aa r X i v : . [ m a t h . GN ] D ec BETWEEN HOMEOMORPHISM TYPE AND TUKEY TYPE
DAVID MILOVICH
Abstract.
Call a compact space X pin homogeneous if every two points a, b are pin equivalent , meaning that there exists a compact space Y , a quotientmap f : Y → X , and a homeomorphism g : Y → Y such that gf − { a } = f − { b } . We will prove a representation theorem for pin equivalence; transi-tivity of pin equivalence will be a corollary.Pin homogeneity is strictly weaker than homogeneity and pin equivalence isstrictly stronger than Tukey equivalence. Just as with topological homogeneity,no infinite compact F -space is pin homogeneous. On the other hand, X × χ ( X ) is pin homogeneous for every compact X . And there is a compact pinhomogeneous space with points of different π -character. Introduction
In this paper, all spaces are assumed to be Hausdorff.Products, but apparently not much else, preserve both compactness and homo-geneity of topological spaces. Meanwhile, compact topological groups are ccc andˇCech-Stone remainders of infinite discrete spaces are not homogeneous. For essen-tially these reasons, “large” homogeneous compact spaces are hard to come by. VanDouwen’s Problem, asked no later than 1980 and still open in all models of ZFC [1],asks whether there is a homogeneous compact space with c + -many disjoint opensets. This a special case of a very natural question: Question . Is every compact space X a continuous image of some homogeneouscompact space Y ? [7]Even for some important spaces X without c + -many disjoint open sets, including ω + 1, βω , and βω \ ω , the above question is open as far as I know. Two significantpartial results are:(1) (Motorov [11]) If X is first countable, compact, and zero-dimensional, then X ω is homogeneous.(2) (Kunen [6]) No product of one or more infinite compact F -spaces and zeroor more spaces with character less than c is homogeneous.Approaching Question 1.1 less directly, we can consider weaker forms of homo-geneity. For example, say that a space is Tukey homogeneous if every two pointshave Tukey equivalent neighborhood filters. This is a much weaker than homo-geneity. For example, the compact space 2 ω × ω lex is Tukey homogeneous yet has Date : December 20, 2019.2010
Mathematics Subject Classification.
Primary: 54A25, 54G05; Secondary: 54D30, 06E05,03E04. Keywords: pin equivalence, pin homogeneous, F-space, compact, homogeneous, Tukeyequivalence. A space X is homogeneous if for each pair ( a, b ) ∈ X some autohomeomorphism f : X → X sends a to b . Dow and Pearl [3] showed that compactness is not needed here. points with different π -characters. And, in striking contrast to Kunen’s theorem, X × χ ( X ) is Tukey homogeneous for every infinite space X . [8] And though βω \ ω is Tukey inhomogeneous under the assumption of d = c (and more generally inany model of set theory where βω \ ω has a P-point), whether ZFC alone provesthis inhomogeneity is a significant open problem in its own right, equivalent [9] toIsbell’s Problem: Question . Is it consistent with ZFC that ( U , ⊃ ) is Tukey equivalent to ([ c ] < ℵ , ⊂ )for every free ultrafilter U on ω ? [4, 2]This paper introduces pin equivalence, a strict strengthening of Tukey equiva-lence of points in compact spaces. Pin equivalence enjoys a representation theoremin terms of closed binary relations and, through Stone duality, an appealing Booleanalgebraic interpretation. We will also show that, as in the case of topological ho-mogeneity, no infinite compact F -space is pin homogeneous. On the other hand,as in the Tukey case, X × χ ( X ) is pin homogeneous for all compact X , as is everyfirst countable crowded compact X . We will also show that 2 ω × ω lex is pin homo-geneous despite having points with different π -characters. Thus, 2 ω × ω lex is a pinhomogeneous space with all points of Tukey type ω × ω , which I count as a tinybit of progress towards answering a question I have asked before: Question . Is there a compact homogeneous space with points of Tukey type ω × ω ? [10]In every known example of a compact homogeneous space X , the (neighborhoodfilters of) points are Tukey equivalent to ([ χ ( X )] < ℵ , ⊂ ). [8]Without further ado, pin equivalence defined: Definition 1.4. • Call closed sets
A, B in a compact space X pin equivalent and write A ≡ p B if there exist a compact space Y , a continuous surjection f : Y → X , anda homeomorphism g : Y → Y such that gf − A = f − B . • Call points a, b in a compact space X pin equivalent and write a ≡ p b if { a } ≡ p { b } . • Call a compact space pin homogeneous if all pairs of points are pin equiva-lent.(Pin equivalence is transitive, but not obviously so. Wait for the proof.)Observe that, without loss of generality, f may be assumed invertible at a and b because we may replace Y with its quotient where f − { a } and f − { b } are collapsedto points. This leads to my motivation for “pin.” I visualize X inflated to acontinuous preimage Y , but with a and b pinned down. Example 1.5.
