2. Topology for Tukey

Transcription

1 2. Topology for Tukey Paul Gartside BLAST 2018 University of Pittsburgh

2 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

3 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

4 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

5 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

6 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 Σω 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

7 How the 11 are related Σω ω 1 CNWD ω ω E µ l 1 ω Z 0 1 ω ω 1 [ω1 ] <ω ω 1 6

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

9 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

10 Topology fundamentals (1) (X, d) metric space. B ɛ (x) = {y X : d(x, y) < ɛ} ɛ-ball. U X open iff a union of ɛ-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

11 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 T 3 space. TFAE: (a) X is separable metrizable, (b) X is Lindelof metrizable, and (c) X has a countable base. 10

12 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

13 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

14 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

15 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

16 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

17 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

18 Similarly Σω ω1 T [ω 1 ] <ω... Give ω the discrete topology. Give ω ω 1 the product topology, and Σω ω 1 the subspace topology. Lemma (a) Σω ω 1 is Frechet-Urysohn (implied by 1 o ) (b) closed discrete subspaces of Σω ω 1 are countable Hence: every uncountable subset of Σω ω 1 (c) Σω ω 1 is CSB. contains a convergent sequence. 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

19 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

20 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 Σω 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

21 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

22 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

23 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

24 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

25 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 ΣP λ = { p λ λ : p λ 0 λ for only countably many λ} is calibre (ω 1, ω). 23

26 Example Theorem There is a directed set P such that: (i) the Σ-product, P Σω ω 1 does not have calibre (ω 1, ω), but every countable subproduct does have calibre (ω 1, ω); and (ii) ΣP 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 Σω ω 1 contains an uncountable closed discrete subspace. (b) All discrete subspaces of P ω are countable. 24

27 Distinguishing ω ω and Σω ω 1 Proposition Let Q be a separable metric and CSBS topological directed set. Then Q T Σω ω 1. Let P = K(X ) be previous example. If Q T Σω ω 1 then P Q T P Σω ω 1. We know P Σω 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

28 Where calibre (ω 1, 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 (ω 1, ω) then X metrizable. 26

29 Where calibre (ω 1, 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 (ω 1, ω) and P T K(X 2 \ ) then X metrizable. 26

30 Where calibre (ω 1, 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 (ω 1, ω) and P T K(X 2 \ ) then X metrizable. If P not calibre (ω 1, ω) then X compact non-metrizable and P T K(X 2 \ ). 26

31 Where calibre (ω 1, 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 (ω 1, ω) and P T K(X 2 \ ) then X metrizable. If P not calibre (ω 1, ω) then X compact with a base of size ω 1 we have P T K(X 2 \ ). 26

32 Why K(X )? Proposition For every directed set P there is a space X P such that: P = T K(X P ). 27

33 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 T 2, 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. 28

34 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 X P = {φ(p) : p P}. Then C = {φ(p) : p P} is cofinal in K(X p ) and P and C are order isomorphic (under p φ(p)). 29