Projective Fra ss e limits and homogeneity for tuples of points of - PowerPoint PPT Presentation

Projective Fra¨ ıss´ e limits and homogeneity for tuples of points of the pseudoarc S� lawomir Solecki University of Illinois at Urbana–Champaign Research supported by NSF grant DMS-1266189 July 2016

Outline Outline of Topics The pseudoarc and projective Fra¨ ıss´ e limits 1 Partial homogeneity of the pre-pseudoarc 2 Transfer theorem and homogeneity of the pseudoarc 3 S� lawomir Solecki (University of Illinois) Fra¨ ıss´ e limits and homogeneity for tuples July 2016 2 / 40

The pseudoarc and projective Fra¨ ıss´ e limits The pseudoarc and projective Fra¨ ıss´ e limits S� lawomir Solecki (University of Illinois) Fra¨ ıss´ e limits and homogeneity for tuples July 2016 3 / 40

The pseudoarc and projective Fra¨ ıss´ e limits Direct Fra¨ ıss´ e limits S� lawomir Solecki (University of Illinois) Fra¨ ıss´ e limits and homogeneity for tuples July 2016 4 / 40

The pseudoarc and projective Fra¨ ıss´ e limits Direct Fra¨ ıss´ e limits Fix a relational language. F a family of finite structures taken with embeddings . S� lawomir Solecki (University of Illinois) Fra¨ ıss´ e limits and homogeneity for tuples July 2016 4 / 40

The pseudoarc and projective Fra¨ ıss´ e limits Direct Fra¨ ıss´ e limits Fix a relational language. F a family of finite structures taken with embeddings . F is a Fra¨ ıss´ e family if it has Joint Embedding Property and Amalgamation Property : S� lawomir Solecki (University of Illinois) Fra¨ ıss´ e limits and homogeneity for tuples July 2016 4 / 40

The pseudoarc and projective Fra¨ ıss´ e limits Direct Fra¨ ıss´ e limits Fix a relational language. F a family of finite structures taken with embeddings . F is a Fra¨ ıss´ e family if it has Joint Embedding Property and Amalgamation Property : B 2 A B 1 S� lawomir Solecki (University of Illinois) Fra¨ ıss´ e limits and homogeneity for tuples July 2016 4 / 40

The pseudoarc and projective Fra¨ ıss´ e limits Direct Fra¨ ıss´ e limits Fix a relational language. F a family of finite structures taken with embeddings . F is a Fra¨ ıss´ e family if it has Joint Embedding Property and Amalgamation Property : B 2 A C B 1 S� lawomir Solecki (University of Illinois) Fra¨ ıss´ e limits and homogeneity for tuples July 2016 4 / 40

The pseudoarc and projective Fra¨ ıss´ e limits Direct Fra¨ ıss´ e limits Fix a relational language. F a family of finite structures taken with embeddings . F is a Fra¨ ıss´ e family if it has Joint Embedding Property and Amalgamation Property : B 2 A C B 1 Fra¨ ıss´ e : Countable Fra¨ ıss´ e families have unique countable limit structures . S� lawomir Solecki (University of Illinois) Fra¨ ıss´ e limits and homogeneity for tuples July 2016 4 / 40

The pseudoarc and projective Fra¨ ıss´ e limits Two examples S� lawomir Solecki (University of Illinois) Fra¨ ıss´ e limits and homogeneity for tuples July 2016 5 / 40

The pseudoarc and projective Fra¨ ıss´ e limits Two examples 1. The random graph R = the family of finite graphs R is a Fra¨ ıss´ e family. The random graph is the Fra¨ ıss´ e limit of R . S� lawomir Solecki (University of Illinois) Fra¨ ıss´ e limits and homogeneity for tuples July 2016 5 / 40

The pseudoarc and projective Fra¨ ıss´ e limits Two examples 1. The random graph R = the family of finite graphs R is a Fra¨ ıss´ e family. The random graph is the Fra¨ ıss´ e limit of R . 2. The rational Urysohn space U = the family of finite metric spaces with rational distances U is a Fra¨ ıss´ e family. The Fra¨ ıss´ e limit of U is the rational Urysohn space U 0 . S� lawomir Solecki (University of Illinois) Fra¨ ıss´ e limits and homogeneity for tuples July 2016 5 / 40

The pseudoarc and projective Fra¨ ıss´ e limits Two examples 1. The random graph R = the family of finite graphs R is a Fra¨ ıss´ e family. The random graph is the Fra¨ ıss´ e limit of R . 2. The rational Urysohn space U = the family of finite metric spaces with rational distances U is a Fra¨ ıss´ e family. The Fra¨ ıss´ e limit of U is the rational Urysohn space U 0 . The metric completion U of U 0 is the Urysohn space , the unique universal separable, complete metric space that is ultrahomogeneous with respect to finite subspaces. S� lawomir Solecki (University of Illinois) Fra¨ ıss´ e limits and homogeneity for tuples July 2016 5 / 40

