1. The Tukey Order Paul Gartside BLAST 2018 University of Pittsburgh
Origins of the Tukey order Topological space X . A neighborhood of x is any subset of X containing an open set containing x . Write N x = all neighborhoods of x . How do we verify a sequence converges ? Definition: ‘( x n ) n converges to x ’ iff for all (open) neighborhoods V of x there is an n s.t. for all n ≥ N we have x n ∈ V . Or a function is continuous ? Definition: ‘ f : X → Y is continuous at x ’ iff or all (open) neighborhoods V of f ( x ) there is an open neighborhood U of x s.t. f ( U ) ⊆ V . 2
What happens cofinally matters most N x = all neighborhoods of x , ordered by reverse inclusion ⊇ . K ( Y ) = all compact subsets of Y , ordered by inclusion ⊆ . Both directed (partially ordered) sets. We want to compare directed sets ‘cofinally’ . . . Let P = ( P , ≤ ) be a directed set. A subset C is cofinal in P iff for all p in P there is a c in C such that p ≤ c . 3
Definition of Tukey order We want to compare directed sets ‘cofinally’ . . . Let P , Q be directed sets. Then P ≥ T Q (‘ P Tukey quotients to Q ’) iff there is a φ : P → Q (a Tukey quotient) such that for all C cofinal in P we have φ ( C ) cofinal in Q . Write: Q ≤ T P iff P ≥ T Q and P = T Q iff Q ≥ T P and P ≥ T Q . 4
Definition of Tukey order Then P ≥ T Q iff there is a φ : P → Q (a Tukey quotient) such that φ order-preserving and φ ( P ) cofinal in Q , provided Q Dedekind complete: ‘bounded sets have a least upper bound’. N x and K ( Y ) are Dedekind complete. 4
Basic examples and lemmas 1 = { 0 } , ω , ω 1 , [ ω ] <ω and [ ω 1 ] <ω = all finite subsets of ω 1 . Example Show: 1. ω ≥ T 1 and 2. 1 ≥ T P iff P . . . ?. . . Deduce: 1. 1 < T ω and 2. ∄ P : 1 < T P < T ω . Example Show: 1. ω ≥ T [ ω ] <ω and 2. [ ω ] <ω ≥ T P iff P . . . ?. . . Deduce: 1. [ ω ] <ω = T ω , generalize: 2. [ κ ] <ω ≥ T P iff P . . . ?. . . Example Solve: 1. P ≥ T ω iff P . . . ?. . . and 2. P ≥ T ω 1 iff P . . . ?. . . What are the relations under the Tukey order of our 5 directed sets? (Draw a diagram. List any open questions.) Example Show: 1. if C cofinal in P then C = T P and 2. P ≥ T cof ( P ). 5
Tukey maps (and calibres) Let P , Q be directed sets. A map ψ : Q → P is a Tukey map iff for all unbounded U in Q we have ψ ( U ) unbounded in P . Lemma There is a Tukey quotient φ : P → Q iff there is a Tukey map ψ : Q → P. Example Let κ be a regular cardinal. Then: P �≥ T κ iff for all S ⊆ P of size ≥ κ there is a bounded subset S ′ of S of size ≥ κ . P �≥ T [ κ ] <ω iff for all S ⊆ P of size ≥ κ there is a bounded subset S ′ of S of size ≥ ω . 6
Products Products are ordered co-ordinatewise. P × Q ordered: ( p , q ) ≤ ( p ′ , q ′ ) iff p ≤ p ′ and q ≤ q ′ . � λ ∈ Λ P λ ordered: � p λ � λ ≤ � p ′ λ � λ iff for all λ we have p λ ≤ p ′ λ . Example Show: 1. P × Q ≥ T P and 2. P ≥ T P × P . Example How are ω × ω 1 and ω ω Tukey related to the others: 1 , ω , ω 1 and [ ω 1 ] <ω ? (Draw a diagram. List any open questions.) 7
Basic questions about the Tukey order • How many Tukey types are there such that . . . ? • How are the Tukey types related? Fix P . • What lies below? P ≥ T Q iff Q . . . ? . . . • What lies above? Q ≥ T P [or Q �≥ T P ] iff Q . . . ? . . . 8
How to show ω ω �≥ T [ ω 1 ] <ω ? Recall: P �≥ T [ ω 1 ] <ω iff every uncountable subset S of P contains an infinite bounded subset. Give ω the discrete topology. Give ω ω the product topology. It is a separable metric space. Lemma Every uncountable subset of a separable metric space contains a convergent sequence. 9
How to show ω ω �≥ T [ ω 1 ] <ω ? Recall: P �≥ T [ ω 1 ] <ω iff every uncountable subset S of P contains an infinite bounded subset. Give ω the discrete topology. Give ω ω the product topology. It is a separable metric space. Lemma Every uncountable subset of a separable metric space contains a convergent sequence. Lemma Every convergent sequence in ω ω is bounded. 9
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