Closed intervals are pin homogeneous. To see why, let us show that0 ≡ p X = [ − , A be the hollow diamond { ( x, y ) ∈ [ − , | | x | + | y | = 2 } . Truncate A to B = A ∩ [ − , and then extend to Y = B ∪ [ − , − . Then f ( x, y ) = x defines a continuous surjection from Y to X and g ( x, y ) = ( y, x ) definesa continuous involution of Y such that gf − { } = g { (0 , } = { (2 , } = f − { } . Between compact Hausdorff spaces, all continuous surjections are quotient maps.
ETWEEN HOMEOMORPHISM TYPE AND TUKEY TYPE 3
We will show later that every instance of pin equivalence in an arbitrary compact X is also witnessed by a symmetric subspace of X . Definition 1.6.
A space is
Boolean if is compact and has a base consisting ofclopen sets.Restricting the definition of pin equivalence to Boolean spaces and applyingStone duality, we obtain and algebraic version of pin equivalence that is very natu-ral: two filters are pin equivalent if they generate isomorphic filters in some largerBoolean algebra. More precisely:
Definition 1.7.
Given two filters
F, G of a Boolean algebra A , we say that F and G are pin equivalent in A and write F ≡ p G if there is a Boolean algebra B extending A and there is a (Boolean) automorphism h of B that sends the filter of B generated by F to the filter of B generated by G .Next observe that if f in Definition 1.4 is required to be a homeomorphism in-stead of a mere continuous surjection, then pin homogeneity becomes homogeneity.This suggests a strategy for incremental progress towards solving the open problemof whether every compact space is a quotient of a homogeneous compact space:start with the positive solution to the analogous problem for pin homogeneouscompacts and incrementally require more of f . Question . How much can we strengthen pin homogeneity before the analogof Question 1.1 for this intermediate homogeneity concept becomes as hard asQuestion 1.1 itself?A natural strengthening of “continuous surjection” is “open continuous surjec-tion.” So, let us define open pin equivalence and open pin homogeneity by therequirement that f to also be an open map. Open pin homogeneous compactspaces appear more difficult to obtain. In particular, open pin equivalence is easilyseen to preserve π -character. Question . Is every compact space a continuous image of an open pin homoge-neous compact space? 2.
A representation theorem
To my mind, the best evidence so far that pin equivalence is worth studying isthe following representation theorem.
Theorem 2.1.
Points a, b in compact space X are pin equivalent iff there is asymmetric binary relation R with domain X such that R is closed in X and, forall x ∈ X , we have aRx ⇔ x = b and bRx ⇔ x = a . Corollary 2.2. If a, b are pin equivalent in a compact space X , then this is wit-nessed by f, g, Y where Y is a closed symmetric subset of X , f : Y → X is the firstcoordinate projection, and g : Y → Y is the continuous involution ( x, y ) ( y, x ) . Before proving Theorem 2.1, we establish a version of the tube lemma for fibers.
Lemma 2.3.
Suppose Y is compact, f : Y → X is continuous, x ∈ X , and V is aneighborhood of f − { x } . Then is a neighborhood U of x such that f − U ⊂ V . DAVID MILOVICH
Proof.
Let C be the complement of the interior of V , which is compact and disjointfrom f − { x } . Then f C is compact and disjoint from { x } . Let U be the complementof f C , which is a neighborhood of x . Then f − U is disjoint from C and, therefore,a subset of the interior of V . (cid:3) Definition 2.4.
Given a binary relation R : • Let R − denote the converse relation. • Given also a set A , let RA denote the set of all b such that aRb for some a ∈ A . • Given also a binary relation S , let SR denote the set of all pairs ( a, c ) suchthat aRbSc for some b . Proof of Theorem 2.1.
Suppose a = b , Y is compact, f : Y → X is a quotient map, g : Y → Y is a homeomorphism, { c } = f − { a } , { d } = f − { b } , and g ( c ) = d . Letus construct R .Let A , B be disjoint neighborhoods of a, b . Choose a neighborhood C of c such that C ⊂ f − A and g ( C ) ⊂ f − B . Applying Lemma 2.3, choose a closedneighborhood A of a such that f − A ⊂ C . Let B = f gf − A , which is compactbecause Y is compact. Applying Lemma 2.3 again, B is a neighborhood of b . Also, A and B are disjoint because A ⊂ f C ⊂ A and B = f gf − A ⊂ f gC ⊂ B . Let T be the all ( p, q, r ) ∈ X × X × Y such that f ( r ) = p and f ( g ( r )) = q . Thisset is compact. Let S be the set of all ( p, q ) ∈ A × B such that ( p, q, r ) ∈ T forsome r ∈ Y . This set is also compact. And, since B = f gf − A , the domain andrange of S are A and B . Moreover, a is the unique p satisfying pSb and b is theunique q satisfying aSq . Finally, let D be the closure of the complement of A ∪ B .Then R = S ∪ S − ∪ D is as desired. (cid:3) Definition 2.5.
