Aronszajn Compacta

Aronsza jn Compacta ∗ Joan E. Hart † and Kenneth Kunen ‡ § June 24, 2018 Abstract W e consider a class of compacta X suc h that the maps fr om X on to metric compacta define an Ar onsza jn tree of closed subsets of X . 1 In tro du ction All top ologies discusse d in this pap er are assumed to b e Hausdorff. W e b egin by defining an A r onszajn c om p actum , along with a na tural tree structure, b y considering a space em b edded in to a cub e. An equiv alen t definition, in terms of elemen tary submo dels, is considered in Section 2. Notation 1.1 Given a pr o duct Q ξ <λ K ξ : If α ≤ β ≤ λ , then π β α denotes the n a tur al pr oje ction fr om Q ξ <β K ξ onto Q ξ <α K ξ . If we ar e studying a sp ac e X ⊆ Q ξ <λ K ξ then X α denotes π λ α ( X ) , and σ β α denotes the r estricte d map π β α ↾ X β ; so σ β α : X β ։ X α . Definition 1.2 An em b edded Aronsza jn compactum is a close d subsp ac e X ⊆ [0 , 1] ω 1 with w ( X ) = ℵ 1 and χ ( X ) = ℵ 0 such that for some club C ⊆ ω 1 : for e ach α ∈ C L α := { x ∈ X α : | ( σ ω 1 α ) − 1 { x }| > 1 } is c ountable. F or e ach such X , define T = T ( X ) := S {L α : α ∈ C } , and l e t ⊳ denote the fol lowing or der: if α, β ∈ C , α < β , x ∈ L α and y ∈ L β , t hen x ⊳ y iff x = π β α ( y ) . The σ ω 1 α for whic h |L α | ≤ ℵ 0 are called c ountable r ank maps in [1, 8]. Observ e that h T ( X ) , ⊳ i is a tree. Eac h lev el L α is countable by definition, and is non-empt y b ecause w ( X ) = ℵ 1 ; then T is Aronsza jn b ecause χ ( X ) = ℵ 0 . Of course, a compactum of weigh t ℵ 1 ma y b e em b edded in to [0 , 1] ω 1 in man y wa ys, but: ∗ 2000 Mathematics Sub ject Classification: Primary 54D30, 0 3E35 . Key W ords and Phrases: Aronsza jn tree, hereditar ily sepa rable, her e dita rily L indel¨ of. † Univ ersity of Wisco nsin, Oshkosh, WI 5 4901 , U.S.A., hartj@uw o sh.edu ‡ Univ ersity of Wisco nsin, Madison, WI 5 3706 , U.S.A., kunen@math.wisc.edu § Both authors par tially suppo rted by NSF Grant DMS-045 6653. 1 1 INTR OD UCTION 2 Lemma 1.3 If X , Y ⊆ [0 , 1] ω 1 , X is an em b e dde d A r onszajn c omp actum, and Y is home omorphic t o X , then Y is an em b e dde d Ar onszajn c omp actum. Pro of. Let f : X → Y b e a homeomorphism. Then use the fact that there is a club D ⊆ ω 1 on whic h f comm utes with pro jection; that is, for α ∈ D , there is a homeomorphism f α : X α ։ Y α suc h t ha t π ω 1 α ◦ f = f α ◦ π ω 1 α . K The pro of o f t his lemma sho ws t ha t the Aronsza jn trees deriv ed from X and fro m Y are isomorphic on a club. Definition 1.4 An Aro nsza j n compactum is a c omp act X such that w ( X ) = ℵ 1 and χ ( X ) = ℵ 0 and for some ( e quivalently, fo r al l ) Z ⊆ [0 , 1] ω 1 home omorphic to X , Z is an emb e dde d Ar onszajn c omp actum. The next lemma is immediate from the definition. F urther closure prop erties of the class of Aronsza jn compacta are considered in Section 4. Lemma 1.5 A close d subset of an A r onszajn c omp actum is either se c ond c ountable or an Ar onszajn c o m p actum. The Dedekind completion of an Aronsza jn line is an Aronsza jn compactum (see Section 2) , and t he asso ciated tree is essen tially the same as the standard tree of closed in terv als. A sp ecial case of t his is a compact Suslin line, whic h is a w ell-kno wn compact L-space; that is, it is HL (hereditarily Lindel¨ of ) and not HS (hereditarily separable). The line deriv ed f r o m a sp ecial Aronsza jn tree is muc h differen t t o p ologically , since it is no t ev en ccc. In Section 5 w e shall pro v e: Theorem 1.6 Assuming ♦ , ther e is an Ar on szajn c omp actum which is b oth HS and HL. Our construction is flexible enough to build in additional prop erties f o r the space and its a sso ciated t ree, whic h ma y b e either Suslin o r special; see Theorem 5.8. The form of the t r ee is (up to club-isomorphism) a top o logical inv aria n t of X , but seems to b e unrelated to more con ve n tional top ological pro p erties of X ; for example, X ma y b e totally