Lelek fan and generalizations of finite Gowers’ FIN k Theorem Dana Bartoˇ sov´ a (USP) Aleksandra Kwiatkowska (UCLA) SETTOP 2014 Novi Sad, Serbia August 18–21, 2014 This work was supported by the grant FAPESP 2013/14458-9. Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
Topological structures L = { f i , R j } - first-order language Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
Topological structures L = { f i , R j } - first-order language X is a topological L -structure if X - second-countable, compact, 0-dimensional X - L -structure f i - continuous R j - closed Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
Topological structures L = { f i , R j } - first-order language X is a topological L -structure if X - second-countable, compact, 0-dimensional X - L -structure f i - continuous R j - closed � Y is an epimorphism if φ : X φ - continuous φ - surjective homomorphism � ∃ ( x 1 , . . . , x n ) ∈ R X ( y 1 , . . . , y n ) ∈ R Y j φ ( x i ) = y i j Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
Projective Fra¨ ıss´ e theory F - countable class of finite topological L -structures Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
Projective Fra¨ ıss´ e theory F - countable class of finite topological L -structures F - projective Fra¨ ıss´ e class if � A and C � B JPP ∀ A, B ∈ F ∃ C ∈ F and epi C � A and C � A ∃ D ∈ F AP ∀ A, B, C ∈ F and epi f : B � B and l : D � C such that f ◦ k = g ◦ l and epi k : D Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
Projective Fra¨ ıss´ e theory F - countable class of finite topological L -structures F - projective Fra¨ ıss´ e class if � A and C � B JPP ∀ A, B ∈ F ∃ C ∈ F and epi C � A and C � A ∃ D ∈ F AP ∀ A, B, C ∈ F and epi f : B � B and l : D � C such that f ◦ k = g ◦ l and epi k : D F - projective Fra¨ ıss´ e limit of F if � A PU ∀ A ∈ F ∃ epi F � S ∃ A ∈ F , R ∀ S finite discrete space and surjection f : F � A and function f ′ : A � S such that f = f ′ ◦ φ epi φ : F � F such that � A ∃ iso ψ : F H ∀ A ∈ F and epi φ 1 , φ 2 : F φ 2 = φ 1 ◦ ψ Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
Projective Fra¨ ıss´ e theory F - countable class of finite topological L -structures F - projective Fra¨ ıss´ e class if � A and C � B JPP ∀ A, B ∈ F ∃ C ∈ F and epi C � A and C � A ∃ D ∈ F AP ∀ A, B, C ∈ F and epi f : B � B and l : D � C such that f ◦ k = g ◦ l and epi k : D F - projective Fra¨ ıss´ e limit of F if � A PU ∀ A ∈ F ∃ epi F � S ∃ A ∈ F , R ∀ S finite discrete space and surjection f : F � A and function f ′ : A � S such that f = f ′ ◦ φ epi φ : F � F such that � A ∃ iso ψ : F H ∀ A ∈ F and epi φ 1 , φ 2 : F φ 2 = φ 1 ◦ ψ Theorem (Irwin, Solecki) Every projective Fra¨ ıss´ e class has a projective Fra¨ ıss´ e limit Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
Projective Fra¨ ıss´ e theory F - countable class of finite topological L -structures F - projective Fra¨ ıss´ e class if � A and C � B JPP ∀ A, B ∈ F ∃ C ∈ F and epi C � A and C � A ∃ D ∈ F AP ∀ A, B, C ∈ F and epi f : B � B and l : D � C such that f ◦ k = g ◦ l and epi k : D F - projective Fra¨ ıss´ e limit of F if � A PU ∀ A ∈ F ∃ epi F � S ∃ A ∈ F , R ∀ S finite discrete space and surjection f : F � A and function f ′ : A � S such that f = f ′ ◦ φ epi φ : F � F such that � A ∃ iso ψ : F H ∀ A ∈ F and epi φ 1 , φ 2 : F φ 2 = φ 1 ◦ ψ Theorem (Irwin, Solecki) Every projective Fra¨ ıss´ e class has a projective Fra¨ ıss´ e limit which is unique up to an isomorphism. Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
Finite trees a, b ∈ ( T, < T ) - a finite tree Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
Finite trees a, b ∈ ( T, < T ) - a finite tree ( a, b ) ∈ R T ← → ( a = b or b < T a & ∄ c ∈ T b < T c < T a) Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
Finite trees a, b ∈ ( T, < T ) - a finite tree ( a, b ) ∈ R T ← → ( a = b or b < T a & ∄ c ∈ T b < T c < T a) Projective Fra¨ ıss´ e classes F t - finite trees with R Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
