Choice-free Stone duality Let A be a Boolean algebra. Let X A be the space of all proper filters of A . We generate a topology by sets � a = { x ∈ X A | a ∈ x } for a ∈ A . Then X A is a spectral space, i.e., compact, T 0 , coherent (compact open sets are closed under intersection and form a basis), sober . The specialization order � is the inclusion ⊆ on proper filters. Second topology to consider : Up ( X A ) , the � -upset topology. Let Int � be the interior operation associated with Up ( X A ) : Int � ( U ) = { x ∈ X | ∀ x ′ � x x ′ ∈ U } . U ⊆ X is � -regular open if it is regular open in Up ( X A ) : an upset s.th. if y �∈ U , then ∃ z � y : z ∈ Int � ( X A \ U ) .
Choice-free Stone duality Let A be a Boolean algebra. Let X A be the space of all proper filters of A . We generate a topology by sets � a = { x ∈ X A | a ∈ x } for a ∈ A . Then X A is a spectral space, i.e., compact, T 0 , coherent (compact open sets are closed under intersection and form a basis), sober . The specialization order � is the inclusion ⊆ on proper filters. Second topology to consider : Up ( X A ) , the � -upset topology. Let Int � be the interior operation associated with Up ( X A ) : Int � ( U ) = { x ∈ X | ∀ x ′ � x x ′ ∈ U } . U ⊆ X is � -regular open if it is regular open in Up ( X A ) : an upset s.th. if y �∈ U , then ∃ z � y : z ∈ Int � ( X A \ U ) . ( X A , � ) is a separative poset, i.e., every principal upset is regular open
Choice-free Stone duality Let A be a Boolean algebra. Let X A be the space of all proper filters of A . We generate a topology by sets � a = { x ∈ X A | a ∈ x } for a ∈ A . Then X A is a spectral space, i.e., compact, T 0 , coherent (compact open sets are closed under intersection and form a basis), sober . The specialization order � is the inclusion ⊆ on proper filters. Second topology to consider : Up ( X A ) , the � -upset topology. Let Int � be the interior operation associated with Up ( X A ) : Int � ( U ) = { x ∈ X | ∀ x ′ � x x ′ ∈ U } . U ⊆ X is � -regular open if it is regular open in Up ( X A ) : an upset s.th. if y �∈ U , then ∃ z � y : z ∈ Int � ( X A \ U ) . ( X A , � ) is a separative poset, i.e., every principal upset is regular open: if y � � x , then ∃ z � y : z ∈ Int � ( X A \ ↑ x ) .
Choice-free Stone duality CO ( X ) = { compact open subsets of X } .
Choice-free Stone duality CO ( X ) = { compact open subsets of X } . Let CO RO ( X A ) be the set of compact open � -regular open sets.
Choice-free Stone duality CO ( X ) = { compact open subsets of X } . Let CO RO ( X A ) be the set of compact open � -regular open sets. If U ∈ CO RO ( X A ) , then Int � ( X \ U ) ∈ CO RO ( X A ) .
Choice-free Stone duality CO ( X ) = { compact open subsets of X } . Let CO RO ( X A ) be the set of compact open � -regular open sets. If U ∈ CO RO ( X A ) , then Int � ( X \ U ) ∈ CO RO ( X A ) . Then CO RO ( X A ) is a Boolean algebra, where U ∧ V = U ∩ V , ¬ U = Int � ( X \ U ) .
Choice-free Stone duality Theorem (Choice-free representation of BAs). Each Boolean algebra A is isomorphic to the Boolean algebra CO RO ( X A ) .
Choice-free Stone duality Theorem (Choice-free representation of BAs). Each Boolean algebra A is isomorphic to the Boolean algebra CO RO ( X A ) . The isomorphism ϕ : A → CO RO ( X A ) is given by ϕ ( a ) = � a .
Choice-free Stone duality Theorem (Choice-free representation of BAs). Each Boolean algebra A is isomorphic to the Boolean algebra CO RO ( X A ) . The isomorphism ϕ : A → CO RO ( X A ) is given by ϕ ( a ) = � a . To show that ϕ is injective we do not need the PFT.
Choice-free Stone duality Theorem (Choice-free representation of BAs). Each Boolean algebra A is isomorphic to the Boolean algebra CO RO ( X A ) . The isomorphism ϕ : A → CO RO ( X A ) is given by ϕ ( a ) = � a . To show that ϕ is injective we do not need the PFT. What kind of space is X A ?