Given a, b, X, R as in Theorem 2.1, we say that R represents a ≡ p b in X .Our representation theorem helps us prove several nice properties of pin equiv-alence, starting with the next lemma, which we will use many times. This lemmaallows us to use a relation R as above like a function that is continuous at a and b . Lemma 2.6. If R represents a ≡ p b in compact space X and V is a neighborhoodof b , then a has a neighborhood U such that RU ⊂ V .Proof. Suppose not. Then there are nets ( p i ) i ∈ I , ( q i ) i ∈ I in X such that p i → a , p i Rq i , and q i V . Since X is compact, ( q i ) i ∈ I has a cluster point c . And c = b since c is not in the interior of V . Since R is closed, aRc . But this contradicts aRx ⇔ x = b . (cid:3) Corollary 2.7.
Suppose R represents a ≡ p b and net ( x i ) i ∈ I converges to a . If x i Ry i for all i ∈ I , then ( y i ) i ∈ I converges to b . Corollary 2.8.
Suppose, in a compact space, that R represents a ≡ p b , U is anultrafilter on a set I , and lim i →U x i = a . If x i Ry i for all i ∈ I , then lim i →U y i = b . Theorem 2.9.
Pin equivalence is transitive.
ETWEEN HOMEOMORPHISM TYPE AND TUKEY TYPE 5
Proof.
Suppose a ≡ p b ≡ p c in compact space X . Let us show that a ≡ p c . We mayassume a, b, c are distinct. Let R, S represent a ≡ p b and b ≡ p c . Let A , B , C bedisjoint closed neighborhoods of a, b, c . Let A , C be open neighborhoods of a, c such that A ⊂ A , C ⊂ C , RA ⊂ B , and SC ⊂ B . Define relations ˆ R, ˆ S, T as follows. ˆ R = R ∩ ( X \ C ) ˆ S = S ∩ ( X \ A ) T = ˆ R ˆ S ∪ ˆ S ˆ R ∪ B Then T is symmetric, aT x ⇔ x = c , and cT x ⇔ x = a . By compactness, T is alsoclosed. We just need to show that T has domain X .Fix x ∈ X . First, suppose x A ∪ B . We then have x ˆ Sy for some y .Moreover, y C because SC ⊂ B and ySx . Therefore, y ˆ Rz for some z . Thus, x ∈ dom( ˆ R ˆ S ). If instead x C ∪ B , then x ∈ dom( ˆ S ˆ R ) by analogous reasoning.In the only remaining case, x ∈ B , we have x ∈ dom( B ). Thus, X = dom( T ). (cid:3) The theorem says that pin equivalence is a local property.
Theorem 2.10.
Let
Y, Z be closed subspaces of a compact space X . Suppose a, b are in the interior of Y ∩ Z and a ≡ p b in Y . Then a ≡ p b in Z .Proof. Let R represent a ≡ p b in Y . Let A , B be closed neighborhoods of a, b in Y ∩ Z . Let A , B be open neighborhoods of a, b such that A ⊂ A , B ⊂ B , RA ⊂ B , and RB ⊂ A . Let S = R ∩ ( A × B ), T = R ∩ ( B × A ), and U = S ∪ T ∪ C where C = Z \ ( A ∪ B ). Then U represents a ≡ p b in Z . (cid:3) The next lemma isolates a recurring technique from the proofs of the abovetheorems.
Lemma 2.11.
Given distinct a, b in a compact space X , there exists R that repre-sents a ≡ p b iff there exists a closed binary relation S on X such that dom( S ) is aneighborhood of a , ran( S ) is a neighborhood of b disjoint from dom( S ) , xSb ⇔ x = a , and aSy ⇔ y = b .Proof. Given R , let S = R ∩ (( U × RU ) ∪ ( RV × V )) for sufficiently small closedneighborhoods U, V of a, b . Given instead S , let R = S ∪ S − ∪ (cid:16) X \ (dom( S ) ∪ ran( S )) (cid:17) . (cid:3) Pin equivalence vs. Tukey equivalence
Here we show that pin equivalence strictly implies Tukey equivalence.
