A Lelek-like compact metric space Joint ongoing work with R. Camerlo Gianluca Basso 12 June 2018 Université de Lausanne and Università di Torino
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A Lelek-like compact metric space Joint ongoing work with R. Camerlo Gianluca Basso 12 June 2018 Université de Lausanne and Università di Torino
Overview 1. An introduction to projective Fraïssé theory 2. A universal space and its characterization 3. Open problems 2
An introduction to projective Fraïssé theory
Approximate a space with a projective sequence of open covers Let Y be a compact metric space. Let ( U n ) n ∈ ω be a sequence of finite open covers of (dense subsets of) Y such that: • for each n , U n + 1 refines U n , that is, for each U ∈ U n + 1 there is U ′ ∈ U n such that U ⊆ U ′ . • lim n →∞ mesh ( U n ) = 0, where mesh ( U ) = max { diam ( U ) | U ∈ U} . Define: { � } � ( U n ) n ∈ ω ∈ ∏ � U n + 1 ⊆ U n U ∞ = U n � � n ∈ ω Then U ∞ , the projective limit of the U n ’s, is a closed subset of the product with the product topology where each U n is given the discrete topology. Then q Y : U ∞ − → Y ∩ ( U n ) n ∈ ω �→ U n n ∈ ω is a continuous surjective function. 3
From covers to graphs... We associate to each finite open cover U n a graph G n = ( U n , R n ) where UR n U ′ if and only if U ∩ U ′ ̸ = ∅ . Then we can associate a graph G ∞ = ( U ∞ , R ∞ ) also to the projective limit U ∞ by letting ( U n ) n ∈ ω R ∞ ( U ′ n ) n ∈ ω iff ∀ n , U n R n U ′ n Proposition ( U n ) n ∈ ω R ∞ ( U ′ ⇒ q Y (( U n ) n ∈ ω ) = q Y (( U ′ n ) n ∈ ω ⇐ n ) n ∈ ω ) Proof. ∩ ∩ ∀ n , U n ∩ U ′ U n = U ′ n ̸ = ∅ ⇐ ⇒ n n ∈ ω n ∈ ω So R ∞ is an equivalence relation and q Y : U ∞ → U ∞ / R ∞ ≃ Y . 4
...to L -structures Let L be a relational language containing a distinguished binary relation symbol R . If Y has an L -structure on it such that the interpretation of each relation symbol is closed and R Y is equality, then to each finite open cover U n we can associate an L -structure G n = ( U n , R G n , . . . ) , where we let ( U 1 , . . . , U s ) ∈ S G n if and only if, there are x i ∈ U i , 1 ≤ i ≤ s , such that ( x 1 , . . . , x s ) ∈ S Y , for each S ∈ L . ∈ S G ∞ if and only if for If, as before, we let ( U 1 n ) n ∈ ω , . . . , ( U s ( ) n ) n ∈ ω ∈ S G n if and only if there are x i U 1 n , . . . , U s each n ∈ ω , n ∈ U i n , ( ) n n ) ∈ S Y , if and only if, since S Y is closed, 1 ≤ i ≤ s , such that ( x 1 n , . . . , x s ( )) ( U 1 ( U s ∈ S Y . q Y , . . . , q Y ( ) ( n ) n ∈ ω n ) n ∈ ω 5
Modeling the refinement relation Definition Let G , G ′ be L -structures. An epimorphism ϕ : G ′ → G is a continuous surjective function such that: ( a 1 , . . . , a s ) ∈ S G iff ∃ ( a ′ s ) ∈ S G ′ , such that ϕ ( a ′ 1 , . . . , a ′ i ) = a i , ∀ 1 ≤ i ≤ s . φ 0 φ 1 So, given a sequence G 0 − G 1 − G 2 · · · , we can define the ← ← projective limit of ( G n , ϕ n ) as { � } � G ∞ = ( a n ) n ∈ ω ∈ ∏ G n � ∀ n , ϕ n ( a n + 1 ) = a n . � � n ∈ ω 6
Modeling mesh → 0 φ 0 φ 1 A sequence G 0 − G 1 − G 2 · · · is fine if it models the mesh going to ← ← 0, that is for each n ∈ ω and each a , a ′ ∈ G n , if d R ( a , a ′ ) ≥ 2 then there is m ≥ n such that ϕ − 1 m − 1 · · · ϕ − 1 n ( a ) , ϕ − 1 m − 1 · · · ϕ − 1 d R ( n ( a ′ ) ) ≥ 3 . If a sequence ( G n , ϕ n ) is fine then R G ∞ is an equivalence relation. Say that ( G n , ϕ n ) approximates G ∞ / R G ∞ . 7
Topological → combinatorial Given a class C of compact metric spaces (with L -structure) we can look at a class Γ of finite L - structures such that each Y ∈ C is approximated by a fine sequence of Γ . In some cases one can determine combinatorial properties Γ on the basis of the topological properties of the class C . Proposition A compact metric space is connected if and only if it can be approximated by a sequence of connected graphs. Theorem (Irwin-Solecki, 2006) A compact metric space is chainable and connected if and only if it can be approximated by a sequence of finite connected linear graphs. 8
Universal sequences Let Γ be a class of finite L -structures. χ 0 χ 1 A sequence H 0 − H 1 − H 2 · · · in Γ is called universal for Γ if for ← ← φ 0 φ 1 any other sequence G 0 − G 1 − G 2 · · · from Γ there are an ← ← increasing subsequence χ 0 ˆ χ 1 ˆ H i 0 − H i 1 − H i 2 · · · , ← ← where ˆ χ n = χ i n χ i n + 1 · · · χ i n + 1 − 1 , and epimorphisms f n : H i n → G n such that ϕ n f n + 1 = f n ˆ χ n . χ 0 χ 1 If H 0 − H 1 − H 2 · · · is a universal fine sequence for Γ it follows ← ← that H ∞ / R H ∞ is projectively universal for all compact metric spaces (with L -structure) approximated by sequences in Γ , since f ∞ = ( f n ) n ∈ ω induces a continuous surjection (epimorphism) on the quotients: q ∗ ( f ∞ ) : X = H ∞ / R H ∞ → G ∞ / R G ∞ = Y x �→ q Y f ∞ q − 1 X ( x ) . 9