UV-spaces A UV-space is a T 0 space X such that:
UV-spaces A UV-space is a T 0 space X such that: CO RO ( X ) is closed under ∩ and Int � ( X \ · ) ; 1
UV-spaces A UV-space is a T 0 space X such that: CO RO ( X ) is closed under ∩ and Int � ( X \ · ) ; 1 x � � y ⇒ there is a U ∈ CO RO ( X ) s.t. x ∈ U and y / ∈ U ; 2
UV-spaces A UV-space is a T 0 space X such that: CO RO ( X ) is closed under ∩ and Int � ( X \ · ) ; 1 x � � y ⇒ there is a U ∈ CO RO ( X ) s.t. x ∈ U and y / ∈ U ; 2 every proper filter in CO RO ( X ) is CO RO ( x ) for some x ∈ X . 3
UV-spaces A UV-space is a T 0 space X such that: CO RO ( X ) is closed under ∩ and Int � ( X \ · ) ; 1 x � � y ⇒ there is a U ∈ CO RO ( X ) s.t. x ∈ U and y / ∈ U ; 2 every proper filter in CO RO ( X ) is CO RO ( x ) for some x ∈ X . 3 Proposition. Every UV-space is a spectral space.
Choice-free representation of BAs Theorem (Choice-free representation of BAs). For each BA A there is a UV-space X such that A is isomorphic to CO RO ( X ) .
Choice-free representation of BAs Theorem (Choice-free representation of BAs). For each BA A there is a UV-space X such that A is isomorphic to CO RO ( X ) . We extend this correspondence to a full duality of the corresponding categories.
Vietoris space of a Stone space Theorem (Vietoris 1922, Stone version). For every Stone space X its Vietoris space, i.e., the space of nonempty closed sets equipped with the hit-and-miss topology, is again a Stone space. Leopold Vietoris (1891 – 2002)
Vietoris space of a Stone space Let X be a Stone space.
Vietoris space of a Stone space Let X be a Stone space. Let F ( X ) be the set of all nonempty closed subsets of X .
Vietoris space of a Stone space Let X be a Stone space. Let F ( X ) be the set of all nonempty closed subsets of X . The upper Vietoris topology has the basis � U = { F ∈ F ( X ) | F ⊆ U } , U ∈ Ω( X ) . The lower Vietoris topology has the subbasis ♦ V = { F ∈ F ( X ) | F ∩ V � = ∅ } , V ∈ Ω( X ) .
Vietoris space of a Stone space Let X be a Stone space. Let F ( X ) be the set of all nonempty closed subsets of X . The upper Vietoris topology has the basis � U = { F ∈ F ( X ) | F ⊆ U } , U ∈ Ω( X ) . The lower Vietoris topology has the subbasis ♦ V = { F ∈ F ( X ) | F ∩ V � = ∅ } , V ∈ Ω( X ) . The Vietoris topology is the join of the upper and lower Vietoris topologies.
Examples of UV-spaces Let X be a Stone space and UV ( X ) its upper Vietoris space.
Examples of UV-spaces Let X be a Stone space and UV ( X ) its upper Vietoris space. Then UV ( X ) is a UV-space.
Examples of UV-spaces Let X be a Stone space and UV ( X ) its upper Vietoris space. Then UV ( X ) is a UV-space. Assuming the PFT, every UV-space is homeomorphic to UV ( X ) for a Stone space X .
Examples of UV-spaces Let X be a Stone space and UV ( X ) its upper Vietoris space. Then UV ( X ) is a UV-space. Assuming the PFT, every UV-space is homeomorphic to UV ( X ) for a Stone space X . We can prove a localic version of this result choice free.
UV-maps Let X and X ′ be spectral spaces.
UV-maps Let X and X ′ be spectral spaces. A map f : X → Y is called spectral if f − 1 [ U ] is compact open for each compact open U .
UV-maps Let X and X ′ be spectral spaces. A map f : X → Y is called spectral if f − 1 [ U ] is compact open for each compact open U . A UV-map between UV-spaces X and X ′ is a spectral map f : X → X ′
UV-maps Let X and X ′ be spectral spaces. A map f : X → Y is called spectral if f − 1 [ U ] is compact open for each compact open U . A UV-map between UV-spaces X and X ′ is a spectral map f : X → X ′ that is also a p-morphism: if f ( x ) � ′ y ′ , then ∃ y : x � y and f ( y ) = y ′ . y ′ y ′ ∃ y ⇒ x x f ( x ) f ( x )
Duality Theorem (Choice-free UV-duality for BAs). The category of UV-spaces with UV-maps is dually equivalent to the category of Boolean algebras with Boolean homomorphisms.
