

n=0

dn(xn, yn), x = (xn), xn ∈ Xn, and y = (yn), yn ∈ Yn, is Polish.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 6 / 31

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Trees and the spaces of Baire and Cantor The tree Aˆ<N and the space AˆN

The tree A<N

Definition 2.1.3

Given A = ∅, define A<N to be the set of all finite sequences of elements of A, i.e. functions t of the form t : n − → A, where n ∈ N. Given n ∈ N, the set of all functions t : n − → A is sometimes denoted An. Thus we can express A<N as A<N =

∞

n=0

An.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 7 / 31

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Trees and the spaces of Baire and Cantor The tree Aˆ<N and the space AˆN

The tree A<N

Definition 2.1.3

Given A = ∅, define A<N to be the set of all finite sequences of elements of A, i.e. functions t of the form t : n − → A, where n ∈ N. Given n ∈ N, the set of all functions t : n − → A is sometimes denoted An. Thus we can express A<N as A<N =

∞

n=0

An.

Definition 2.1.5

Let s, t ∈ A<N. We write s t if t is an extension of s (extension is not defined in a strict sense, so s is an extension of itself, i.e. s s). The pair (A<N, ) is a tree. Usually we just write A<N.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 7 / 31

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Trees and the spaces of Baire and Cantor The tree Aˆ<N and the space AˆN

The metric space AN

Definition 2.1.7

Given A = ∅, define AN to be the set of all infinite sequences of elements of A: AN = {x : N − → A}.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 8 / 31

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Trees and the spaces of Baire and Cantor The tree Aˆ<N and the space AˆN

The metric space AN

Definition 2.1.7

Given A = ∅, define AN to be the set of all infinite sequences of elements of A: AN = {x : N − → A}.

Definition 2.1.8

Define d on AN by d(x, y) = if x = y 2−n if x = y and where n is minimal, subject to x(n) = y(n).

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 8 / 31

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Trees and the spaces of Baire and Cantor The tree Aˆ<N and the space AˆN

The metric space AN

Definition 2.1.7

Given A = ∅, define AN to be the set of all infinite sequences of elements of A: AN = {x : N − → A}.

Definition 2.1.8

Define d on AN by d(x, y) = if x = y 2−n if x = y and where n is minimal, subject to x(n) = y(n).

Proposition 2.1.9

The function d is a complete metric on AN.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 8 / 31

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Trees and the spaces of Baire and Cantor The tree Aˆ<N and the space AˆN

Open balls and 2nd countability of AN

Definition 2.1.10

The open balls of AN are precisely the sets Ws =

x ∈ AN : s ≺ x
,

s ∈ A<N.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 9 / 31

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Trees and the spaces of Baire and Cantor The tree Aˆ<N and the space AˆN

Open balls and 2nd countability of AN

Definition 2.1.10

The open balls of AN are precisely the sets Ws =

x ∈ AN : s ≺ x
,

s ∈ A<N.

Proposition 2.1.11

If A is countable then AN is 2nd countable. Moreover, it is a Polish space.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 9 / 31

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Trees and the spaces of Baire and Cantor Baire’s space, Cantor’s space and Lusin schemes

Lusin schemes

Definition 2.2.2

Let A = ∅ and let X be a Polish space with compatible metric d. A Lusin scheme on X is a system (Fs)s∈A<N of subsets of X satisfying

1

Fs⌢a ⊆ Fs for all s ∈ A<N and a ∈ A;

2

for all x ∈ AN, we have diam

Fx|n
→ 0 as n → ∞;

3

Fs ∩ Ft = ∅ whenever s ⊥ t. A Lusin scheme is called a Cantor scheme if A = {0, 1} = 2.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 10 / 31

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Trees and the spaces of Baire and Cantor Baire’s space, Cantor’s space and Lusin schemes

Associated maps of Lusin schemes

Lemma 2.2.4

Given a Lusin scheme (Fs)s∈A<N, define D =

x ∈ AN : Fx|n = ∅ for all n ∈ N
.

1

D is a closed subset of AN.

2

Let x ∈ D. By Definition 2.2.2 (1) and (2), and Theorem 1.2.17, the inter- section ∞

n=0 Fx|n = ∞ n=0 Fx|n is a singleton.

Define the associated map f : D − → X by letting f(x) be the unique element satisfying f(x) ∈

∞

n=0

Fx|n.

3

f is injective.

4

f is continuous.

