Polishable Borel equivalence relations S lawomir Solecki Cornell - PowerPoint PPT Presentation

Polishable Borel equivalence relations S� lawomir Solecki Cornell University Research supported by NSF grant DMS–1700426 June 2018

Outline Outline of Topics Introduction 1 Polishable equivalence relations 2 Canonical approximations 3 Polishable equivalence relations continued 4 S� lawomir Solecki (Cornell University) Polishable equivalence relations June 2018 2 / 28

Introduction Introduction S� lawomir Solecki (Cornell University) Polishable equivalence relations June 2018 3 / 28

Introduction Scope : E a Borel equivalence relation on a Polish space E induced by a continuous action of a Borel group Aim : introduce a notion of Polishable equivalence relations Requirements : (1) orbit equivalence relations of continuous Polish group actions ⊆ Polishable equivalence relations ⊆ idealistic equivalence relations; (2) approximability by transfinite sequences of “simple” Polishable equivalence relations. S� lawomir Solecki (Cornell University) Polishable equivalence relations June 2018 4 / 28

Introduction Recall the notion of uniformity V on a set X : V is a closed upwards family of symmetric sets, whose intersection is the diagonal, and such that for each V ∈ V there exists W ∈ V with W ◦ W ⊆ V . V induces a topology t ( V ) whose neighborhood basis (not necessarily open) at x ∈ X is { V x : V ∈ V} . Weil : t ( V ) is metrizable if and only if V has a countable basis. S� lawomir Solecki (Cornell University) Polishable equivalence relations June 2018 5 / 28

Introduction The following basic framework will show up on several occasions. X a set, τ a topology on X , V a uniformity on X , and Γ a group of transformations of X . We say that τ , V , and Γ are compatible if — τ is Polish; — t ( V ) is completely metrizable; — Γ is countable; — τ ⊆ t ( V ); — functions in Γ are τ -homeomorphisms; — functions in Γ are V -uniform. S� lawomir Solecki (Cornell University) Polishable equivalence relations June 2018 6 / 28

Introduction Let a topology τ and a unformity V , both on X , be compatible. We say that V has a Borel basis ( with respect to τ ) if it has a basis consisting of subsets of X × X that are Borel with respect to τ × τ . Lemma If V has a Borel basis and τ is Polish, then V has a Borel basis consisting of sets that are open with respect to t ( V ) × t ( V ) . S� lawomir Solecki (Cornell University) Polishable equivalence relations June 2018 7 / 28

Introduction Let a uniformity V and a group of transformations Γ, both on X , be compatible. For V ∈ V and γ ∈ Γ, let γ ≤ V iff γ x ∈ V x for each x ∈ X . Γ is dense in V if, for each V ∈ V , there exists W ∈ V such that, for each x ∈ X , { γ x : γ ∈ Γ , γ ≤ V } is t ( V )-dense in W x . S� lawomir Solecki (Cornell University) Polishable equivalence relations June 2018 8 / 28

Introduction Examples 1. Let ( X , d ) be a metric space. The uniformity induced by d is the upward closure of the family of all sets of the form { ( x , y ) ∈ X × X : d ( x , y ) < r } for r ∈ R , r > 0. S� lawomir Solecki (Cornell University) Polishable equivalence relations June 2018 9 / 28

Introduction 2. Let a be a continuous action of a Polish group G on a Polish space ( X , τ ). The action a induces the uniformity V : 1 ∈ V = V − 1 open in G } , V a = the upward closure of { ˆ where ˆ V = { ( x , y ) ∈ X × X : ∃ g ∈ V gx = y } . If Γ a is the group of transformations of X induced by a countable dense subgroup of G , then τ , V a and Γ a are compatible. Γ a is dense in V a . If E a is the orbit equivalence relation, then E a ∈ V a . S� lawomir Solecki (Cornell University) Polishable equivalence relations June 2018 10 / 28

Polishable equivalence relations Polishable equivalence relations S� lawomir Solecki (Cornell University) Polishable equivalence relations June 2018 11 / 28

Polishable equivalence relations E an equivalence relation on a Polish space ( X , τ ) E is Polishable if there is a uniformity V on X and a group Γ of transformations of X such that — τ , V and Γ are compatible; — V has a Borel basis with respect to τ ; and, for each x ∈ X , — [ x ] E is G δ with respect to t ( V ); — Γ x is a t ( V )-dense subset of [ x ] E . So t ( V ) is a Polish topology when restricted to each E -class; the assignment of the Polish topologies to E -classes (the witnessing of completeness and separability) is global/uniform. S� lawomir Solecki (Cornell University) Polishable equivalence relations June 2018 12 / 28