The pseudoarc and projective Fra¨ ıss´ e limits Aim: By analogy with the above approach, develop a logic/combinatorics-based point of view to: — find canonical/combinatorial models for some topological spaces , for example, the pseudoarc, the Menger compacta, the Brouwer curve etc.; — find a unified approach to topological homogeneity results for these spaces. S� lawomir Solecki (University of Illinois) Fra¨ ıss´ e limits and homogeneity for tuples July 2016 6 / 40

The pseudoarc and projective Fra¨ ıss´ e limits There will be three important objects : S� lawomir Solecki (University of Illinois) Fra¨ ıss´ e limits and homogeneity for tuples July 2016 7 / 40

The pseudoarc and projective Fra¨ ıss´ e limits There will be three important objects : the pseudoarc P = a certain compact, connected, second countable space the pre-pseudoarc P = the Cantor set and a certain compact equivalence relation R on it with P / R = P and with a certain relationship to a family of finite structures the augmented pre-pseudoarc P RU = P with additional structure S� lawomir Solecki (University of Illinois) Fra¨ ıss´ e limits and homogeneity for tuples July 2016 7 / 40

The pseudoarc and projective Fra¨ ıss´ e limits The pseudoarc S� lawomir Solecki (University of Illinois) Fra¨ ıss´ e limits and homogeneity for tuples July 2016 8 / 40

The pseudoarc and projective Fra¨ ıss´ e limits K ([0 , 1] 2 ) = compact subsets of [0 , 1] 2 with the Vietoris topology K ([0 , 1] 2 ) is compact S� lawomir Solecki (University of Illinois) Fra¨ ıss´ e limits and homogeneity for tuples July 2016 9 / 40

The pseudoarc and projective Fra¨ ıss´ e limits K ([0 , 1] 2 ) = compact subsets of [0 , 1] 2 with the Vietoris topology K ([0 , 1] 2 ) is compact C = all connected sets in K ([0 , 1] 2 ) C is compact S� lawomir Solecki (University of Illinois) Fra¨ ıss´ e limits and homogeneity for tuples July 2016 9 / 40

The pseudoarc and projective Fra¨ ıss´ e limits K ([0 , 1] 2 ) = compact subsets of [0 , 1] 2 with the Vietoris topology K ([0 , 1] 2 ) is compact C = all connected sets in K ([0 , 1] 2 ) C is compact Bing: There exists a (unique up to homeomorphism) P ∈ C such that { P ′ ∈ C : P ′ homeomorphic to P } is a dense G δ in C . S� lawomir Solecki (University of Illinois) Fra¨ ıss´ e limits and homogeneity for tuples July 2016 9 / 40

The pseudoarc and projective Fra¨ ıss´ e limits K ([0 , 1] 2 ) = compact subsets of [0 , 1] 2 with the Vietoris topology K ([0 , 1] 2 ) is compact C = all connected sets in K ([0 , 1] 2 ) C is compact Bing: There exists a (unique up to homeomorphism) P ∈ C such that { P ′ ∈ C : P ′ homeomorphic to P } is a dense G δ in C . This P is called the pseudoarc . S� lawomir Solecki (University of Illinois) Fra¨ ıss´ e limits and homogeneity for tuples July 2016 9 / 40

The pseudoarc and projective Fra¨ ıss´ e limits K ([0 , 1] N ) = compact subsets of [0 , 1] N with the Vietoris topology K ([0 , 1] N ) is compact C = all connected sets in K ([0 , 1] N ) C is compact Bing: There exists a (unique up to homeomorphism) P ∈ C such that { P ′ ∈ C : P ′ homeomorphic to P } is a dense G δ in C . This P is called the pseudoarc . S� lawomir Solecki (University of Illinois) Fra¨ ıss´ e limits and homogeneity for tuples July 2016 9 / 40

The pseudoarc and projective Fra¨ ıss´ e limits Continuum = compact and connected S� lawomir Solecki (University of Illinois) Fra¨ ıss´ e limits and homogeneity for tuples July 2016 10 / 40

The pseudoarc and projective Fra¨ ıss´ e limits Continuum = compact and connected The pseudoarc is a hereditarily indecomposable continuum, that is, if C 1 , C 2 ⊆ P are continua with C 1 ∩ C 2 � = ∅ , then C 1 ⊆ C 2 or C 2 ⊆ C 1 . S� lawomir Solecki (University of Illinois) Fra¨ ıss´ e limits and homogeneity for tuples July 2016 10 / 40

The pseudoarc and projective Fra¨ ıss´ e limits Projective Fra¨ ıss´ e limits S� lawomir Solecki (University of Illinois) Fra¨ ıss´ e limits and homogeneity for tuples July 2016 11 / 40

The pseudoarc and projective Fra¨ ıss´ e limits Fix a relational language. F a family of finite structures taken with epimorphisms between structures in F . S� lawomir Solecki (University of Illinois) Fra¨ ıss´ e limits and homogeneity for tuples July 2016 12 / 40

The pseudoarc and projective Fra¨ ıss´ e limits Fix a relational language. F a family of finite structures taken with epimorphisms between structures in F . F is called a projective Fra¨ ıss´ e family if it has Joint Epimorphism Property and Projective Amalgamation Property . B 1 A C B 2 S� lawomir Solecki (University of Illinois) Fra¨ ıss´ e limits and homogeneity for tuples July 2016 12 / 40