5

If F∅ = X and Fs =

a∈A Fs⌢a for all s ∈ A<N, then f is also surjective.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 11 / 31

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Trees and the spaces of Baire and Cantor Baire’s space, Cantor’s space and Lusin schemes

Embedding 2N and NN into uncountable Polish spaces

Theorem 2.2.7

If X is an uncountable Polish space then there is a map f : 2N − → X, such that f(2N) is closed in X (moreover, f(2N) is sequentially compact) and f is a homeomorphism between 2N and f(2N).

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 12 / 31

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Trees and the spaces of Baire and Cantor Baire’s space, Cantor’s space and Lusin schemes

Embedding 2N and NN into uncountable Polish spaces

Theorem 2.2.7

If X is an uncountable Polish space then there is a map f : 2N − → X, such that f(2N) is closed in X (moreover, f(2N) is sequentially compact) and f is a homeomorphism between 2N and f(2N).

Corollary 2.2.8

If X is an uncountable Polish space then there is a map f : NN − → X, such that f(NN) is a Gδ in X and f is a homeomorphism between NN and f(NN).

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 12 / 31

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Trees and the spaces of Baire and Cantor Baire’s space, Cantor’s space and Lusin schemes

Polish spaces and closed subsets of NN

Theorem 2.2.12

Let X be a Polish space. Then there exists a closed subset D of NN and a continuous bijection f : D − → X.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 13 / 31

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Trees and the spaces of Baire and Cantor Trees and retractions

Every Polish space is a continuous image of NN

Proposition 2.3.6

Any non-empty closed subset D ⊆ AN is a retract of AN.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 14 / 31

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Trees and the spaces of Baire and Cantor Trees and retractions

Every Polish space is a continuous image of NN

Proposition 2.3.6

Any non-empty closed subset D ⊆ AN is a retract of AN.

Corollary 2.3.7

If X is Polish then there exists a continuous surjection h : NN − → X.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 14 / 31

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Borel sets sigma-algebras and Borel sets and maps

Generating σ-algebras and the Borel σ-algebra

Definition 3.1.5

Let X be a set and let E ⊆ P(X). Then the σ-algebra generated by E, denoted σ(E), is the intersection of all the σ-algebras containing E, i.e. σ(E) = {E ⊆ X : if E ⊆ S and S is a σ-algebra on X, then E ∈ S} .

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 15 / 31

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Borel sets sigma-algebras and Borel sets and maps

Generating σ-algebras and the Borel σ-algebra

Definition 3.1.5

Let X be a set and let E ⊆ P(X). Then the σ-algebra generated by E, denoted σ(E), is the intersection of all the σ-algebras containing E, i.e. σ(E) = {E ⊆ X : if E ⊆ S and S is a σ-algebra on X, then E ∈ S} .

Definition 3.1.6

Let X be a metric space and let T be the family of all open subsets of X. The Borel σ-algebra, denoted B(X), is the σ-algebra σ(T ) generated by T . We say that a subset E ⊆ X is Borel measurable, or simply Borel, if E ∈ B(X).

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 15 / 31

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Borel sets sigma-algebras and Borel sets and maps

Borel maps

Definition 3.1.9

Let (X, S) and (Y, A) be measurable spaces. We say that f : X − → Y is measurable if f −1(E) ∈ S whenever E ∈ A. In particular, a map f : X − → Y between metric spaces is Borel measurable,

r Borel, if f −1(E) ∈ B(X) whenever E ∈ B(Y).

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 16 / 31

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Borel sets sigma-algebras and Borel sets and maps

Borel maps

Definition 3.1.9

Let (X, S) and (Y, A) be measurable spaces. We say that f : X − → Y is measurable if f −1(E) ∈ S whenever E ∈ A. In particular, a map f : X − → Y between metric spaces is Borel measurable,

r Borel, if f −1(E) ∈ B(X) whenever E ∈ B(Y).

Proposition 3.1.10

Let f : X − → Y be a map, such that f −1(V) ∈ B(X) whenever V ⊆ Y is open. Then f is a Borel map. In particular, continuous maps are Borel.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 16 / 31

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Borel sets sigma-algebras and Borel sets and maps

Graphs and monotonically closed families

Proposition 3.1.11

α is transitive;

2

if ξ ∈ α then ξ is transitive and ξ / ∈ ξ;

3

if J ⊆ α is non-empty, then there exists ξ ∈ J with the property that whenever η ∈ J, then either ξ ∈ η or ξ = η.

