2. Topology for Tukey Paul Gartside BLAST 2018 University of - - PowerPoint PPT Presentation

• 2. Topology for Tukey

Paul Gartside BLAST 2018

University of Pittsburgh

The 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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The 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’.

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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 . . . ? . . . Behavior under operations.

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Basic examples and lemmas

1 = {0}, ω, ω1, ω × ω1, [ω1]<ω = all finite subsets of ω1 and ωω. Products are ordered co-ordinatewise.

• λ∈Λ Pλ ordered: pλλ ≤ p′

λλ iff for all λ we have pλ ≤ p′ λ.

Lemma Always: P ≥T P × P.

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Isbell’s 7 of 10

Tukey (1940). Schmidt (1950). Isbell (1972). 1 = {0}, ω, ω1, ω × ω1, and [ω1]<ω ωω Σωω1 = all elements of ωω1 with countable support (= 0) CNWD, Z0, ℓ1 Fremlin (1991): Eµ = all compact measure 0 subsets of [0, 1]

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How the 11 are related

1 ω ω1 ω × ω1 [ω1]<ω ωω Eµ

CNWD

Z0 ℓ1 Σωω1

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Thanks to. . .

Fremlin (1991) Louveau & Velickovich (1999) Matrai (2010) Solecki & Todorcevic (2011) and see Solecki & Todorcevic (2004)

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Enter topology. . .

Many directed sets have a topology naturally connected with the order. This is useful to:

• Show a directed set has certain Tukey invariants
• Show that is a Tukey quotient exists between two directed

sets then there must be a ‘nice’ one

• Construct interesting directed sets

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Topology fundamentals (1)

(X, d) – metric space. Bǫ(x) = {y ∈ X : d(x, y) < ǫ} – ǫ-ball. U ⊆ X open iff a union of ǫ-balls. Collection of open subsets of X: contains ∅ and X, and is closed under finite intersections and arbitrary unions. A topological space is a set with a collection of subsets with the above properties. ‘Closed’ is complement of open. Continuity and convergence of sequence in terms of open sets. The discrete topology on a set X is the powerset of X.

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Topology fundamentals (2)

Let X be a (topological) space. A collection B of open sets is a base if every open set is the union of a subcollection of B. A collection U of open sets is a cover if its union, U, is X. A space X is: compact iff every open cover has a finite subcover Lindelof iff every open cover has a countable subcover separable iff there is a countable D ⊆ X such that X is the smallest closed set containing it Theorem (Urysohn Metrization) Let X be a T3 space. TFAE: (a) X is separable metrizable, (b) X is Lindelof metrizable, and (c) X has a countable base.

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Topology fundamentals (3)

Space X is: T2 if for every x = y in X there are disjoint open U and V s.t. x ∈ U and y ∈ V T3 if T2 and for every x not in closed C there are disjoint open U and V s.t. x ∈ U and C ⊆ V 10 (‘first countable’) if for every x the set Nx = all open sets containing x, has countable cofinality Let X be a space and A ⊆ X. The subspace topology on A is: {U ∩ A : U open in X}.

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Topology fundamentals (4)

Fix, for each λ in Λ, a space Xλ. Then

λ Xλ has the product topology which has base all:

• λ Uλ where

for all λ we have Uλ is open in Xλ, and Uλ = Xλ except for finitely many λ Fact: Product of compact is compact. Every separable metric space embeds in [0, 1]ω.

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Tukey maps (and an application)

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. Lemma P ≥T [ω1]<ω iff for all uncountable S ⊆ P there is an infinite bounded subset S′ of S.

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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 (non-trivial) convergent sequence.

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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 (non-trivial) convergent sequence. Lemma Every convergent sequence in ωω is bounded.

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Topological directed sets

Let P be both a directed set and a space (topological directed set). P is CSB iff every convergent sequence in P is bounded P is CSBS iff every convergent sequence in P contains a bounded subsequence Proposition If a topological directed set P is separable metric and CSBS then P ≥T [ω1]<ω. CNWD, Z0, ℓ1 and Eµ are separable metric and CSBS.

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Similarly Σωω1 ≥T [ω1]<ω. . .

Give ω the discrete topology. Give ωω1 the product topology, and Σωω1 the subspace topology. Lemma (a) Σωω1 is ‘Frechet-Urysohn’ (implied by 1o) (b) closed discrete subspaces of Σωω1 are countable Hence: every uncountable subset of Σωω1 contains a convergent sequence. (c) Σωω1 is CSB. Proposition Let P be a Frechet-Urysohn CSBS topological directed set. Then P ≥T [ω1]<ω iff all closed discrete subspaces of P are countable.

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How many of size ≤ ω1?

Todorcevic (1985). Theorem Under (PFA): there are exactly 5 directed sets of size ≤ ω1, up to Tukey type. Theorem In (ZFC): there are at least 2ω1 Tukey types of directed sets of size c. Hence, under (CH): there are exactly 2ω1 directed sets of size ≤ ω1, up to Tukey type.

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Todorcevic’s examples

For a space X write K(X) = all compact subspaces of X, ordered by ⊆. Give ω1 the order topology. Example Show: 1. K(ω1) =T ω1 and 2. K(S0) =T [ω1]<ω, where S0 = {α + 1 : α ∈ ω1} Show: 1. K(ω1 \ {ω}) =T ω × ω1 and 2. K(S1) =T Σωω1, where S1 = S0∪ all limits of limits. S ⊆ ω1 is stationary iff C ∩ S = ∅ for all closed unbounded C Proposition If S and T are unbounded and S \ T is stationary then K(S) ≥T K(T)

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Chain conditions, calibres

P is calibre ω1 iff every uncountable subset S of P contains an uncountable bounded subset S′. P is calibre (ω1, ω) iff every uncountable subset S of P contains an infinite bounded subset S′. Lemma Let P be a directed set. Then:

• P ≥T ω1 iff P is calibre ω1
• P ≥T [ω1]<ω iff P is calibre (ω1, ω).

