A locally finite characterization of AE(0) and related classes of compacta David Milovich Texas A&M International University david.milovich@tamiu.edu http://www.tamiu.edu/ ∼ dmilovich/ March 13, 2014 Spring Topology and Dynamics Conference University of Richmond 1 / 9
Stone duality notation ◮ A compactum is a compact Hausdorff space. ◮ A boolean space is a compactum with a clopen base. ◮ Clop is the contravarient functor from boolean spaces and continuous maps to boolean algebras and homomorphisms. ◮ Clop( X ) is ( { K ⊆ X : K clopen } , ∩ , ∪ , K �→ X \ K } ). ◮ Clop( f )( K ) = f − 1 [ K ]. ◮ Modulo isomorphism, the inverse of Clop is the functor Ult: ◮ Ult( A ) is { U ⊆ A : U ultrafilter } with clopen base {{ U ∈ Ult( A ) : a ∈ U } : a ∈ A } ; ◮ Ult( φ )( U ) = φ − 1 [ U ]. 2 / 9
Open is dual to relatively complete. ◮ A boolean subalgebra A of B is called relatively complete if every b ∈ B has a least upper bound in A . ◮ Let A ≤ rc B abbreviate “ A is relatively complete in B .” ◮ A boolean homomorphism φ : A → B is called relatively complete if φ [ A ] ≤ rc B . ◮ A boolean homomorphism φ is relatively complete iff Ult( φ ) is open. 3 / 9
AE(0) spaces Definition A boolean space X is an absolute extensor of dimension zero , or AE(0) for short, if, for every continuous f : Y → X with Y ⊆ Z boolean, f extends to a continuous g : Z → X . 4 / 9
AE(0) spaces Definition A boolean space X is an absolute extensor of dimension zero , or AE(0) for short, if, for every continuous f : Y → X with Y ⊆ Z boolean, f extends to a continuous g : Z → X . Given a boolean space X of weight ≤ κ , the following are known to be equivalent: ◮ X is AE(0). ◮ X is Dugundji, i.e. , a retract of 2 κ . ◮ X × 2 κ ∼ = 2 κ . = Y ⊆ 2 κ and, for all α < β < κ , ◮ There exists Y such that X ∼ the projection π α,β : Y ↾ β → Y ↾ α is open. ◮ Clop( X ) has an additive rc-skeleton, i.e. , if n < ω , θ is a regular cardinal, and Clop( X ) ∈ N i ≺ H ( θ ) for all i < n , then � � Clop( X ) ∩ � i < n N i ≤ rc Clop( X ). 4 / 9
Multicommutativity ◮ A poset diagram of boolean spaces is pair of sequences ( � X ,� f ) with ◮ dom( � X ) a poset, ◮ X i a boolean space for all i ∈ dom( � X ), ◮ f j , i : X i → X j continuous for all j < i , and ◮ f k , i = f k , j ◦ f j , i for all k < j < i . ◮ Given a poset diagram ( � X ,� f ) and I ⊆ dom( � X ), let � � � lim( X i : i ∈ I ) = p ∈ X i : ∀{ j < i } ⊆ I p ( j ) = f j , i ( p ( i )) . i ∈ I ◮ Call a poset diagram ( � X ,� f ) multicommutative if, for all i ∈ dom( � X ), � j < i f j , i maps X i onto lim( X j : j < i ). 5 / 9
A new characterization of AE(0) ◮ A poset P is called locally finite if every lower cone is finite. ◮ A poset diagram ( � X ,� f ) is locally finite if dom( � X ) is locally finite and every X i is finite . ◮ A locally finite poset is a lattice iff every nonempty finite subset has a least upper bound. ◮ A poset diagram ( � X ,� f ) is called a lattice diagram if dom( � X ) is a lattice. Theorem Given a boolean space X, the following are equivalent. ◮ X is AE(0). ◮ X is homeomorphic to the limit of a multicommutative locally finite poset diagram. ◮ X is homeomorphic to the limit of a multicommutative locally finite lattice diagram. 6 / 9
Long ω 1 -approximation sequences ◮ For every ordinal α , let ◮ ⌊ α ⌋ = max { β ≤ α : β < ω 1 or ∃ γ | α | · γ = β } ; ◮ α = ⌊ α ⌋ + [ α ]; 7 / 9
Long ω 1 -approximation sequences ◮ For every ordinal α , let ◮ ⌊ α ⌋ = max { β ≤ α : β < ω 1 or ∃ γ | α | · γ = β } ; ◮ α = ⌊ α ⌋ + [ α ]; ◮ [ α ] 0 = α ; ◮ [ α ] n +1 = [[ α ] n ]; ◮ ⌊ α ⌋ n = � i < n ⌊ [ α ] i ⌋ ; 7 / 9
Long ω 1 -approximation sequences ◮ For every ordinal α , let ◮ ⌊ α ⌋ = max { β ≤ α : β < ω 1 or ∃ γ | α | · γ = β } ; ◮ α = ⌊ α ⌋ + [ α ]; ◮ [ α ] 0 = α ; ◮ [ α ] n +1 = [[ α ] n ]; ◮ ⌊ α ⌋ n = � i < n ⌊ [ α ] i ⌋ ; ◮ � ( α ) = min { n < ω : [ α ] n = 0 } . ◮ If 1 ≤ k < ω and α ≤ ω k , then � ( α ) ≤ k . ◮ Given θ regular and uncountable, a long ω 1 -approximation sequence is a transfinite sequence ( M α ) α<η of countable elementary substructures of H ( θ ) such that ( M β ) β<α ∈ M α for all α < η . ◮ (Milovich, 2008) If � M is a long ω 1 -approximation sequence and α, β ∈ dom( � M ), then ◮ M β ∈ M α ⇔ β ∈ α ∩ M α ⇔ M β � M α ; ◮ for all i < � ( α ), M i α = � { M γ : ⌊ α ⌋ i ≤ γ < ⌊ α ⌋ i +1 } is a directed union; hence, M i α ≺ H ( θ ). 7 / 9