Definition 3.1. A directed set is a nonempty set S equipped with a transitivereflexive relation ≤ such that for all x, y ∈ S there exists z ∈ S such that x, y ≤ z . Definition 3.2.
Given two directed sets
P, Q : • We say P is Tukey below Q and write P ≤ T Q if there exists f : Q → P that is convergent , that is, for every p ∈ P there exists q ∈ Q such that f ( q ) ≥ p for all q ≥ q . • We say P is Tukey equivalent to Q and write P ≡ T Q if P ≤ T Q ≤ T P . • A subset U of P is unbounded if has no upper bound in P . • A subset C of P is cofinal if for every p ∈ P has an upper bound in C . DAVID MILOVICH • The cofinality cf( P ) of P is least of the cardinalities of cofinal subsets of P . • Given a cardinal κ , we say P is κ -directed if every subset of P size less than κ has an upper bound in P . • Given a cardinal κ , we say P is κ -OK if, for each f : ω → P there exists g : κ → P such that for every n < ω , every increasing n -tuple ξ < · · · <ξ n < κ , and every upper bound b ∈ P of { g ( ξ ) , . . . , g ( ξ n ) } , we have f ( n ) ≤ b .Below are some elementary consequences of the above definitions. • Composition preserves convergence. • If C is a cofinal subset of P , then C ≡ T P . • If P ≤ T Q , then cf( P ) ≤ cf( Q ). • If Q is κ -directed and P ≤ T Q , then P is κ -directed. • If P is λ -OK and κ ≤ λ , then P is κ -OK. • If cf( P ) ≤ κ , then P ≤ T [ κ ] < ℵ where [ S ] < ℵ denotes the finite subsets of S ordered by inclusion ( ⊂ ). Lemma 3.3. If P is κ -OK but not ω -directed, then [ κ ] < ℵ ≤ T P .Proof. Let f map ω to an unbounded subset of P . Let g be as in the definitionof κ -OK. Then g maps each infinite subset of κ to an unbounded set. To obtain aconvergent map from P to [ κ ] < ℵ , map each p ∈ P to the set of all ξ < κ satisfying g ( ξ ) ≤ p . (cid:3) Through neighborhood filters, the order concepts defined above induce the topo-logical concepts defined next.
Definition 3.4.
Given points a, b in space X : • We denote by N X ( a ) the neighborhood filter of a , that is, the set of all N ⊂ X with a in the interior of N . We make N X ( a ) a directed set byordering it by containment ( ⊃ ). • We say a is Tukey below (resp., Tukey equivalent to) b if N X ( a ) ≤ T N X ( b )(resp., N X ( a ) ≡ T N X ( b )). • We denote by χ ( a, X ), the character of a , which is the cofinality cf( N X ( a ))of a ’s neighborhood filter. • Given a cardinal κ , we say a is κ -OK if its neighborhood filter is. Theorem 3.5. If a ≡ p b in compact space X , then a ≡ T b .Proof. Let R represent a ≡ p b . It suffices to show that b ≤ T a . Define r : N X ( a ) →N X ( b ) by r ( U ) = RU . By Lemma 2.6, r is convergent. (cid:3) Definition 3.6.
Given a point a in a space X : • We say a is a P-point if N X ( a ) is ω -directed. • We say a is a weak P-point if X \ C ∈ N X ( a ) for every countable C ⊂ X \{ a } . Theorem 3.7.
In a compact space X , if a ≡ p b and a is not a weak P-point, thenneither is b .Proof. Let a ∈ { x n | n < ω } but x n = a for all n < ω . Letting some R represent a ≡ p b , choose y n such that x n Ry n , for each n < ω . Then y n = b for all n < ω .For each N ∈ N X ( a ), choose x ϕ ( N ) ∈ N , thus defining a net converging to a . Then y ϕ ( N ) → b by Corollary 2.7. Hence, b ∈ { y n | n < ω } . (cid:3) ETWEEN HOMEOMORPHISM TYPE AND TUKEY TYPE 7
Kunen proved that the ˇCech-Stone remainder ω ∗ has weak P-points, a fact thatpreviously was merely known to be consistent with ZFC. His proof consists of aneasy result followed by a hard result: Lemma 3.8 (Kunen [5]) . In a space, if a point is ω -OK, then it is also a weakP-point. Lemma 3.9 (Kunen [5]) . In the ˇCech-Stone remainder ω ∗ , there is a c -OK pointthat is not a P-point. Definition 3.10.
