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 . 2

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’. 2

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. 3

Basic examples and lemmas 1 = { 0 } , ω , ω 1 , ω × ω 1 , [ ω 1 ] <ω = all finite subsets of ω 1 and ω ω . Products are ordered co-ordinatewise. λ ∈ Λ P λ ordered: � p λ � λ ≤ � p ′ λ � λ iff for all λ we have p λ ≤ p ′ � λ . Lemma Always: P ≥ T P × P. 4

Isbell’s 7 of 10 Tukey (1940). Schmidt (1950). Isbell (1972). 1 = { 0 } , ω , ω 1 , ω × ω 1 , and [ ω 1 ] <ω ω ω Σ ω ω 1 = all elements of ω ω 1 with countable support ( � = 0) CNWD, Z 0 , ℓ 1 Fremlin (1991): E µ = all compact measure 0 subsets of [0 , 1] 5

How the 11 are related Σ ω ω 1 CNWD ω ω E µ ℓ 1 Z 0 ω 1 ω × ω 1 [ ω 1 ] <ω ω 1 6

Thanks to. . . Fremlin (1991) Louveau & Velickovich (1999) Matrai (2010) Solecki & Todorcevic (2011) and see Solecki & Todorcevic (2004) 7

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 8

Topology fundamentals (1) ( X , d ) – metric space. B ǫ ( 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 . 9

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 . if its union, � U , is X . A collection U of open sets is a cover 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 T 3 space. TFAE: (a) X is separable metrizable, (b) X is Lindelof metrizable, and (c) X has a countable base. 10

Topology fundamentals (3) Space X is: T 2 if for every x � = y in X there are disjoint open U and V s.t. x ∈ U and y ∈ V T 3 if T 2 and for every x not in closed C there are disjoint open U and V s.t. x ∈ U and C ⊆ V 1 0 (‘first countable’) if for every x the set N x = 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 } . 11

Topology fundamentals (4) Fix, for each λ in Λ, a space X λ . Then � λ X λ has the product topology which has base all: � λ U λ where for all λ we have U λ is open in X λ , and U λ = X λ except for finitely many λ Fact: Product of compact is compact. Every separable metric space embeds in [0 , 1] ω . 12

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 [ ω 1 ] <ω iff for all uncountable S ⊆ P there is an infinite bounded subset S ′ of S. 13

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 (non-trivial) convergent sequence. 14

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 (non-trivial) convergent sequence. Lemma Every convergent sequence in ω ω is bounded. 14

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 [ ω 1 ] <ω . CNWD, Z 0 , ℓ 1 and E µ are separable metric and CSBS. 15

Similarly Σ ω ω 1 �≥ T [ ω 1 ] <ω . . . Give ω the discrete topology. Give ω ω 1 the product topology, and Σ ω ω 1 the subspace topology. Lemma (a) Σ ω ω 1 is ‘Frechet-Urysohn’ (implied by 1 o ) (b) closed discrete subspaces of Σ ω ω 1 are countable Hence: every uncountable subset of Σ ω ω 1 contains a convergent sequence. (c) Σ ω ω 1 is CSB. Proposition Let P be a Frechet-Urysohn CSBS topological directed set. Then P �≥ T [ ω 1 ] <ω iff all closed discrete subspaces of P are countable. 16

How many of size ≤ ω 1 ? Todorcevic (1985). Theorem Under (PFA): there are exactly 5 directed sets of size ≤ ω 1 , up to Tukey type. Theorem In (ZFC): there are at least 2 ω 1 Tukey types of directed sets of size c . Hence, under (CH): there are exactly 2 ω 1 directed sets of size ≤ ω 1 , up to Tukey type. 17

Todorcevic’s examples For a space X write K ( X ) = all compact subspaces of X , ordered by ⊆ . Give ω 1 the order topology. Example Show: 1. K ( ω 1 ) = T ω 1 and 2. K ( S 0 ) = T [ ω 1 ] <ω , where S 0 = { α + 1 : α ∈ ω 1 } Show: 1. K ( ω 1 \ { ω } ) = T ω × ω 1 and 2. K ( S 1 ) = T Σ ω ω 1 , where S 1 = S 0 ∪ all limits of limits. S ⊆ ω 1 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 ) 18

Chain conditions, calibres P is calibre ω 1 iff every uncountable subset S of P contains an uncountable bounded subset S ′ . P is calibre ( ω 1 , ω ) iff every uncountable subset S of P contains an infinite bounded subset S ′ . Lemma Let P be a directed set. Then: • P �≥ T ω 1 iff P is calibre ω 1 • P �≥ T [ ω 1 ] <ω iff P is calibre ( ω 1 , ω ) . 19

Calibres in products Example Let P and Q be directed sets. Show: 1. if P and Q calibre ω 1 then P × Q calibre ω 1 , and 2. if P calibre ( ω 1 , ω ) then P × P is calibre ( ω 1 , ω ). ‘Weak’ chain conditions productive consistent and independent. ‘Strong’ chain conditionsproductive in ZFC. 20

Calibre ( ω 1 , ω ) not productive Proposition Let S and T = ω 1 \ S be stationary. Then K ( S ) and K ( T ) are calibre ( ω 1 , ω ) , but K ( S ) × K ( T ) not calibre ( ω 1 , ω ) . 21

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. 22

Σ -products Theorem Let { P λ : λ ∈ Λ } be separable metric CSBS directed sets. Let – wlog – 0 λ be the minimum element of P λ . Then their Σ -product Σ P λ = {� p λ � λ : p λ � = 0 λ for only countably many λ } is calibre ( ω 1 , ω ) . 23

Example Theorem There is a directed set P such that: (i) the Σ -product, P × Σ ω ω 1 does not have calibre ( ω 1 , ω ) , but every countable subproduct does have calibre ( ω 1 , ω ) ; and (ii) Σ P ω 1 does not have calibre ( ω 1 , ω ) , but P ω has calibre ( ω 1 , ω ) . Let X = { x α : α < ω 1 } ⊆ R . For each α let U α = { x β : β ≥ α } . Refine the subspace topology on X by adding all U α . P = K ( X ). (a) P × Σ ω ω 1 contains an uncountable closed discrete subspace. (b) All discrete subspaces of P ω are countable. 24

Distinguishing ω ω and Σ ω ω 1 Proposition Let Q be a separable metric and CSBS topological directed set. Then Q �≥ T Σ ω ω 1 . Let P = K ( X ) be previous example. If Q ≥ T Σ ω ω 1 then P × Q ≥ T P × Σ ω ω 1 . We know P × Σ ω ω 1 ≥ T [ ω 1 ] <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 ( ω 1 , ω ) i.e. P × Q �≥ T [ ω 1 ] <ω . 25