SLIDE 1
Paul Gartside BLAST 2018
University of Pittsburgh
SLIDE 2 Origins of the Tukey order
Topological space X. A neighborhood of x is any subset of X containing an open set containing x. Write Nx = all neighborhoods of x. How do we verify a sequence converges? Definition: ‘(xn)n converges to x’ iff for all (open) neighborhoods V of x there is an n s.t. for all n ≥ N we have xn ∈ V . Or a function is continuous? Definition: ‘f : X → Y is continuous at x’ iff
- r all (open) neighborhoods V of f (x)
there is an open neighborhood U of x s.t. f (U) ⊆ V .
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SLIDE 3
What happens cofinally matters most
Nx = 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.
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SLIDE 4
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.
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SLIDE 5
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’. Nx and K(Y ) are Dedekind complete.
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SLIDE 6
Basic examples and lemmas
1 = {0}, ω, ω1, [ω]<ω and [ω1]<ω = all finite subsets of ω1. 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 ω1 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).
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SLIDE 7
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.
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SLIDE 8 Products
Products are ordered co-ordinatewise. P × Q ordered: (p, q) ≤ (p′, q′) iff p ≤ p′ and q ≤ q′.
λλ iff for all λ we have pλ ≤ p′ λ.
Example Show: 1. P × Q ≥T P and 2. P ≥T P × P. Example How are ω × ω1 and ωω Tukey related to the others: 1, ω, ω1 and [ω1]<ω? (Draw a diagram. List any open questions.)
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SLIDE 9 Basic questions about the Tukey order
- How many Tukey types are there such that . . . ?
- How are the Tukey types related?
Fix P.
P ≥T Q iff Q . . . ? . . .
Q ≥T P [or Q ≥T P] iff Q . . . ? . . .
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SLIDE 10
How to show ωω ≥T [ω1]<ω?
Recall: P ≥T [ω1]<ω iff every uncountable subset S of P contains an infinite bounded subset. Give ω the discrete topology. Give ωω the product topology. It is a separable metric space. Lemma Every uncountable subset of a separable metric space contains a convergent sequence.
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SLIDE 11
How to show ωω ≥T [ω1]<ω?
Recall: P ≥T [ω1]<ω iff every uncountable subset S of P contains an infinite bounded subset. Give ω the discrete topology. Give ωω 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 ωω is bounded.
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