We say a space X is an F-space if every two disjoint open F σ -setshave disjoint closures. ω ∗ is the quintessential example of a compact F-space. Indeed, a Stone space ofBoolean algebra is an F-space iff the algebra has the countable separable property :every two countably generated ideals I, J with I ∩ J = { } extend to principalideals I ′ , J ′ with I ′ ∩ J ′ = { } . Now ω ∗ is homeomorphic to the Stone space of P ( ω ) / [ ω ] < ℵ . It is an easy exercise to show that this algebra has the countableseparable property. Theorem 3.11.
In the ˇCech-Stone remainder ω ∗ , there exist a, b such that a ≡ T b but a p b .Proof. Let X = ω ∗ . We identify each point e ∈ X with the ultrafilter { U ⊂ ω | e ∈ U } where the closure U is computed in the ˇCech-Stone compactification βω = ω ∪ X .The map E E \ ω surjects from the above ultrafilter to the set of the clopenneighborhoods of e . And for U, V ⊂ ω , we have U \ ω ⊂ V \ ω iff U ⊂ ∗ V where ⊂ ∗ is inclusion modulo finite sets. Therefore, e ≡ T N X ( e ) provided e is ordered by ⊃ ∗ .Let a ∈ X be c -OK but not a P-point. Then [ c ] < ℵ ≤ T a by Lemma 3.3 and a is a weak P-point by Lemma 3.8. Let ( c n ) n<ω be a discrete sequence in X and let b be the ultralimit lim n → a c n . Then a p b because b is not a weak P-point. Claim. a ≤ T b . Proof.
We will show that ϕ ( V ) = { n < ω | c n ∈ V } defines a convergent map from( b, ⊃ ∗ ) to ( a, ⊃ ), noting that the identity map from ( a, ⊃ ) to ( a, ⊃ ∗ ) is convergent.Since ( c n ) n<ω is discrete, there are disjoint open F σ sets ( O n ) n<ω such that c n ∈ O n for each n . Suppose U ∈ a . Since X is an F-space, S n ∈ U O n and S n U O n havedisjoint closures. Since also b ∈ S n ∈ U O n , we may choose V ∈ b such that V isdisjoint from { c n | n U } . Therefore, ϕ ( V ) ⊂ U for all V ⊂ ∗ V . (cid:3) Moreover, b ≤ T [ c ] < ℵ since b has cardinality c . Therefore, a ≡ T b ≡ T [ c ] < ℵ . (cid:3) Remark . In the above proof, the justification of a ≤ T b works for any a ∈ ω ∗ .It really shows that if a is strictly below b in the Rudin-Frol´ık order, then ( a, ⊃ ) isTukey below ( b, ⊃ ∗ ). DAVID MILOVICH Pin inhomogeneity in other F-spaces
Besides ω ∗ , another simply defined example of a compact F-space is the absoluteof 2 ω , that is, the Stone space Ξ of the algebra of regular open subsets of 2 ω . This isan F-space because any regular open algebra is complete. On the other hand, Ξ hasa countable π -base because 2 ω does. Therefore, Ξ lacks weak P-points. Hence, ourconstruction of pin-inequivalent points in ω ∗ , which relied on Kunen’s constructionof a weak P-point in ω ∗ , cannot generalize to all infinite F-spaces. Nevertheless, wecan use a lemma from another paper of Kunen’s to show that every infinite F-spacehas pin-inequivalent points. Definition 4.1.
Given ultrafilters U , V on ω , we say U is Rudin-Keisler below V and write U ≤ RK V if there exists f : ω → ω such that βf ( V ) = U , that is, suchthat E ∈ U iff f − E ∈ V , for all E ⊂ ω . Theorem 4.2 (Kunen [5]) . There are Rudin-Keisler incomparable weak P-pointsin ω ∗ . Lemma 4.3 (Kunen [6]) . Suppose U , V are Rudin-Keisler incomparable weak P-points in ω ∗ . Also suppose that, in a compact F-space X , a is the U -limit of adiscrete ω -sequence. Then a is not the V -limit of any ω -sequence in X \ { a } . Theorem 4.4.
Let X be an infinite compact F-space. Then there exist a, b ∈ X such that a p b .Proof. Let U , V ∈ ω ∗ be Rudin-Keisler incomparable weak P-points. Let ( a n ) n<ω be a discrete sequence in X , let a = lim n →U a n , and let b = lim n →V a n . ByLemma 4.3, b is not the U -limit of any ω -sequence in X \ { b } . Seeking a contra-diction, suppose that R represents a ≡ p b . For each n < ω , choose b n such that a n Rb n . Then b n = b for all n < ω . Also, by Corollary 2.8, lim n →U b n = b . Thus,we have a contradiction. (cid:3) Pin homogeneity
Definition 5.1.