Comparison Theorem (Choice-free UV-duality for BAs). The category of UV-spaces with UV-maps is dually equivalent to the category of Boolean algebras with Boolean homomorphisms. Theorem (Stone duality). Assuming the PFT, the category of Stone spaces with continuous maps is dually equivalent to the category of Boolean algebras with Boolean homomorphisms.
Comparison Theorem (Choice-free UV-duality for BAs). The category of UV-spaces with UV-maps is dually equivalent to the category of Boolean algebras with Boolean homomorphisms. Theorem (Stone duality). Assuming the PFT, the category of Stone spaces with continuous maps is dually equivalent to the category of Boolean algebras with Boolean homomorphisms. Let’s further compare these with a localic Stone duality.
Stone locales A locale is a complete lattice L satisfying the join-infinite distributive law for each a ∈ L and S ⊆ L : � � a ∧ S = { a ∧ s | s ∈ S } . For any space X , Ω( X ) ordered by ⊆ is a locale.
Stone locales A locale is a complete lattice L satisfying the join-infinite distributive law for each a ∈ L and S ⊆ L : � � a ∧ S = { a ∧ s | s ∈ S } . For any space X , Ω( X ) ordered by ⊆ is a locale. A Stone locale is a locale that is: compact – � S = 1 implies � T = 1 for a finite T ⊆ S . zero-dimensional – every element is a join of complemented elements.
Stone locales A locale is a complete lattice L satisfying the join-infinite distributive law for each a ∈ L and S ⊆ L : � � a ∧ S = { a ∧ s | s ∈ S } . For any space X , Ω( X ) ordered by ⊆ is a locale. A Stone locale is a locale that is: compact – � S = 1 implies � T = 1 for a finite T ⊆ S . zero-dimensional – every element is a join of complemented elements. Theorem (Choice-free localic Stone duality). The category of Stone locales with localic maps is dually equivalent to the category of Boolean algebras with Boolean homomorphisms.
Comparison Theorem (Choice-free UV-duality for BAs). The category of UV-spaces with UV-maps is dually equivalent to the category of Boolean algebras with Boolean homomorphisms. Theorem (Stone duality). Assuming the PFT, the category of Stone spaces with continuous maps is dually equivalent to the category of Boolean algebras with Boolean homomorphisms. Theorem (Choice-free localic Stone duality). The category of Stone locales with localic maps is dually equivalent to the category of Boolean algebras with Boolean homomorphisms.
Comparison Theorem (Choice-free UV-duality for BAs). The category of UV-spaces with UV-maps is dually equivalent to the category of Boolean algebras with Boolean homomorphisms. Theorem (Stone duality). Assuming the PFT, the category of Stone spaces with continuous maps is dually equivalent to the category of Boolean algebras with Boolean homomorphisms. Theorem (Choice-free localic Stone duality). The category of Stone locales with localic maps is dually equivalent to the category of Boolean algebras with Boolean homomorphisms. Let’s now relate the first and third approaches.
Vietoris space of a Stone locale The Vietoris space of X may be defined using the open sets instead of the closed sets, which led Johnstone to define for any Stone locale L the Vietoris space of L .
Vietoris space of a Stone locale The Vietoris space of X may be defined using the open sets instead of the closed sets, which led Johnstone to define for any Stone locale L the Vietoris space of L . Similarly, we can define the upper Vietoris space of L .
Upper Vietoris space of a Stone locale The upper Vietoris space of L has as its set of points L − = { a ∈ L | a � = 1 } with the topology generated by the sets � a = { b ∈ L − | a ∨ b = 1 } , a ∈ L .
Upper Vietoris space of a Stone locale The upper Vietoris space of L has as its set of points L − = { a ∈ L | a � = 1 } with the topology generated by the sets � a = { b ∈ L − | a ∨ b = 1 } , a ∈ L . Idea : switch from nonempty closed sets to their open complements.
Upper Vietoris space of a Stone locale The upper Vietoris space of L has as its set of points L − = { a ∈ L | a � = 1 } with the topology generated by the sets � a = { b ∈ L − | a ∨ b = 1 } , a ∈ L . Idea : switch from nonempty closed sets to their open complements. Since the basic opens of upper Vietoris are � U = { F ∈ F ( X ) | F ⊆ U } for U ∈ Ω( X ) , when we switch to open complements we look at � U = { V ∈ Ω( X ) \{ X } | V c ⊆ U } = { V ∈ Ω( X ) \{ X } | U ∪ V = X } .
