Introduction A proof of a rigidity result Further work Homeomorphisms of ˇ Cech-Stone remainders: the zero-dimensional case Paul McKenney Joint work with Ilijas Farah BLAST 2018 Homeomorphisms of ˇ Paul McKenney Cech-Stone remainders: the zero-dimensional
Introduction A proof of a rigidity result Further work Given a topological space X , let X ∗ = β X \ X denote its ˇ Cech-Stone remainder. Homeomorphisms of ˇ Paul McKenney Cech-Stone remainders: the zero-dimensional
Introduction A proof of a rigidity result Further work Given a topological space X , let X ∗ = β X \ X denote its ˇ Cech-Stone remainder. Question If X ∗ ≃ Y ∗ , how similar do X and Y have to be? Homeomorphisms of ˇ Paul McKenney Cech-Stone remainders: the zero-dimensional
Introduction A proof of a rigidity result Further work Given a topological space X , let X ∗ = β X \ X denote its ˇ Cech-Stone remainder. Question If X ∗ ≃ Y ∗ , how similar do X and Y have to be? Theorem (Parovicenko) Assume the Continuum Hypothesis. Then for every zero-dimensional, locally compact, noncompact, Hausdorff space X, X ∗ ≃ ω ∗ . Homeomorphisms of ˇ Paul McKenney Cech-Stone remainders: the zero-dimensional
Introduction A proof of a rigidity result Further work Theorem (Parovicenko) Assume the Continuum Hypothesis. Then for every zero-dimensional, locally compact, noncompact, Hausdorff space X, X ∗ ≃ ω ∗ . Homeomorphisms of ˇ Paul McKenney Cech-Stone remainders: the zero-dimensional
Introduction A proof of a rigidity result Further work Theorem (Parovicenko) Assume the Continuum Hypothesis. Then for every zero-dimensional, locally compact, noncompact, Hausdorff space X, X ∗ ≃ ω ∗ . Sketch. Let C ( X ) denote the Boolean algebra of clopen subsets of X , and K ( X ) its ideal of compact-open sets. Homeomorphisms of ˇ Paul McKenney Cech-Stone remainders: the zero-dimensional
Introduction A proof of a rigidity result Further work Theorem (Parovicenko) Assume the Continuum Hypothesis. Then for every zero-dimensional, locally compact, noncompact, Hausdorff space X, X ∗ ≃ ω ∗ . Sketch. Let C ( X ) denote the Boolean algebra of clopen subsets of X , and K ( X ) its ideal of compact-open sets. Then by Stone duality, it’s enough to prove that C ( X ) / K ( X ) ≃ P ( ω ) / fin. Homeomorphisms of ˇ Paul McKenney Cech-Stone remainders: the zero-dimensional
Introduction A proof of a rigidity result Further work Theorem (Parovicenko) Assume the Continuum Hypothesis. Then for every zero-dimensional, locally compact, noncompact, Hausdorff space X, X ∗ ≃ ω ∗ . Sketch. Let C ( X ) denote the Boolean algebra of clopen subsets of X , and K ( X ) its ideal of compact-open sets. Then by Stone duality, it’s enough to prove that C ( X ) / K ( X ) ≃ P ( ω ) / fin. These Boolean algebras are both countably saturated and have size c = ℵ 1 . Homeomorphisms of ˇ Paul McKenney Cech-Stone remainders: the zero-dimensional
Introduction A proof of a rigidity result Further work Theorem (Parovicenko) Assume the Continuum Hypothesis. Then for every zero-dimensional, locally compact, noncompact, Hausdorff space X, X ∗ ≃ ω ∗ . Sketch. Let C ( X ) denote the Boolean algebra of clopen subsets of X , and K ( X ) its ideal of compact-open sets. Then by Stone duality, it’s enough to prove that C ( X ) / K ( X ) ≃ P ( ω ) / fin. These Boolean algebras are both countably saturated and have size c = ℵ 1 . A back-and-forth argument finishes the proof. Homeomorphisms of ˇ Paul McKenney Cech-Stone remainders: the zero-dimensional
Introduction A proof of a rigidity result Further work So under CH, ˇ Cech-Stone remainders are very malleable . Homeomorphisms of ˇ Paul McKenney Cech-Stone remainders: the zero-dimensional
Introduction A proof of a rigidity result Further work So under CH, ˇ Cech-Stone remainders are very malleable . Theorem (Farah-McKenney) Assume OCA and MA ℵ 1 . Let X and Y be zero-dimensional, locally compact Polish spaces, and suppose ϕ : X ∗ → Y ∗ is a homeomorphism. Then there are cocompact subsets of X and Y which are homeomorphic, and moreover ϕ is induced by such a homeomorphism. Homeomorphisms of ˇ Paul McKenney Cech-Stone remainders: the zero-dimensional
Introduction A proof of a rigidity result Further work So under CH, ˇ Cech-Stone remainders are very malleable . Theorem (Farah-McKenney) Assume OCA and MA ℵ 1 . Let X and Y be zero-dimensional, locally compact Polish spaces, and suppose ϕ : X ∗ → Y ∗ is a homeomorphism. Then there are cocompact subsets of X and Y which are homeomorphic, and moreover ϕ is induced by such a homeomorphism. This says that under OCA and MA ℵ 1 , ˇ Cech-Stone remainders (in a certain class) are very rigid . Homeomorphisms of ˇ Paul McKenney Cech-Stone remainders: the zero-dimensional