Given a point a in a space X , a subset S of N X ( a ) is a neighborhoodsubbase at a if N X ( a ) is the smallest filter containing S . Definition 5.2.
A set E of sets is independent if, for each pair of finite nonempty F , G ⊂ E , if
F ∩ G = ∅ , then T F 6⊂ S G . Lemma 5.3.
In a compact space X , if there is a bijection from an independentneighborhood subbase at a to an independent neighborhood subbase at b , then a ≡ p b .Proof. We may assume a = b . Let f : A → B biject from an independentneighborhood subbase at a to an independent neighborhood subbase at b . First,we construct modified f , A , B for which S A and S B are disjoint. Choosefinite nonempty C ⊂ A and D ⊂ B such that T C and T D are disjoint. Let C = C ∪ f − D and D = f C ∪ D . Let A = { U ∩ \ C | U ∈ A \ C } ; B = { U ∩ \ D | U ∈ B \ D } . ETWEEN HOMEOMORPHISM TYPE AND TUKEY TYPE 9
Then A = S A and B = S B are disjoint. Moreover, because A and B areeach independent, A and B are too and f ( U ∩ T C ) = f ( U ) ∩ T D defines abijection from A to B .For each N ∈ A , let T N = ( N × f ( N )) ∪ (( A \ N ) × ( B \ f ( N ))) . Because A and B are each independent, if F ⊂ A is finite and nonempty, then S F = T N ∈F T N has domain A and range B . By compactness, S F has domain A and range B ; so does S = T F S F . Moreover, if xT N b for some x, N , then x ∈ N .Therefore, xSb implies x = a . Likewise, aSy implies y = b . By Lemma 2.11, a ≡ p b . (cid:3) Theorem 5.4. If X is a compact space, κ is a cardinal, and χ ( x, X ) ≤ κ for all x ∈ X , then X × κ is pin homogeneous.Proof. Given ( a, b ) ∈ X × κ , it suffices to find an independent local subbase at( a, b ) of cardinality κ . Let { A α | α < κ } be a neighborhood subbase at a . Foreach α < κ , let U α = A α × { y ∈ κ | y ( α ) = b ( α ) } . Then α = β ⇒ U α = U β and U = { U α | α < κ } is a local subbase at ( a, b ).To see that U is independent, suppose σ, τ ∈ [ κ ] < ℵ are disjoint. Define c ∈ κ by c ( α ) = b ( α ) iff α ∈ σ . Then ( a, c ) ∈ U α for all α ∈ σ and ( a, c ) U α for all α ∈ τ . (cid:3) I was not able to adapt the above proof to show that X κ is pin homogeneous. Question . Does every compact space have a pin homogeneous power?
Definition 5.6.
A space is crowded if it has no isolated points.
Theorem 5.7.
Suppose X is a first countable crowded compact space. Then X ispin homogeneous.Proof. Let a, b be distinct points in X . Let { A n | n < ω } and { B n | n < ω } beneighborhood bases at a and b such that A n ) A n +1 and B n ) B n +1 . Let S = { ( a, b ) } ∪ [ n<ω (cid:16) A n \ A n +1 × B n \ B n +1 (cid:17) . By Lemma 2.11, a ≡ p b . (cid:3) Proposition 5.8.
Pin homogeneity is productive.Proof.
Suppose that for each i in some set I we have Y i g i / / Y i f i / / X i witness-ing a ( i ) ≡ p b ( i ) in X i . Then, letting X = Q i X i and Y = Q i Y i , we have Y g / / Y f / / X witnessing a ≡ p b in X where f ( y )( i ) = f i ( y ( i )) and g ( y )( i ) = g i ( y ( i )). (cid:3) Pin equivalence and Boolean algebras
The proofs of Lemma 5.3 and Theorem 5.7 implicitly used Boolean isomorphismsbetween Boolean closures of neighborhood bases. Thus, these results are actuallyspecial cases of the following theorem.
Definition 6.1. A neighborhood subbase of a subset E of a space X is family S ofsubsets of X such that smallest filter containing S is the set of neighborhoods of E . Definition 6.2.
Given a subset E of a Boolean algebra A , h E i denotes the Booleanclosure of E . Theorem 6.3.