disconnected, or it ma y b e connected and lo cally connected, with dim( X ) finite o r infinite. Question 1.7 Is ther e , in ZF C, an HL Ar on szajn c o mp actum? W e w ould exp ect a Z F C example to b e b oth HS and HL. Note that an Aronsza jn compactum is dissipated in the sense of [9], so it cannot b e an L- space if t here are no Suslin lines by Corolla ry 5.3 o f [9]. T o refute the existence of a n HL Aro nsza jn compactum, one needs more than just an Aronsza jn tree of closed sets, since this m uch exists in the Cantor set: 2 ELEMENT AR Y SUBMODELS 3 Prop osition 1.8 The r e is an Ar o nszajn tr e e T whose no des ar e close d subsets of the Cantor set 2 ω . The tr e e or dering is ⊃ , with r o ot 2 ω . Each level of T c onsists o f a p airwise disjoint family of sets. The pro of is like that of Theorem 4 of Galvin and Miller [4], whic h is attributed there to T o dor ˇ cevi ´ c. 2 Elemen t ary Sub mo de l s W e consider Aronsza jn compacta from the p oin t of view o f elemen tary submodels. Assume that X is compact, with X ( and its top ology) in some suitably large H ( θ ). If X is first countable, so that | X | ≤ c and its top o logy is a set of size ≤ 2 c , then θ can b e an y regular car dina l larger than 2 c , assuming that the set X is c ho sen so that its t ransitiv e clos ure has size ≤ c . If X ∈ M ≺ H ( θ ), then there is a natural quotien t map π = π M : X ։ X/ M obtained by iden tifying t w o points of X iff they are not separated by an y function in C ( X , R ) ∩ M . F urthermore, X/ M is second countable whenev er M is coun table. Lemma 2.1 Assume that X is c o mp act, w ( X ) = ℵ 1 , an d χ ( X ) = ℵ 0 . Then the fol lowing a r e e quiva l e n t: 1. X is an A r onszajn c omp actum. 2. Whenever M is c ountable and X ∈ M ≺ H ( θ ) , ther e ar e onl y c o untably many y ∈ X/ M such that π − 1 { y } is not a singleton. 3. (2) holds for al l M in some club of c ountable elementary submo dels of H ( θ ) . Pro of. F or (1) → (2), note tha t X ∈ M ≺ H ( θ ) implies that M contains some club satisfying D efinition 1.2. K F or example, sa y that X is a compact first coun table LOTS. Then the equiv alence classes a re all con vex ; and, if x < y then π ( x ) = π ( y ) iff [ x, y ] ∩ M = ∅ . No w consider Aronsza jn lines: Definition 2.2 A compacted Aronsza jn line is a c omp act LO TS X such that w ( X ) = ℵ 1 and χ ( X ) = ℵ 0 and the closur e of every c ountable set is se c ond c o untable. By χ ( X ) = ℵ 0 , there are no increasing or decreasing ω 1 –sequence s. No t e tha t our definition a llo ws for the p ossibilit y that X con tains uncoun tably man y disjoint in terv als isomorphic to [0 , 1]. The term “ c omp act Aronsza jn line” is not common in the literature. An Ar on szajn line is usually define d to b e a LOTS of size ℵ 1 with no increasing or decreasing ω 1 –sequence s and no uncoun table subsets o f r e al typ e (that is, order-isomorphic to a s ubset of R ). Suc h a LOTS cannot b e compact; the Dedekind completions of suc h LO TSes are the compact e d Aronsza jn lines of Definition 2 .2 . 3 NORMALIZING AR ONSZAJN COMP AC T A 4 Lemma 2.3 A LOT S X is an Ar onszajn c omp actum i ff X is a c omp a c te d Ar onszajn line. Pro of. F o r ← : suppose that X ∈ M ≺ H ( θ ) and M is countable. Then X/ M is a compact metric LOTS, and is hence or der-em b eddable into [0 , 1]. Supp ose there w ere an uncoun ta ble E ⊆ X/ M suc h that | π − 1 { y }| ≥ 2 fo r all y ∈ E . Say π − 1 { y } = [ a y , b y ] ⊂ X f o r y ∈ E , where a y < b y . If D is a countable dense subset of E then cl( { a y : y ∈ D } ) ⊆ X w ould not b e second coun t able, a con tradiction. K W e use the standard definition of a Suslin line as any LOTS whic h is ccc and no t separable; this is alw a ys an L-space. Then a c om p act Suslin l i n e is just a Suslin line whic h happ ens to b e compact. A compacted Aronsza jn line may b e a Sus lin line, but a compact Suslin line need not be a compacte d Aronsza jn line. F or example, we ma y form X from