Polishable equivalence relations Kechris–Louveau : E an equivalence relation on a Polish space X . E is idealistic if there is an assignment C → I ( C ) that with each equivalence class C of E associates a σ -ideal I ( C ) of subsets of C such that C �∈ I ( C ) and, for each Borel set A ⊆ X × X , the set { x ∈ X : A x ∩ [ x ] E ∈ I ([ x ] E ) } is Borel. S� lawomir Solecki (Cornell University) Polishable equivalence relations June 2018 13 / 28

Polishable equivalence relations Theorem (i) If E be a Polishable Borel equivalence relation, then E is idealistic. (ii) Let a be a continuous action of a Polish group on a Polish space. If the orbit equivalence relation E a is Borel, then it is Polishable. S� lawomir Solecki (Cornell University) Polishable equivalence relations June 2018 14 / 28

Polishable equivalence relations For (i) : if a uniformity V witnesses that E is Polishable, then the assignment C → { M ⊆ C : M meager in C with respect to t ( V ) } , where C is an E -equivalence class, witnesses that E is idealistic. For (ii) : The uniformity V a and the group Γ a witness Polishability of E a . Proof uses work of Becker–Kechris. S� lawomir Solecki (Cornell University) Polishable equivalence relations June 2018 15 / 28

Canonical approximations Canonical approximations S� lawomir Solecki (Cornell University) Polishable equivalence relations June 2018 16 / 28

Canonical approximations ( X , τ ) a topological space An operation on subsets of X × X For A ⊆ X × X symmetric, let A τ = { ( x , y ) ∈ X × X : y ∈ A x and x ∈ A y } . A τ is symmetric. An operation on families of symmetric subsets of X × X For an upward closed family U of symmetric subsets of X × X , let U τ = the upward closure of { A τ : A ∈ U} . S� lawomir Solecki (Cornell University) Polishable equivalence relations June 2018 17 / 28

Canonical approximations τ a topology on X , V a uniformity on X ; assume they are compatible. Let α ≤ ω 1 . ( V ξ ) 0 <ξ<α , where each V ξ is a uniformity on X with t ( V ξ ) completely metrizable, is called a canonical approximation of V if — V 1 = V τ and V ξ = ( V ) t ( V ξ − 1 ) , for all successor ξ < α ; — V λ = � ξ<λ V ξ , for all limit λ < α . Notation: τ 0 = τ , τ ξ = t ( V ξ ) for 0 < ξ < α . S� lawomir Solecki (Cornell University) Polishable equivalence relations June 2018 18 / 28

Canonical approximations Main issues : — termination at V of canonical approximations to V ; — uniqueness of canonical approximations to V (trivial); — existence of canonical approximations to V . S� lawomir Solecki (Cornell University) Polishable equivalence relations June 2018 19 / 28

Canonical approximations Termination Theorem Let τ, V be compatible. Let ( V ξ ) 0 <ξ<β be a canonical approximation of V , with β ≤ ω 1 , and let α < β . ξ<α Π 0 If V has an open basis consisting of sets in � 1+ ξ with respect to τ , then V = V α . S� lawomir Solecki (Cornell University) Polishable equivalence relations June 2018 20 / 28

Canonical approximations Lemma (Main Lemma) Let τ α , α ≤ ω 1 , be completely metrizable topologies on X such that — τ α ⊆ τ β for α < β ≤ ω 1 and — for each α < ω 1 , if F is τ ξ -closed for some ξ < α , then int τ α ( F ) = int τ ω 1 ( F ) . ξ<α Π 0 If τ ω 1 has an open basis consisting of sets in � 1+ ξ with respect to τ 0 , then τ ω 1 = τ α . S� lawomir Solecki (Cornell University) Polishable equivalence relations June 2018 21 / 28

Canonical approximations Existence Theorem Assume that τ , V , and Γ are compatible, and Γ is dense in V . There exists a canonical approximation ( V ξ ) 0 <ξ<ω 1 of V . Moreover, for each 0 < ξ < ω 1 , — V ξ has a basis consisting of Π 0 1+ ξ · 2 sets with respect to τ × τ ; — τ , V ξ , and Γ are compatible and Γ is dense in V ξ . S� lawomir Solecki (Cornell University) Polishable equivalence relations June 2018 22 / 28

Canonical approximations Lemma Assume that τ , V , Γ are compatible and Γ is dense in V . Then V τ is a uniformity. S� lawomir Solecki (Cornell University) Polishable equivalence relations June 2018 23 / 28

Canonical approximations Summary Corollary Assume : τ , V , and Γ compatible, V has a Borel basis with respect to τ , and Γ is dense in V . Then : for some α < ω 1 , there exists a canonical approximation ( V ξ ) 0 <ξ ≤ α of V with V α = V . Moreover : V ξ has a basis consisting of Π 0 1+ ξ · 2 sets with respect to τ × τ , for each 0 < ξ ≤ α . S� lawomir Solecki (Cornell University) Polishable equivalence relations June 2018 24 / 28