Example 3.2.5

1

Every natural number is an ordinal number.

2

N is an ordinal number. In set theory, N is often denoted by ω.

3

If α is an ordinal then α + 1 = α ∪ {α} is an ordinal (Fact 3.2.7 (1)). For example, ω + 1 = ω ∪ {ω} is an ordinal.

4

For every n ∈ N = ω, the set ω + n is an ordinal number, as is ω · 2 = ω + ω = ω ∪ {ω + n : n ∈ ω} .

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 18 / 31

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Borel sets Ordinal numbers

Fundamental facts about ordinal numbers

Fact 3.2.7

Let α be an ordinal.

1

α ∪ {α} is an ordinal (we denote it by α + 1).

2

If ξ ∈ α then ξ is an ordinal.

3

If ξ ⊆ α and ξ is transitive, then ξ ∈ α or ξ = α.

4

Trichotomy of ∈ on ordinals - if β is another ordinal, then exactly one of three possibilities holds: α ∈ β, α = β

r

β ∈ α.

5

If (αi)i∈I is a family of ordinals indexed by a set I, then supi∈I αi =

i∈I αi

is also an ordinal.

6

Transfinite induction on α - suppose that S ⊆ α has the property that, for all ξ ∈ α, if ξ ⊆ S then ξ ∈ S. Then S = α.

7

Suppose that P is a property that is satisfied by at least one ordinal. Then there is a least ordinal satisfying P.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 19 / 31

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Borel sets Ordinal numbers

Representing countable ordinals as subsets of R

Example 3.2.5 revisited

1

Every natural number is an ordinal number. 0 ∈ 1 ∈ 2 ∈ 3 ∈ 4 ∈ . . . . . . ∈ n − 1 n

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 20 / 31

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Borel sets Ordinal numbers

Representing countable ordinals as subsets of R

Example 3.2.5 revisited

1

Every natural number is an ordinal number. 0 ∈ 1 ∈ 2 ∈ 3 ∈ 4 ∈ . . . . . . ∈ n − 1 n

2

N = ω is an ordinal number. 0 ∈ 1 ∈ 2 ∈ 3 ∈ 4 ∈ . . . ω

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 20 / 31

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Borel sets Ordinal numbers

Representing countable ordinals as subsets of R

Example 3.2.5 revisited

1

Every natural number is an ordinal number. 0 ∈ 1 ∈ 2 ∈ 3 ∈ 4 ∈ . . . . . . ∈ n − 1 n

2

N = ω is an ordinal number. 0 ∈ 1 ∈ 2 ∈ 3 ∈ 4 ∈ . . . ω

3

ω + 1 = ω ∪ {ω} is an ordinal. 0 ∈ 1 ∈ 2 ∈ 3 ∈ 4 ∈ . . . . . . ∈ ω ω + 1

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 20 / 31

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Borel sets Ordinal numbers

Representing countable ordinals as subsets of R

Example 3.2.5 revisited

1

Every natural number is an ordinal number. 0 ∈ 1 ∈ 2 ∈ 3 ∈ 4 ∈ . . . . . . ∈ n − 1 n

2

N = ω is an ordinal number. 0 ∈ 1 ∈ 2 ∈ 3 ∈ 4 ∈ . . . ω

3

ω + 1 = ω ∪ {ω} is an ordinal. 0 ∈ 1 ∈ 2 ∈ 3 ∈ 4 ∈ . . . . . . ∈ ω ω + 1

4

ω + ω = ω ∪ {ω + n : n ∈ ω} is an ordinal. 0 ∈ . . . . . . ∈ ω ∈ . . . ω + ω

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 20 / 31

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Borel sets Ordinal numbers

Taxonomy of ordinal numbers

Fact 3.2.8

Let α be an ordinal. There are three possibilities:

1

α = 0

2

α is a successor ordinal, that is, there is an ordinal ξ such that α = ξ + 1. In this case, ξ is the immediate predecessor of α.

3

α is a limit ordinal, that is, α = 0 and is not a successor ordinal. In this case, if ξ < α then ξ + 1 < α.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 21 / 31

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Borel sets Ordinal numbers

The least uncountable ordinal number

Definition 3.2.9

There is a least uncountable ordinal, denoted by ω1. The ordinal ω1 is the set of all countable ordinals.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 22 / 31

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Borel sets Ordinal numbers

The least uncountable ordinal number

Definition 3.2.9

There is a least uncountable ordinal, denoted by ω1. The ordinal ω1 is the set of all countable ordinals.