n-commutativity ◮ Call a lattice diagram ( � X ,� f ) n-commutative if, for all i ∈ dom( � X ) and all j 0 , . . . , j n − 1 < i , � k ∈ K f k , i maps X i onto lim( X k : k ∈ K ) where K = � m < n { k : k ≤ j m } . ◮ Call a boolean space n-commutative if it is homeomorphic to the limit of an n-commutative locally finite lattice diagram. 8 / 9
n-commutativity ◮ Call a lattice diagram ( � X ,� f ) n-commutative if, for all i ∈ dom( � X ) and all j 0 , . . . , j n − 1 < i , � k ∈ K f k , i maps X i onto lim( X k : k ∈ K ) where K = � m < n { k : k ≤ j m } . ◮ Call a boolean space n-commutative if it is homeomorphic to the limit of an n-commutative locally finite lattice diagram. ◮ The Stone dual of “2-commutative boolean space” has been studied under the name of “ strong Freese-Nation property .” ◮ There are 2-commutative boolean spaces of weight ℵ 2 that are known to not be AE(0), e.g. , the symmetric square of 2 ω 2 . 8 / 9
n-commutativity ◮ Call a lattice diagram ( � X ,� f ) n-commutative if, for all i ∈ dom( � X ) and all j 0 , . . . , j n − 1 < i , � k ∈ K f k , i maps X i onto lim( X k : k ∈ K ) where K = � m < n { k : k ≤ j m } . ◮ Call a boolean space n-commutative if it is homeomorphic to the limit of an n-commutative locally finite lattice diagram. ◮ The Stone dual of “2-commutative boolean space” has been studied under the name of “ strong Freese-Nation property .” ◮ There are 2-commutative boolean spaces of weight ℵ 2 that are known to not be AE(0), e.g. , the symmetric square of 2 ω 2 . ◮ (Milovich) Every locally finite lattice of size ℵ n − 1 contains a cofinal suborder that is an n -ladder, i.e. , a locally finite lattice in which every element has at most n maximal strict lower bounds. 8 / 9
n-commutativity ◮ Call a lattice diagram ( � X ,� f ) n-commutative if, for all i ∈ dom( � X ) and all j 0 , . . . , j n − 1 < i , � k ∈ K f k , i maps X i onto lim( X k : k ∈ K ) where K = � m < n { k : k ≤ j m } . ◮ Call a boolean space n-commutative if it is homeomorphic to the limit of an n-commutative locally finite lattice diagram. ◮ The Stone dual of “2-commutative boolean space” has been studied under the name of “ strong Freese-Nation property .” ◮ There are 2-commutative boolean spaces of weight ℵ 2 that are known to not be AE(0), e.g. , the symmetric square of 2 ω 2 . ◮ (Milovich) Every locally finite lattice of size ℵ n − 1 contains a cofinal suborder that is an n -ladder, i.e. , a locally finite lattice in which every element has at most n maximal strict lower bounds. ◮ Hence, a boolean space of weight ℵ n − 1 is AE(0) iff it is n-commutative. ◮ Hence, there are 2-commutative boolean spaces of weight ℵ 2 that are not 3-commutative. 8 / 9
The strong Freese-Nation property is strictly stronger ◮ A boolean algebra A is said to have the Freese-Nation property , or FN, if A ∩ M ≤ rc A whenever A ∈ M ≺ H ( θ ). 9 / 9
The strong Freese-Nation property is strictly stronger ◮ A boolean algebra A is said to have the Freese-Nation property , or FN, if A ∩ M ≤ rc A whenever A ∈ M ≺ H ( θ ). ◮ Heindorf and Shapiro introduced the strong Freese-Nation property, or SFN, and showed that it implied the FN, and asked if the implication was strict. 9 / 9
The strong Freese-Nation property is strictly stronger ◮ A boolean algebra A is said to have the Freese-Nation property , or FN, if A ∩ M ≤ rc A whenever A ∈ M ≺ H ( θ ). ◮ Heindorf and Shapiro introduced the strong Freese-Nation property, or SFN, and showed that it implied the FN, and asked if the implication was strict. ◮ If A is a boolean algebra, ( M α ) α< | A | is a long ω 1 -approximation sequence, and A ∈ M 0 , then, for all α < | A | , i < � ( α ), and a ∈ A ∩ M α \ � β<α M β , set σ i ( a ) = min { b ∈ A ∩ M i α : b ≥ a } if it exists. 9 / 9
Recommend
More recommend