Upper Vietoris space of a Stone locale The upper Vietoris space of L has as its set of points L − = { a ∈ L | a � = 1 } with the topology generated by the sets � a = { b ∈ L − | a ∨ b = 1 } , a ∈ L . Idea : switch from nonempty closed sets to their open complements. Since the basic opens of upper Vietoris are � U = { F ∈ F ( X ) | F ⊆ U } for U ∈ Ω( X ) , when we switch to open complements we look at � U = { V ∈ Ω( X ) \{ X } | V c ⊆ U } = { V ∈ Ω( X ) \{ X } | U ∪ V = X } . Theorem (Representation of UV-spaces). X is a UV-space iff X is homeomorphic to the upper Vietoris space of a Stone locale.
Upper Vietoris locale of a Stone locale Johnstone also defined the notion of the Vietoris locale of a Stone locale L such that its space of points is homeomorphic to the Vietoris space of L .
Upper Vietoris locale of a Stone locale Johnstone also defined the notion of the Vietoris locale of a Stone locale L such that its space of points is homeomorphic to the Vietoris space of L . Similarly, one can define the upper Vietoris locale of a Stone locale L such that its space of points is homeomorphic to the upper Vietoris space of L .
Upper Vietoris locale of a Stone locale Johnstone also defined the notion of the Vietoris locale of a Stone locale L such that its space of points is homeomorphic to the Vietoris space of L . Similarly, one can define the upper Vietoris locale of a Stone locale L such that its space of points is homeomorphic to the upper Vietoris space of L . Theorem . X is a UV-space iff X is homeomorphic to the space of points of the upper Vietoris locale of a Stone locale.
Hyperspace & pointfree approaches related upper Vietoris locale of L locales: locale L of filters upper Vietoris space of L pt pt BA A Stone space X spaces: upper Vietoris space of X UV ( A )
Duality dictionary BA UV Stone BA UV-space Stone space homomorphism UV-map continuous map filter ↑ x , x ∈ X closed set ideal U ∈ O RO ( X ) open set principal filter U ∈ CO RO ( X ) clopen set principal ideal U ∈ CO RO ( X ) clopen set maximal filter { x } , x ∈ Max � ( X ) { x } , x ∈ X maximal ideal X \ ↓ x , x ∈ Max � ( X ) X \ { x } , x ∈ X subspace U ∈ CO RO ( X ) subspace U ∈ Clop ( X ) relativization complete algebra complete UV-space ED Stone space atom isolated point isolated point atomic algebra Cl ( X iso ) = X Cl ( X iso ) = X X iso = ∅ X iso = ∅ atomless algebra homomorphic image subspace induced by ↑ x , x ∈ X closed set subalgebra image under UV-map image under continuous map direct product UV-sum disjoint union canonical completion RO ( X ) ℘ ( X ) MacNeille completion RO ( { x ∈ X | ↑ x ∈ CO RO ( X ) } ) RO ( X ) Table: Dictionary for BA , UV , and Stone .
Example applications: antichains of BAs
Example applications: antichains of BAs By an antichain in a BA, we mean a collection C of elements such that for all x , y ∈ C with x � = y , we have x ∧ y = 0.
Example applications: antichains of BAs By an antichain in a BA, we mean a collection C of elements such that for all x , y ∈ C with x � = y , we have x ∧ y = 0. Proposition . Every infinite BA contains infinite chains and infinite antichains.
Example applications: antichains of BAs By an antichain in a BA, we mean a collection C of elements such that for all x , y ∈ C with x � = y , we have x ∧ y = 0. Proposition . Every infinite BA contains infinite chains and infinite antichains. The standard Stone duality proof uses the fact that if X is an infinite set and U ⊆ X , then either U is infinite or X \ U is infinite.
Example applications: antichains of BAs By an antichain in a BA, we mean a collection C of elements such that for all x , y ∈ C with x � = y , we have x ∧ y = 0. Proposition . Every infinite BA contains infinite chains and infinite antichains. The standard Stone duality proof uses the fact that if X is an infinite set and U ⊆ X , then either U is infinite or X \ U is infinite. Our proof is very similar, but we use the fact that if X is an infinite separative poset and U ∈ RO ( X ) , then either U is infinite or ¬ U = Int � ( X \ U ) = { x ∈ X | ∀ x ′ � x x ′ �∈ U } is infinite.