Fact 3.2.10

Let (αn)∞

n=0 ⊆ ω1 be a sequence of countable ordinals. Then α = supn αn < ω1,

i.e., α is also a countable ordinal.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 22 / 31

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Borel sets The Borel hierarchy

The Borel hierarchy

Definition 3.3.1

For a metric space X and 1 ≤ α < ω1, we define Σ0

α(X), Π0 α(X) ⊆ P(X). Set

Σ0

1(X) = {U ⊆ X : U is open in X}

If Σ0

α(X) has been defined, we define the multiplicative class

Π0

α(X) =

X \ E : E ∈ Σ0

α(X)

.

If Π0

ξ has been defined for all 1 ≤ ξ < α, we define the additive class

Σ0

α(X) =

  

∞

n=0

En : (En) ⊆

1≤ξ<α

Π0

ξ(X)

   . In addition, we define the ambiguous classes ∆0

α(X) = Σ0 α(X) ∩ Π0 α(X).

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 23 / 31

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Borel sets The Borel hierarchy

Properties of the Borel hierarchy

Proposition 3.3.3

The following statements hold for 1 ≤ α < ω1.

1

Σ0

η(X), Π0 η(X) ⊆ ∆0 α(X) whenever 1 ≤ η < α;

2

Σ0

α(X), Π0 α(X) ⊆ B(X).

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 24 / 31

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Borel sets The Borel hierarchy

Properties of the Borel hierarchy

Proposition 3.3.3

The following statements hold for 1 ≤ α < ω1.

1

Σ0

η(X), Π0 η(X) ⊆ ∆0 α(X) whenever 1 ≤ η < α;

2

Σ0

α(X), Π0 α(X) ⊆ B(X).

Corollary 3.3.4

1≤α<ω1

∆0

α(X) =

1≤α<ω1

Σ0

α(X) =

1≤α<ω1

Π0

α(X) = B(X).

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 24 / 31

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Borel sets The Borel hierarchy

Properties of the Borel hierarchy

Proposition 3.3.3

The following statements hold for 1 ≤ α < ω1.

1

Σ0

η(X), Π0 η(X) ⊆ ∆0 α(X) whenever 1 ≤ η < α;

2

Σ0

α(X), Π0 α(X) ⊆ B(X).

Corollary 3.3.4

1≤α<ω1

∆0

α(X) =

1≤α<ω1

Σ0

α(X) =

1≤α<ω1

Π0

Borel sets The Borel hierarchy

Universal sets of higher Borel class

Lemma 3.3.7

If X is a separable metric space and 1 ≤ α < ω1, then there exists A ∈ Σ0

α(X ×

NN) that is universal for all Σ0

α(X) sets. Likewise, there is B ∈ Π0 α(X × NN),

universal for Π0

α(X) sets.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 26 / 31

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Borel sets The Borel hierarchy

Universal sets of higher Borel class

Lemma 3.3.7

If X is a separable metric space and 1 ≤ α < ω1, then there exists A ∈ Σ0

α(X ×

NN) that is universal for all Σ0

α(X) sets. Likewise, there is B ∈ Π0 α(X × NN),

universal for Π0

α(X) sets.

Lemma 3.3.8

Let Γ be a countable set and fix a bijection f : Γ × N − → N. For each γ ∈ Γ, define fγ : NN − → NN by fγ(x)(i) = x(f(γ, i)). The following statements hold.

1

Each fγ is continuous.

2

Imagine that we have some yγ ∈ NN for every γ ∈ Γ. Then there exists y ∈ NN such that fγ(y) = yγ for all γ ∈ Γ.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 26 / 31

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Analytic and coanalytic sets Analytic sets

Analytic sets

Definition 4.1.1

Let X be a Polish space. A subset A ⊆ X is called analytic if there exists another Polish space Y and a continuous map f : Y − → X such that A = f(Y).

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 27 / 31

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Analytic and coanalytic sets Analytic sets

Analytic sets

Definition 4.1.1

Let X be a Polish space. A subset A ⊆ X is called analytic if there exists another Polish space Y and a continuous map f : Y − → X such that A = f(Y).

Proposition 4.1.2

Let X be Polish and let A ⊆ X. The following statements are equivalent.

1

A is analytic.

2

There exists a continuous map h : NN − → X with A = h(NN).

3

There exists a closed subset F ⊆ X × NN, such that A = π(F).