Fraïssé Theory A class Γ of finite L -structures such that: • (JPP) ∀ G , G ′ ∈ Γ , ∃ H ∈ Γ and epimorphisms ϕ : H → G , ϕ ′ : H → G ′ ; • (AP) ∀ G , G ′ , G ′′ ∈ Γ and epimorphisms ϕ : G → G ′′ , ϕ ′ : G ′ → G ′′ , ∃ H ∈ Γ and epimorphisms ψ : H → G , ψ ′ : H → G ′ such that ϕψ = ϕ ′ ψ ′ ; is called a projective Fraïssé class. Theorem (Irwin, Solecki, 2006) If Γ is a projective Fraïssé class then there is a universal sequence χ 0 χ 1 H 0 − H 1 − H 2 · · · for Γ . Moreover (uniqueness) any two universal ← ← sequences for Γ have the same projective limit H ∞ (the Fraïssé limit of Γ ) up to isomorphism, i.e. injective epimorphism, and (ultrahomogeneity) given two epimorphisms ϕ, ϕ ′ : H ∞ → G ∈ Γ there exists an isomorphism α ∞ : H ∞ → H ∞ such that ϕ = ϕ ′ α ∞ . 10
Some consequences Let Γ be a Fraïssé class of finite L -structures whose sequences approximate the compact metric spaces (with L -structure) of a class χ 0 χ 1 C , and H 0 − H 1 − H 2 · · · be a fine universal sequence for Γ . ← ← Denote H ∞ / R H ∞ by X C . Then: • approximate projective homogeneity: let Y ∈ C and f , f ′ : X C → Y be continuous surjections (epimorphisms), then, for any ϵ > 0, there exists a homeomorphism α : X C → X C such that for any x ∈ X C , d ( f ( x ) , f ′ α ( x )) < ϵ ; • any homeomorphism (isomorphism) h : X C → : X C uniformly approximable by homeomorphisms coming from isomorphisms α ∞ : H ∞ → H ∞ . 11
Linear graphs and the pseudo-arc Theorem (Irwin-Solecki, 2006) The class Γ of all finite connected linear graphs is a Fraïssé class. χ 0 χ 1 Therefore it has a universal sequence H 0 − H 1 − H 2 · · · . The ← ← universal sequence is fine thus and H ∞ / R H ∞ is projectively universal and projectively approximately homogeneous for the class of all chainable and connected compact metric spaces. Theorem (Irwin-Solecki, 2006) H ∞ / R H ∞ is homeomorphic to the pseudo-arc. 12
A universal space and its characterization
Hasse Diagrams of Partial Orders Let L = { R , ≤} . An L -structure A is a Hasse diagram of a partial order if ≤ A is a partial order and xR A x ′ if and only if x = x ′ or x is the immediate predecessor or successor of x ′ . Let Π ∇ be the class of all Hasse diagram of finite partial orders which do not contain R -cycles. An order preserving surjection ϕ : A ′ → A between structures in Π ∇ is an epimorphisms if and only if for every maximal linear sub-order M ⊆ A there is a maximal linear sub-order of M ′ ⊆ A ′ such that ϕ [ M ′ ] = M . Theorem (B.- Camerlo) Π ∇ is a projective Fraïssé class, whose universal sequence χ 0 χ 1 P 0 − P 1 − P 2 · · · P ∞ is fine. ← ← 13
Characterization theorem A fence is a compact disjoint union of points and arcs. The Cantor fence is 2 N × [ 0 , 1 ] . An endpoint of a compact metric space Y is a point x such that, for any embedding h : [ 0 , 1 ] → Y such that x ∈ ran ( h ) , x = h ( 0 ) or x = h ( 1 ) . Theorem (B.- Camerlo) Let X be a sub-fence of the Cantor fence such that for any clopen non-empty C ⊆ 2 N and two open subintervals ( a , b ) , ( c , d ) of [ 0 , 1 ] , letting U = C × ( a , b ) ∩ X and V = C × ( c , d ) ∩ X, if U , V are ( ) ( ) nonempty then there is an arc in X whose endpoints lie in U , V, respectively. Then X is homeomorphic to P ∞ / R P ∞ . Notice that the set of endpoints of X is dense. 14
Cofinal families If Γ is a projective Fraïssé class and Γ ′ ⊆ Γ is cofinal, that is, ∀ G ∈ Γ , ∃ G ′ ∈ Γ ′ , ϕ : G ′ → G , then Γ ′ is a projective Fraïssé class with the same limit as Γ , up to isomorphism. The class Π □ of all Hasse diagrams of finite disjoint unions of finite linear orders is cofinal in Π ∇ . 15
Outline of the proof of the characterization • Consider a space X which satisfies the assumptions of the theorem. • Find an appropriate fine projective sequence of Π □ which approximates X . • Prove that such a sequence is a universal sequence for Π □ . • Conclude that X ∞ is isomorphic to P ∞ by uniqueness of the projective Fraïssé limit and thus that their quotients are homeomorphic. 16
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