Topics in Descriptive Set Theory (Ma 191) Lecturer: Alexander Kechris, Notes by: Siddharth Prasad February 21, 2017 1 Lecture 1 (1/5/17) 1.1 Basic Classical Notions Descriptive set theory is the study of “definable” sets and functions in well behaved spaces. Definition 1.1. A Polish space is a topological space given by a complete, separable metric. n Example. R; C; R , separable Hilbert spaces, separable Banach spaces. Definition 1.2. For a Polish space X we are interested in the so called Borel sets in X. We denote by B∼ (X) the class of all Borel subsets of X. These consist of 0 Σ∼ 1(X) = open sets of X( sometimes denoted by G) 0 Π∼ 1(X) = closed sets of X( sometimes denoted by F ) 0 Σ∼ 2(X) = countable unions of closed sets (Fσ) 0 0 Π∼ 2(X) = complements of Σ∼ 2(X) sets = countable intersections of open sets(Gδ) 0 Σ∼ 3(X) = Gδσ sets 0 Π∼ 3(X) = Fσδ sets and so on. This notion can be extended into the transfinite as such: for a countable ordinal ξ > 1 define 0 0 Σ∼ ξ(X) = f[nAn : An 2 Π∼ ξn (X); ξn < ξg 0 0 Π∼ ξ(X) =∼ Σ∼ ξ(X) 0 0 0 ∆∼ ξ(X) = Σ∼ ξ(X) \ Π∼ ξ(X) 1 0 0 0 We have B∼ (X) = [ξ<!1 Σ∼ ξ(X) = [ξ<!1 Π∼ ξ(X) = [ξ<!1 ∆∼ ξ(X), which forms the so called Borel hierarchy. 1 Definition 1.3. We define the analytic sets (A) Σ∼ 1(X) = fprojX (A): A ⊆ X×Y; A Borelg. 1 1 Their complements are the co-analytic sets (CA) Π∼ 1(X) =∼ Σ∼ 1(X). PCA sets are 1 1 1 1 Σ∼ 2(X) = fprojX (A): A ⊆ X ×Y; A is Π∼ 1(X ×Y )g, the CPCA sets are Π∼ 2(X) =∼ Σ∼ 2(X). This hierarchy of sets forms the projective sets or projective hierarchy, denoted by P∼ (X). 1 Remark. A result of Suslin shows that B∼ (X) = ∆∼ 1(X). Definition 1.4. A function f : X ! Y is a Borel (measurable) function if for each open Borel set U ⊆ Y , f −1(U) is borel in X. Effective descriptive set theory: Introduces concepts and methods of computability theory, is a refinement of the classical theory. We introduce a class of so called \transfer theorems", that allow us to work primarily in the so called Baire space N = NN (endowed with product topology). A metric on this space of infinite sequences is given by ( 0 if α = β d(α; β) = 1 2n+1 where n is the least number such that α(n) 6= β(n) The Cantor space is the subset C = 2N ⊆ N . Elements of Baire space are sometimes ∼ referred to irrationals since N = R n Q. Theorem 1.1 (Transfer theorems). N and C denote Baire space and Cantor space respec- tively. • If K is a compact metrizable space, there exists a continuous surjection f : C K. • If X is a Polish space, there exists a continuous surjection f : N X . Moreover, there exists a closed F ⊆ N and there exists a continuous bijection f : F X. • If X is a perfect Polish space, there exists a continuous bijection f : N X. • X and Y are Polish spaces with jXj = jY j () there exists a Borel bijection f : X Y . 1 Example. Analytic sets satisfy the perfect set property, i.e. if A ⊂ X is Σ∼ 1, either jAj ≤ @0 (A is countable) or there is a continuous bijection f : C A. 2 1.2 Review of Computability Theory k Main objects of study are partial computable (recursive) k-ary functions f : N ! N. Recall we can code TMs by numbers, fegk(x1; : : : ; xk) = the function computed by the TM with code e. k ' (e; x¯) = fegk(¯x) is a universal recursive function. m Theorem 1.2 (S-m-n). There is a total recursive function Sn such that fegm+n(¯x; y¯) = m fSn (e; x¯)g(¯y). Theorem 1.3 (Recursion theorem). For each computable function f, there is e such that feg(¯x) = f(e; x¯). k m k ∼ ` A function f : N ! N is computable if its projections are computable. So N =c N m are \computably isomorphic". We have the notion of a coding function hm; ni2 = 2 · 2 (2n + 1) − 1 which gives a bijection h·; ·i : N N. Put hm; n; ki3 = hhm; ni2; ki2. By the nature of the coding function we have that m1; : : : ; mn << hm1; : : : ; mnin. We finally extend this to a bijective coding function on