Example applications: antichains of BAs Proposition . Every infinite BA contains infinite chains and infinite antichains. Proof .
Example applications: antichains of BAs Proposition . Every infinite BA contains infinite chains and infinite antichains. Proof . By duality, it suffices to show that in any infinite UV-space X , there is an infinite descending chain U 0 � U 1 � . . . of sets from CO RO ( X ) , as well as an infinite family of pairwise disjoint sets from CO RO ( X ) .
Example applications: antichains of BAs Proposition . Every infinite BA contains infinite chains and infinite antichains. Proof . By duality, it suffices to show that in any infinite UV-space X , there is an infinite descending chain U 0 � U 1 � . . . of sets from CO RO ( X ) , as well as an infinite family of pairwise disjoint sets from CO RO ( X ) . For this it suffices to show that ( ⋆ ) for any n ∈ N , there is a descending chain U 0 ⊇ U 1 ⊇ · · · ⊇ U n of infinite sets from CO RO ( X ) such that U i ∩ ¬ U i + 1 � = ∅ for i ∈ n .
Example applications: antichains of BAs Proposition . Every infinite BA contains infinite chains and infinite antichains. Proof . By duality, it suffices to show that in any infinite UV-space X , there is an infinite descending chain U 0 � U 1 � . . . of sets from CO RO ( X ) , as well as an infinite family of pairwise disjoint sets from CO RO ( X ) . For this it suffices to show that ( ⋆ ) for any n ∈ N , there is a descending chain U 0 ⊇ U 1 ⊇ · · · ⊇ U n of infinite sets from CO RO ( X ) such that U i ∩ ¬ U i + 1 � = ∅ for i ∈ n . For then by DC, there is an infinite descending chain U 0 ⊇ U 1 ⊇ . . . of sets from CO RO ( X ) with U i ∩ ¬ U i + 1 � = ∅ for each i ∈ N , in which case { U 0 ∩ ¬ U 1 , U 1 ∩ ¬ U 2 , . . . } is our antichain.
Example applications: antichains of BAs ( ⋆ ) for any n ∈ N , there is a descending chain U 0 ⊇ U 1 ⊇ · · · ⊇ U n of infinite sets from CO RO ( X ) such that U i ∩ ¬ U i + 1 � = ∅ for i ∈ n . We prove ( ⋆ ) by induction.
Example applications: antichains of BAs ( ⋆ ) for any n ∈ N , there is a descending chain U 0 ⊇ U 1 ⊇ · · · ⊇ U n of infinite sets from CO RO ( X ) such that U i ∩ ¬ U i + 1 � = ∅ for i ∈ n . We prove ( ⋆ ) by induction. Let U 0 = X .
Example applications: antichains of BAs ( ⋆ ) for any n ∈ N , there is a descending chain U 0 ⊇ U 1 ⊇ · · · ⊇ U n of infinite sets from CO RO ( X ) such that U i ∩ ¬ U i + 1 � = ∅ for i ∈ n . We prove ( ⋆ ) by induction. Let U 0 = X . For the inductive step: Since U n is infinite and X is T 0 , there are x , y ∈ U n such that x � � y .
Example applications: antichains of BAs ( ⋆ ) for any n ∈ N , there is a descending chain U 0 ⊇ U 1 ⊇ · · · ⊇ U n of infinite sets from CO RO ( X ) such that U i ∩ ¬ U i + 1 � = ∅ for i ∈ n . We prove ( ⋆ ) by induction. Let U 0 = X . For the inductive step: Since U n is infinite and X is T 0 , there are x , y ∈ U n such that x � � y . Then by the separation property of UV-spaces, there is a V ∈ CO RO ( X ) such that x ∈ V and y �∈ V ,
Example applications: antichains of BAs ( ⋆ ) for any n ∈ N , there is a descending chain U 0 ⊇ U 1 ⊇ · · · ⊇ U n of infinite sets from CO RO ( X ) such that U i ∩ ¬ U i + 1 � = ∅ for i ∈ n . We prove ( ⋆ ) by induction. Let U 0 = X . For the inductive step: Since U n is infinite and X is T 0 , there are x , y ∈ U n such that x � � y . Then by the separation property of UV-spaces, there is a V ∈ CO RO ( X ) such that x ∈ V and y �∈ V , which with y ∈ U n and U n , V ∈ RO ( X ) implies that there is a z � y such that z ∈ U n ∩ ¬ V .
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