4

For any uncountable Polish space Y, there exists a Gδ set G ⊆ X × Y such that A = π(G).

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 27 / 31

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Analytic and coanalytic sets Analytic sets

Properties of analytic sets

Proposition 4.1.3

Let X and Y be Polish spaces, A ⊆ X analytic and g : A − → Y continuous. Then g(A) is analytic.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 28 / 31

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Analytic and coanalytic sets Analytic sets

Properties of analytic sets

Proposition 4.1.3

Let X and Y be Polish spaces, A ⊆ X analytic and g : A − → Y continuous. Then g(A) is analytic.

Proposition 4.1.4

Let X be a Polish space and (An) a sequence of analytic subsets of X. Then ∞

n=0 An and ∞ n=0 An are analytic.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 28 / 31

SLIDE 54

Analytic and coanalytic sets Analytic sets

Properties of analytic sets

Proposition 4.1.3

Let X and Y be Polish spaces, A ⊆ X analytic and g : A − → Y continuous. Then g(A) is analytic.

Proposition 4.1.4

Let X be a Polish space and (An) a sequence of analytic subsets of X. Then ∞

n=0 An and ∞ n=0 An are analytic.

Corollary 4.1.5

Every Borel subset of a Polish space is analytic.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 28 / 31

SLIDE 55

Analytic and coanalytic sets Analytic sets

Properties of analytic sets

Proposition 4.1.3

Let X and Y be Polish spaces, A ⊆ X analytic and g : A − → Y continuous. Then g(A) is analytic.

Proposition 4.1.4

Let X be a Polish space and (An) a sequence of analytic subsets of X. Then ∞

n=0 An and ∞ n=0 An are analytic.

Corollary 4.1.5

Every Borel subset of a Polish space is analytic.

Theorem 4.1.6

Let X, Y be Polish spaces, f : X − → Y a Borel map, and A ⊆ X, B ⊆ Y analytic subsets. Then f(A) ⊆ Y and f −1(B) are also analytic.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 28 / 31

SLIDE 56

Analytic and coanalytic sets Analytic non-Borel sets, coanalytic sets and the separation theorem

Analytic non-Borel sets

Theorem 4.2.1

Any uncountable Polish space X contains an analytic non-Borel subset.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 29 / 31

SLIDE 57

Analytic and coanalytic sets Analytic non-Borel sets, coanalytic sets and the separation theorem

Theorems of Lusin and Souslin

Definition 4.2.2

Let A and B be disjoint subsets of a Polish space. We say that A and B can be separated by a Borel set if there exists a Borel set E such that A ⊆ E and B ⊆ X \ E.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 30 / 31

SLIDE 58

Analytic and coanalytic sets Analytic non-Borel sets, coanalytic sets and the separation theorem

Theorems of Lusin and Souslin

Definition 4.2.2

Let A and B be disjoint subsets of a Polish space. We say that A and B can be separated by a Borel set if there exists a Borel set E such that A ⊆ E and B ⊆ X \ E.

Theorem 4.2.3

If A and B are disjoint analytic subsets of a Polish space X then they can be separated by a Borel set.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 30 / 31

SLIDE 59

Analytic and coanalytic sets Analytic non-Borel sets, coanalytic sets and the separation theorem

Theorems of Lusin and Souslin

Definition 4.2.2

Let A and B be disjoint subsets of a Polish space. We say that A and B can be separated by a Borel set if there exists a Borel set E such that A ⊆ E and B ⊆ X \ E.

Theorem 4.2.3

If A and B are disjoint analytic subsets of a Polish space X then they can be separated by a Borel set.

Theorem 4.2.5

Let X be a Polish space. Then A ⊆ X is Borel if and only if both A and its complement are analytic.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 30 / 31

SLIDE 60

Analytic and coanalytic sets Analytic non-Borel sets, coanalytic sets and the separation theorem

Coanalytic sets and an application of Souslin’s Theorem

Definition 4.2.6

Let X be a Polish space. We call a subset C ⊆ X coanalytic if X \ A is analytic.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 31 / 31

SLIDE 61

Analytic and coanalytic sets Analytic non-Borel sets, coanalytic sets and the separation theorem

Coanalytic sets and an application of Souslin’s Theorem

Definition 4.2.6

Let X be a Polish space. We call a subset C ⊆ X coanalytic if X \ A is analytic.

Theorem 4.2.7

Let f : X − → Y be a Borel bijection between Polish spaces. Then f −1 is also Borel.

Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 31 / 31