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Calibres in products

Example Let P and Q be directed sets. Show: 1. if P and Q calibre ω1 then P × Q calibre ω1, and

• 2. if P calibre (ω1, ω) then P × P is calibre (ω1, ω).

‘Weak’ chain conditions productive consistent and independent. ‘Strong’ chain conditionsproductive in ZFC.

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Calibre (ω1, ω) not productive

Proposition Let S and T = ω1 \ S be stationary. Then K(S) and K(T) are calibre (ω1, ω), but K(S) × K(T) not calibre (ω1, ω).

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K(X) as a topological space

Let X be any space. Then K(X) has a natural topology – the Vietoris topology. Proposition (Key Properties) (a) X compact iff K(X) compact. (b) If C is a compact subspace of K(X) then C is compact. (c) The map ι : X → K(X) where ι(x) = {x} is a homeomorphism with its image, which is a closed subset of K(X). From (b): K(X) is CSB.

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Σ-products

Theorem Let {Pλ : λ ∈ Λ} be separable metric CSBS directed sets. Let – wlog – 0λ be the minimum element of Pλ. Then their Σ-product ΣPλ = {pλλ : pλ = 0λ for only countably many λ} is calibre (ω1, ω).

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Example

Theorem There is a directed set P such that: (i) the Σ-product, P ×Σωω1 does not have calibre (ω1, ω), but every countable subproduct does have calibre (ω1, ω); and (ii) ΣPω1 does not have calibre (ω1, ω), but Pω has calibre (ω1, ω). Let X = {xα : α < ω1} ⊆ R. For each α let Uα = {xβ : β ≥ α}. Refine the subspace topology on X by adding all Uα. P = K(X). (a) P × Σωω1 contains an uncountable closed discrete subspace. (b) All discrete subspaces of Pω are countable.

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Distinguishing ωω and Σωω1

Proposition Let Q be a separable metric and CSBS topological directed set. Then Q ≥T Σωω1. Let P = K(X) be previous example. If Q ≥T Σωω1 then P × Q ≥T P × Σωω1. We know P × Σωω1 ≥T [ω1]<ω. But Q ‘separable metric and CSBS’ and P ‘all discrete subspaces countable, first countable and CSBS’ = ⇒ P × Q ‘all discrete subspaces countable, first countable and CSBS’. So P × Q calibe (ω1, ω) i.e. P × Q ≥T [ω1]<ω.

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Where calibre (ω1, ω) matters

Let X be compact. Let ∆ = {(x, x) : x ∈ X} ⊆ X 2. Note K(X 2 \ ∆) =T N X 2

∆ = all neighborhoods of ∆ in X 2.

Theorem (PG+ J.Morgan) If X compact and K(X 2 \ ∆) calibre (ω1, ω) then X metrizable.

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Where calibre (ω1, ω) matters

Let X be compact. Let ∆ = {(x, x) : x ∈ X} ⊆ X 2. Note K(X 2 \ ∆) =T N X 2

∆ = all neighborhoods of ∆ in X 2.

Theorem (PG+ J.Morgan) If X compact, P calibre (ω1, ω) and P ≥T K(X 2 \ ∆) then X metrizable.

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Where calibre (ω1, ω) matters

Let X be compact. Let ∆ = {(x, x) : x ∈ X} ⊆ X 2. Note K(X 2 \ ∆) =T N X 2

∆ = all neighborhoods of ∆ in X 2.

Theorem (PG+ J.Morgan) If X compact, P calibre (ω1, ω) and P ≥T K(X 2 \ ∆) then X metrizable. If P not calibre (ω1, ω) then ∃X compact non-metrizable and P ≥T K(X 2 \ ∆).

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Where calibre (ω1, ω) matters

Let X be compact. Let ∆ = {(x, x) : x ∈ X} ⊆ X 2. Note K(X 2 \ ∆) =T N X 2

∆ = all neighborhoods of ∆ in X 2.

Theorem (PG+ J.Morgan) If X compact, P calibre (ω1, ω) and P ≥T K(X 2 \ ∆) then X metrizable. If P not calibre (ω1, ω) then ∀X compact with a base of size ≤ ω1 we have P ≥T K(X 2 \ ∆).

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Why K(X)?

Proposition For every directed set P there is a space XP such that: P =T K(XP).

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The Stone-Cech compactification

Let X be a space. The Stone-Cech compactification of X is the compact space βX: X is a subspace of βX, βX is compact and T2, and for every compact K and continuous f : X → K there is a continuous βf : βX → K with βf ↾ X = f . Lemma Let D be discrete. Let A be a subset of D. Let C = A be the smallest closed set containing A. Then C is open. Consider f : D → {0, 1} where f (x) = 1 iff x ∈ A.

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Fix directed set P. Let D = D(P) the set P with discrete topology. Define φ : P → K(βD) by φ(p) = {p′ : p′ ≤ p}. Set XP = {φ(p) : p ∈ P}. Then C = {φ(p) : p ∈ P} is cofinal in K(Xp) and P and C are order isomorphic (under p ↔ φ(p)).

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