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

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

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

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

Metric spaces Polish spaces Cantor’s intersection theorem Theorem 1.2.17 Let X be completely metrisable, with compatible metric d . Suppose that ( F n ) is a sequence of closed non-empty subsets of X satisfying F n + 1 ⊆ F n (i.e. ( F n ) is decreasing ) 1 diam ( F n ) → 0 as n → ∞ . 2 Then the intersection F = � ∞ n = 1 F n is a singleton. Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 3 / 31

Metric spaces Building new Polish spaces from old Examples of G δ sets Example 1.3.8 Any open subset of a metric space is a G δ . 1 R \ Q is clearly not open, but it is a G δ . Enumerate Q as ( q n ) and put 2 U n = R \ { q n } , then each U n is open and R \ Q = � ∞ n = 0 U n . ( TOP × ) Any closed subset F of a metric space is a G δ . Define 3 x ∈ X : d ( x , y ) < 2 − n for some y ∈ F � � U n = . Each U n is open and F = � ∞ n = 0 U n . The set of points of continuity of an arbitrary function f : R − → R is a G δ 4 (Question 2, Exercise Sheet 1). Q is not a G δ in R (same question). 5 Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 4 / 31

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

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

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 G δ 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

Metric spaces Building new Polish spaces from old Products of Polish spaces Proposition 1.3.11 Let ( X n ) ∞ n = 0 be a sequence of Polish spaces, with corresponding compatible metrics d n ≤ 2 − n . Then the product X = � ∞ n = 0 X n , with metric defined by ∞ � d ( x , y ) = d n ( x n , y n ) , n = 0 x = ( x n ) , x n ∈ X n , and y = ( y n ) , y n ∈ Y n , is Polish. Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 6 / 31

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 . → A is sometimes denoted A n . Thus Given n ∈ N , the set of all functions t : n − we can express A < N as ∞ A < N = � A n . n = 0 Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 7 / 31

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 . → A is sometimes denoted A n . Thus Given n ∈ N , the set of all functions t : n − we can express A < N as ∞ A < N = � A n . n = 0 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

Trees and the spaces of Baire and Cantor The tree Aˆ<N and the space AˆN The metric space A N Definition 2.1.7 Given A � = ∅ , define A N to be the set of all infinite sequences of elements of A : A N = { x : N − → A } . Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 8 / 31

Trees and the spaces of Baire and Cantor The tree Aˆ<N and the space AˆN The metric space A N Definition 2.1.7 Given A � = ∅ , define A N to be the set of all infinite sequences of elements of A : A N = { x : N − → A } . Definition 2.1.8 Define d on A N by � 0 if x = y d ( 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

Trees and the spaces of Baire and Cantor The tree Aˆ<N and the space AˆN The metric space A N Definition 2.1.7 Given A � = ∅ , define A N to be the set of all infinite sequences of elements of A : A N = { x : N − → A } . Definition 2.1.8 Define d on A N by � 0 if x = y d ( 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 A N . Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 8 / 31

Trees and the spaces of Baire and Cantor The tree Aˆ<N and the space AˆN Open balls and 2nd countability of A N Definition 2.1.10 The open balls of A N are precisely the sets x ∈ A N : s ≺ x � � s ∈ A < N . W s = , Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 9 / 31

Trees and the spaces of Baire and Cantor The tree Aˆ<N and the space AˆN Open balls and 2nd countability of A N Definition 2.1.10 The open balls of A N are precisely the sets x ∈ A N : s ≺ x � � s ∈ A < N . W s = , Proposition 2.1.11 If A is countable then A N 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

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 ( F s ) s ∈ A < N of subsets of X satisfying F s ⌢ a ⊆ F s for all s ∈ A < N and a ∈ A ; 1 for all x ∈ A N , we have diam � � F x | n → 0 as n → ∞ ; 2 F s ∩ F t = ∅ whenever s ⊥ t . 3 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

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 ( F s ) s ∈ A < N , define x ∈ A N : F x | n � = ∅ for all n ∈ N � � D = . D is a closed subset of A N . 1 Let x ∈ D . By Definition 2.2.2 (1) and (2), and Theorem 1.2.17, the inter- 2 section � ∞ n = 0 F x | n = � ∞ n = 0 F x | n is a singleton. Define the associated map f : D − → X by letting f ( x ) be the unique element satisfying ∞ � f ( x ) ∈ F x | n . n = 0 f is injective. 3 f is continuous. 4 If F ∅ = X and F s = � a ∈ A F s ⌢ a for all s ∈ A < N , then f is also surjective. 5 Dr Richard Smith (maths.ucd.ie/~rsmith) Intro to Desc Set Theory (MATH40350) Semester 2 2011 – 2012 11 / 31