Introduction A proof of a rigidity result Further work Notation: given a set V we write [ V ] 2 for the set of unordered pairs { v , w } ( v � = w ) of elements of V . Homeomorphisms of ˇ Paul McKenney Cech-Stone remainders: the zero-dimensional
Introduction A proof of a rigidity result Further work Notation: given a set V we write [ V ] 2 for the set of unordered pairs { v , w } ( v � = w ) of elements of V . The Open Coloring Axiom states: Homeomorphisms of ˇ Paul McKenney Cech-Stone remainders: the zero-dimensional
Introduction A proof of a rigidity result Further work Notation: given a set V we write [ V ] 2 for the set of unordered pairs { v , w } ( v � = w ) of elements of V . The Open Coloring Axiom states: For every separable metric space V and every open G ⊆ [ V ] 2 , (where [ V ] 2 is identified with V × V minus the diagonal), either Homeomorphisms of ˇ Paul McKenney Cech-Stone remainders: the zero-dimensional
Introduction A proof of a rigidity result Further work Notation: given a set V we write [ V ] 2 for the set of unordered pairs { v , w } ( v � = w ) of elements of V . The Open Coloring Axiom states: For every separable metric space V and every open G ⊆ [ V ] 2 , (where [ V ] 2 is identified with V × V minus the diagonal), either there is an uncountable A ⊆ V such that [ A ] 2 ⊆ G ( G has an uncountable complete subgraph ), or Homeomorphisms of ˇ Paul McKenney Cech-Stone remainders: the zero-dimensional
Introduction A proof of a rigidity result Further work Notation: given a set V we write [ V ] 2 for the set of unordered pairs { v , w } ( v � = w ) of elements of V . The Open Coloring Axiom states: For every separable metric space V and every open G ⊆ [ V ] 2 , (where [ V ] 2 is identified with V × V minus the diagonal), either there is an uncountable A ⊆ V such that [ A ] 2 ⊆ G ( G has an uncountable complete subgraph ), or there is a partition V = � ∞ n =1 B n such that for all n , [ B n ] 2 ∩ G = ∅ . ( G is countably chromatic .) Homeomorphisms of ˇ Paul McKenney Cech-Stone remainders: the zero-dimensional
Introduction A proof of a rigidity result Further work Notation: given a set V we write [ V ] 2 for the set of unordered pairs { v , w } ( v � = w ) of elements of V . The Open Coloring Axiom states: For every separable metric space V and every open G ⊆ [ V ] 2 , (where [ V ] 2 is identified with V × V minus the diagonal), either there is an uncountable A ⊆ V such that [ A ] 2 ⊆ G ( G has an uncountable complete subgraph ), or there is a partition V = � ∞ n =1 B n such that for all n , [ B n ] 2 ∩ G = ∅ . ( G is countably chromatic .) Note: the set-theoretic strength of OCA is in the “for every separable metric V ” part. For instance, OCA is true in ZFC for analytic V ⊆ R (by a Cantor-Bendixon style argument). Homeomorphisms of ˇ Paul McKenney Cech-Stone remainders: the zero-dimensional
Introduction A proof of a rigidity result Further work OCA and MA ℵ 1 have been used to prove similar things before, most prominently Homeomorphisms of ˇ Paul McKenney Cech-Stone remainders: the zero-dimensional
Introduction A proof of a rigidity result Further work OCA and MA ℵ 1 have been used to prove similar things before, most prominently Theorem (Veliˇ ckovi´ c, 1993) Assume OCA and MA ℵ 1 . Then every homeomorphism of ω ∗ is induced by a bijection e : ω \ F 1 → ω \ F 2 where F 1 , F 2 are finite. Homeomorphisms of ˇ Paul McKenney Cech-Stone remainders: the zero-dimensional
Introduction A proof of a rigidity result Further work OCA and MA ℵ 1 have been used to prove similar things before, most prominently Theorem (Veliˇ ckovi´ c, 1993) Assume OCA and MA ℵ 1 . Then every homeomorphism of ω ∗ is induced by a bijection e : ω \ F 1 → ω \ F 2 where F 1 , F 2 are finite. Theorem (Farah, 1996) (Same as our result but restricted to countable X and Y .) Homeomorphisms of ˇ Paul McKenney Cech-Stone remainders: the zero-dimensional
Introduction A proof of a rigidity result Further work OCA and MA ℵ 1 have been used to prove similar things before, most prominently Theorem (Veliˇ ckovi´ c, 1993) Assume OCA and MA ℵ 1 . Then every homeomorphism of ω ∗ is induced by a bijection e : ω \ F 1 → ω \ F 2 where F 1 , F 2 are finite. Theorem (Farah, 1996) (Same as our result but restricted to countable X and Y .) Theorem (Farah, 2011) Assume OCA. Then every automorphism of B ( ℓ 2 ) / K ( ℓ 2 ) is induced by a linear isometry between closed subspaces of ℓ 2 with finite codimension. Homeomorphisms of ˇ Paul McKenney Cech-Stone remainders: the zero-dimensional
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