Finite trees a, b ∈ ( T, < T ) - a finite tree ( a, b ) ∈ R T ← → ( a = b or b < T a & ∄ c ∈ T b < T c < T a) Projective Fra¨ ıss´ e classes F t - finite trees with R F - finite fans - coinitial in F t Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
Finite trees a, b ∈ ( T, < T ) - a finite tree ( a, b ) ∈ R T ← → ( a = b or b < T a & ∄ c ∈ T b < T c < T a) Projective Fra¨ ıss´ e classes F t - finite trees with R F - finite fans - coinitial in F t F < - finite fans with linearly ordered branches Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
Lelek fan L - limit of F t = limit of F Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
Lelek fan L - limit of F t = limit of F s - symmetrized R L - equivalence relation with 1 and 2-point R L classes Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
Lelek fan L - limit of F t = limit of F s - symmetrized R L - equivalence relation with 1 and 2-point R L classes Theorem L /R L s is the Lelek fan. Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
Lelek fan L - limit of F t = limit of F s - symmetrized R L - equivalence relation with 1 and 2-point R L classes Theorem L /R L s is the Lelek fan. Lelek fan = unique non-trivial subcontinuum of the Cantor fan with a dense set of endpoints (Bula-Oversteegen, Charatonik) Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
Lelek fan L - limit of F t = limit of F s - symmetrized R L - equivalence relation with 1 and 2-point R L classes Theorem L /R L s is the Lelek fan. Lelek fan = unique non-trivial subcontinuum of the Cantor fan with a dense set of endpoints (Bula-Oversteegen, Charatonik) Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
� Homeo Aut Aut( L , R L s ) and Homeo( L ) + the compact-open topology Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
� Homeo Aut Aut( L , R L s ) and Homeo( L ) + the compact-open topology � L /R L s ∼ π : L = L Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
� Homeo Aut Aut( L , R L s ) and Homeo( L ) + the compact-open topology � L /R L s ∼ π : L = L induces a continuous embedding Aut( L , R L s ) ֒ → Homeo( L ) Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
� Homeo Aut Aut( L , R L s ) and Homeo( L ) + the compact-open topology � L /R L s ∼ π : L = L induces a continuous embedding Aut( L , R L s ) ֒ → Homeo( L ) with a dense image Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
� Homeo Aut Aut( L , R L s ) and Homeo( L ) + the compact-open topology � L /R L s ∼ π : L = L induces a continuous embedding Aut( L , R L s ) ֒ → Homeo( L ) with a dense image h ∗ h �→ h ∗ ◦ π. π ◦ h = Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
Homeo( L ) Polish group with the compact-open topology Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
Homeo( L ) Polish group with the compact-open topology is totally disconnected i.e., for every f, g ∈ Homeo( L ) there exists a clopen U ⊂ Homeo( L ) such that f ∈ U and g / ∈ U. Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
Homeo( L ) Polish group with the compact-open topology is totally disconnected i.e., for every f, g ∈ Homeo( L ) there exists a clopen U ⊂ Homeo( L ) such that f ∈ U and g / ∈ U. is generated by every neighbourhood of the identity i.e., for every g ∈ Homeo( L ) and every ε > 0 there exist f 1 , . . . , f n ∈ Homeo( L ) such that g = f n ◦ . . . ◦ f 1 and d sup (id , f i ) < ε. Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
Homeo( L ) Polish group with the compact-open topology is totally disconnected i.e., for every f, g ∈ Homeo( L ) there exists a clopen U ⊂ Homeo( L ) such that f ∈ U and g / ∈ U. is generated by every neighbourhood of the identity i.e., for every g ∈ Homeo( L ) and every ε > 0 there exist f 1 , . . . , f n ∈ Homeo( L ) such that g = f n ◦ . . . ◦ f 1 and d sup (id , f i ) < ε. does not contain any open subgroup, in particular it is not non-archimedean. Dana Bartoˇ sov´ a, Aleksandra Kwiatkowska Lelek fan and Gowers’ FIN k Theorem
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