N given by hi = 0, hmi = h0; m + 1i2, and hm0; : : : ; mk−1i = hk − 1; hm0; : : : ; mk−1iki2. k Recall that a set A ⊆ N is computable () the characteristic function χA is com- putable. Recall the following equivalent characterizations for recursively enumerable sets. A ⊆ N is recursively enumerable () A = ; or A = f(N); f total computable () A = f(N) is computable () A = dom(f); f computable 2 () A = proj(B);B ⊆ N ;B computable: k The relation W (e; x¯) () fegk(¯x) # is universal of r.e. subsets of N . Given R r.e., there is e such that W (e; x¯) () R(e; x¯). Proposition 1.4. A function f is recursive () graph(f) is r.e. Proposition 1.5. A set A is recursive () A and ∼ A are r.e. 1.3 Effective Topology We will be primarily working with product spaces over the Baire space N and the naturals k N. Let s = (s0; : : : ; sk−1) 2 N . The neighborhoods Ns = fα 2 N : s ⊆ αg are the so called basic open sets of the Baire space. Then, open sets are of the form [i2NNsi for some sequence (si). The basic open sets for N are singleton sets fng, and likewise the basic open sets in the n m product space N × N are sets of the form Ns1 × · · · × Nsn × fn1g × · · · × fnmg. 3 n m 0 Definition 1.5. A ⊆ N ×N is effectively open or Σ1 iff A = ; or there are total recursive functions f1; : : : ; fn; g1; : : : ; gm such that [ A = Nf1(k) × · · · × Nfn(k) × fg1(k)g · · · × fgm(k)g: k 0 Remark. Note that there are thus only countably many Σ1 sets. n m 0 Proposition 1.6. A ⊆ N × N is Σ1 if and only if there exists a recursive relation n+m R ⊆ N such that (α1; : : : ; αn; n1; : : : ; nm) 2 A () 9kR(¯α1(k);:::; α¯n(k); n1; : : : ; nm); where α¯(k) = hα ki = hα(0); : : : ; α(k − 1)i. 2 Lecture 2 (1/10/17) We prove the proposition from last time: n m 0 Proposition 2.1. A ⊆ N × N is Σ1 if and only if there exists a recursive relation n+m R ⊆ N such that (α1; : : : ; αn; n1; : : : ; nm) 2 A () 9kR(¯α1(k);:::; α¯n(k); n1; : : : ; nm): Proof. For simplicity we will take n = m = 1, so A ⊆ N × N. For the forward direction, if 0 A is Σ1 we can write A = [pNf(p) × fg(p)g for total recursive functions f and g. Then we have (α; n) 2 A () 9p(f(p) ⊆ α&n = g(p)) () 9p9q(f(p) ⊆ α q&n = g(p)) () 9k(f((k)0) ⊆ α &n = g((k)0)) (k)1 9kR(¯α(k); n) where R is clearly recursive. < For the other direction, suppose (α; n) 2 A () 9kR(¯α(k); n), where R ⊆ N N × N is recursive. Then, there are recursive functions f; g such that R = f(f(p); g(p)) : p 2 Ng, so we can write A = [pNf(p) × fg(p)g. < Definition 2.1. A Tree on N is a set of finite sequences T ⊆ N N that is closed under the subsequence relation, i.e. if t 2 T and s ⊆ t, then s 2 T . The set of all infinite branches of T is called the body of T and is denoted by [T ], i.e. [T ] = fα 2 N : 8n(α n 2 T )g. 0 Remark. fA ⊆ N : A 2 Π∼ 1g = f[T ]: T a tree on Ng. 4 Proof. See page 138 of [1]. 0 Proposition 2.2. A ⊆ N is Π1 () 9T (T is recursive &A = [T ]). 0 Proof. For the forward direction assume A ⊆ N is Π1. Then, there exists a recursive R such that α 2 A () 8nR(¯α(n)): Then, take T to be the tree such that s 2 T () R(hsi). Then A = [T ] and we are done. For the reverse implication take R = hT i. Then, α 2 A () 8nR(¯α(n)). m m < Definition 2.2. Trees on N : A tree T on N is a set of (s1; : : : ; sm), si 2 N N, lh(s1) = ··· = lh(sm), such that if (t1; : : : ; tm) ⊆ (s1; : : : ; sm) (meaning ti ⊆ si8i), then (t1; : : : ; tm) 2 T . m 0 So the effective version of the remark is as follows: A ⊆ N is Π1 () 9 a recursive m tree T on N such that A = [T ]. 0 N N Proposition 2.3. Σ1 is closed under ^; _; 9 ; 9 ; 8m ≤ n. N 0 Proof. We will prove closure under 9 . Suppose P ⊆ N × N × N is Σ1. Then, we can write P (α; β; n) () 9tR(¯α(t); β¯(t); n), for some recursive R. We have 9αP (α; β; n) () 9α9tR(¯α(t); β¯(t); n) () 9k9s(R(s; β¯(k); n)&jsj = k) () 9uR0(β¯(u); n); where R0 is recursive. n m Definition 2.3.