Given closed disjoint subsets
H, K of a compact space X , we have H ≡ p K if H and K have neighborhood subbases U and V such that there is a map f : U → V that extends to a Boolean isomorphism ϕ : hUi → hVi of the Booleanclosures of U and V in P ( X ) .Proof. Let ϕ : hUi → hVi be as above. Choose U ∈ hUi and V ∈ hVi such that H ⊂ U , K ⊂ V , and U ∩ V = ∅ . Letting A = U ∩ ϕ − ( V ), we obtain H ⊂ A , K ⊂ ϕ ( A ), and A ∩ ϕ ( A ) = ∅ . Let A be the Boolean subalgebra hUi ∩P ( A ) of P ( A )(not a Boolean subalgebra of P ( X )); let B be the Boolean subalgebra hVi ∩ P ( B )of P ( B ) where B = ϕ ( A ); let ψ be the restriction of ϕ to A . Then ψ is a Booleanisomorphism to B .For each finite partition E ⊂ A , the relation T E = [ E ∈E ( E × ψ ( E ))has domain A and range B . Moreover, if F refines E , then T F ⊂ T E . By com-pactness, each T E has domain A and range B ; so does S = T E T E . For any x , if xT E y for some y ∈ K , then x ∈ E for the unique E ∈ E with H ⊂ E . Therefore, xSy ∈ K ⇒ x ∈ H . Analogously, H ∋ xSy ⇒ y ∈ K . Let R = S ∪ S − ∪ (cid:16) X \ ( A ∪ B ) (cid:17) . Then the involution g : R → R given by g ( x, y ) = ( y, x ) and the coordinate projec-tion f : R → X given by f ( x, y ) = x witness that H ≡ p K . (cid:3) Definition 6.4. • Two filters
F, G of a Boolean algebra are incompatible if x ∧ y = 0 for some( x, y ) ∈ F × G . • A subset E of a filter F of a Boolean algebra A generates F in A is F isthe smallest filter of A that contains E . Corollary 6.5.
Suppose that F and G are incompatible filters of a Boolean algebraand that they are generated by sets D and E . If there is a map from D to E thatextends to a Boolean isomorphism from h D i to h E i , then F ≡ p G . When compared to Definition 1.7, the converse of Corollary 6.5 looks too goodto be true. But I have not yet found a counterexample.
Problem . Find a Boolean algebra with pin equivalent and incompatible filters F , G such that for all bijections ϕ : D → E , if D generates F and E generates G ,then ϕ does not extend to a Boolean isomorphism from h D i to h E i .We next use the above theorem to show that pin equivalence does not preserve π -character. ETWEEN HOMEOMORPHISM TYPE AND TUKEY TYPE 11
Definition 6.7.
The π -character πχ ( a, X ) of a point a in a space X is the leastof the cardinalities of families F of nonempty open subsets of X such that everyneighborhood of a contains an element of F . Such a family is called a local π -base at a . Definition 6.8.
Given an ordinal α , 2 α lex is the set of all f : α → L = 2 ω lex , every monotone ω -sequence is eventually constant. But every pointis the limit of a strictly increasing ω -sequence or the limit of a strictly decreasing ω -sequence (or both). Moreover, topologically, there are exactly three types ofpoints in L , as shown in the illustration below.II ω o o ω / / I ω o o ω / / III ω o o ω / / III ω o o ω / / II2 ℵ -many points of L are simultaneously the limit of a strictly increasing ω -sequence and the limit of a strictly decreasing ω -sequence. Call these points type I .Call the two endpoints and the 2 ℵ -many points of L with either an immediate pre-decessor or immediate successor type II . All points of type I or II are P-points withTukey type ω and π -character ω . The remainder of L , the set of type III points,consists of 2 ℵ -many limits of strictly increasing or strictly decreasing ω -sequences.These have Tukey type ω × ω and have π -character ω because the nonempty openintervals with endpoints from the ω -sequence form a local π -base.In the product space K = 2 ω × L , there are no P-points. But K inherits both π -characters of L ; indeed, πχ (( p, q ) , K ) = πχ ( q, L ). On the other hand, every pointin K has Tukey type ω × ω . Interestingly, K is also pin homogeneous. Definition 6.9.
Subalgebras A , . . . A n − of a Boolean algebra B are independent if, for all x ∈ Q i K, L , suppose that x = ( p , q ), and x = ( p , q ) are distinct points in K . For each i < 2, conditionally define sets P in , Q iα , R in , S iα , T in as follows. Let { P in | n < ω } be a neighborhood base at p i suchthat P in ) P in +1 . If q i is type I or II, then let ( Q iα ) α<ω be a sequence of intervalssuch that Q iα ) Q iβ for α < β , q i is in the interior of each Q iα , and { q i } = T α Q iα .If q i is type III, let ( Q iα ) α<ω be a sequence of rays of L such that Q iα ) Q iβ for α < β and q i is the interior of each Q iα but on the boundary of T α Q iα . In all cases,let S iα = 2 ω × Q iα . If q i is type I or II, let T in = P in × K . If q i is type III, let T in = P in × R in where ( R in ) n<ω is a sequence of rays of L such that R in ) R in +1 and q i is the interior of each R in but on the boundary of T n R in . In all cases, B i = { S iα | α < ω } ∪ { T in | n < ω } is a neighborhood subbase at x i .Let S i be the Boolean closure of the set of all sets of the form S iα . Let T i bethe Boolean closure of the set of all sets of the form T in . Let U i = (cid:10) S i ∪ T i (cid:11) . Since S iα ) S iβ for α < β , the map S α S α extends uniquely to an isomorphism of σ : S → S . Likewise, the map T n T n extends uniquely to an isomorphism of Types I and II are topologically distinguishable: each type I point a is in the closure of eachof two disjoint topological copies of ω in L \ { a } ; the type II points lack this property. τ : T → T . Moreover, for each i < S i and T i are independent because, foreach α < ω and n < ω , the intersection of S iα \ S iα +1 and T in \ T in +1 is nonemptybecause in all cases it contains( P in \ P in +1 ) × ( Q iα \ Q iα +1 ) . Therefore, σ ∪ τ extends uniquely to an isomorphism from U onto U . Therefore,by Theorem 6.3, x ≡ p x . (cid:3) It is not too hard to generalize the above theorem to Q m ≤ n X m where X m = 2 ω m lex and n < ω . The points of this product space attain all π -characters in [ ω, ω n ]and have Tukey type Q m ≤ n ω m . (Note that the product and lexicographic ordertopologies on 2 ω are identical.) Theorem 6.11. For each n < ω , Q m ≤ n ω m lex is pin homogeneous.Proof. For convenience, let ω − = 1. Using the above X m notation, for each m ≤ n and x ∈ X m there is a least s ( x ) ∈ {− , , . . . , m } for which there are two strictlydecreasing sequences of rays ( P α ( x ) | α < ω m ) and ( Q β ( x ) | α < ω s ( x ) ) suchthat each of these rays has x in its interior and { x } = \ α<ω m P α ( x ) ∩ \ α<ω s ( x ) Q α ( x ) . Given y ∈ Y = Q m ≤ n X m , it suffices to show that y has a neighborhood subbaseconsisting of the union of n +1 strictly decreasing chains ( R mα | α < ω m ) for m ≤ n whose respective Boolean closures A , . . . , A n are independent. Letting y i = y ( i ), s i = s ( y i ), P iα = P α ( y i ), and Q iα = Q α ( y i ) for each i ≤ n , define R mα = Q i ≤ n S m,iα where S m,iα = X i : i < mP mα ∩ Q m : i = m ; s m = − P mα : i = m ; 0 ≤ s m < mP mα ∩ Q mα : i = m ; s m = mX i : i > m ; s i = mQ iα : i > m ; s i = m, thus making ( R mα | α < ω m ) strictly decreasing for each m ≤ n and the unionof these n + 1 chains a neighborhood subbase at y . Moreover, A , . . . , A n areindependent because if α ( m ) < ω m for each m ≤ n , then T m ≤ n ( R mα ( m ) \ R mα ( m )+1 )is nonempty because it contains Q m ≤ n ( P mα ( m ) \ P mα ( m )+1 ). (In verifying this, a keyobservation is that P mα ( m ) \ P mα ( m )+1 = P mα ( m ) ∩ Q mβ \ P mα ( m )+1 for all β .) (cid:3) On the other hand, it is shown in [10] that if X is a compact space and P and Q are directed sets such that cf( P ) , cf( Q ) ≥ ω and Q is cf( P ) ++ -directed, then X has a point not of Tukey type P × Q . In particular, we cannot have a compactspace, pin homogeneous or otherwise, with all points of Tukey type ω × ω .7. Acknowledgements This research was conducted in part while I was an associate professor at TexasA&M International University. ETWEEN HOMEOMORPHISM TYPE AND TUKEY TYPE 13 References [1] A. Arhangel ′ ski˘ı, J. van Mill. Topological Homogeneity. Recent Progress in General Topol-ogy III (K. Hart, J. van Mill, and P. Simon, eds.), Atlantis Press, 2014.[2] N. Dobrinen. Survey on the Tukey theory of ultrafilters. Selected Topics in Combinato-rial Analysis, Zbornik Radova, Mathematical Institutes of the Serbian Academy of Sciences, (2015), 53–80.[3] A. Dow, E. Pearl. Homogeneity in powers of zero-dimensional first-countable spaces. Proc.Amer. Math. Soc. (1997), 2503–2510.[4] J. Isbell. The category of cofinal types II. Trans. Amer. Math. Soc. (1965), 394–416.[5] K. Kunen. Weak P-points in N ∗ , Colloq. Math. Soc. 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