a connected compact Suslin line Y by doubling uncountably man y p oints lying in some Cantor subset of Y . More generally , Lemma 2.4 L et X b e a c o mp act Suslin line. Then X is a c om p acte d Ar ons z a jn line iff D := { x ∈ X : ∃ y > x ([ x, y ] = { x, y }} do es not c ontain an unc ountable subset of r e al typ e. Pro of. Note that D is the set of all p oints with a right nearest neighbor. If D con ta ins an uncountable set E real ty p e, let B ⊆ E b e coun table and dense in E . Then whenev er M is coun table and X , B ∈ M ≺ H ( θ ), there are uncoun tably man y y ∈ X/ M s uc h that | π − 1 { y }| ≥ 2, so that X is not an Aronsza j n compactum. Con ve rsely , if X is not an Aronsza jn compactum, consider an y coun table M with X ∈ M ≺ H ( θ ) and A := { y ∈ X/ M : | π − 1 { y }| ≥ 2 } uncoun table. Let A ′ := { y ∈ X/ M : | π − 1 { y }| > 2 } . Since eac h π − 1 { y } is conv ex, A ′ is coun table by the ccc, and the left p o ints of the π − 1 { y } for y ∈ A \ A ′ yield an uncoun table subset o f D of real t yp e. K A zero dimens ional compact Suslin line formed in the usual w a y fro m a binary Suslin tree will a lso b e a compac ted Aronsza jn line. 3 Normalizing Aronsz a jn Compacta The club C and tree T deriv ed from an Aronsza jn compactum X in Definition 1.2 can dep end on the em b edding of X in to [0 , 1] ω 1 . T o standardize the tree, we choose a nice em b edding. F or X ⊆ [0 , 1 ] ω 1 , C cannot in g eneral b e ω 1 , since C = ω 1 implies that dim( X ) ≤ 1. Replacing [0 , 1 ] b y the Hilbert cub e, how ev er, we can assume C = ω 1 , whic h simplifies our tree not a tion. In particular, the lev els will be indexe d b y ω 1 , s o that L α will b e lev el α of the tree in the usual se nse. 3 NORMALIZING AR ONSZAJN COMP AC T A 5 Definition 3.1 Q d e notes the Hilb ert cub e, [0 , 1] ω . If X ⊆ Q ω 1 is clos e d and α < ω 1 , then L α = L α ( X ) = { x ∈ X α : | ( σ ω 1 α ) − 1 { x }| > 1 } . W ( X ) = { α < ω 1 : |L α | ≤ ℵ 0 } . So, X is an Aronsza jn compactum iff W ( X ) contains a club; W ( X ) itself need not b e closed, and W ( X ) dep ends on ho w X is em b edded in to Q ω 1 . No w, using the f a cts that Q ∼ = Q ω and that a n Aronsza jn tree can ha v e only coun tably man y finite lev els: Lemma 3.2 Every Ar on s z ajn c omp actum is home omorph i c to some X ⊆ Q ω 1 such that W ( X ) = ω 1 and |L α | = ℵ 0 for al l α > 0 . Of course, L 0 = X 0 = {∅} = Q 0 , and ∅ is the ro ot no de of the tree. Definition 3.3 If X ⊆ Q ω 1 is an Ar onszajn c omp actum and W ( X ) = ω 1 , let b L α = { x ∈ L α : w (( σ ω 1 α ) − 1 { x } ) = ℵ 1 } , and let b T = S α b L α . Since X is not se cond coun table, eac h b L α 6 = ∅ and b T is an Aronsza jn subtree of T . Repeating the ab ov e argumen t , w e get Lemma 3.4 Every Ar on s z ajn c omp actum is home omorph i c to some X ⊆ Q ω 1 such that W ( X ) = ω 1 , and | b L α | = ℵ 0 for al l α > 0 , and e ach x ∈ L α \ b L α is a le af, and e ach x ∈ b L α has ℵ 0 imme diate suc c ess o rs in b L α +1 . This normalization can a lso b e obtained with elemen tary submodels. Sta r t with a con tin uous c hain of elemen tary submodels, M α ≺ H ( θ ), for α < ω 1 , with X ∈ M 0 and eac h M α ∈ M α +1 . L et X α = X/ M α , let π α : X ։ X α b e the natural map, and let L α = { y ∈ X α : | π − 1 α { y }| > 1 } . W e may view eac h X α as em b edded top ologically into Q α , in whic h case L α has the same meaning a s b efore. If π − 1 α { y } is second coun ta ble, then (since M α ∈ M α +1 ), all the p oints in π − 1 α { y } are separated by functions in C ( X ) ∩ M α +1 , so y ∈ L α \ b L α is a leaf. If X is a compacted Aronsza jn line, then X α +1 is formed b y replacing eac h y ∈ L α b y a compact interv al I y of size at least 2. If y ∈ L α \ b L α , then π − 1 α { y } is second coun ta ble and is isomorphic to I y . Note tha t the tree may ha v e uncoun tably man y lea ves ; w e do no t obtain the conv en tiona l normalization o f an Aronsza jn tree, whe re the t ree is uncountable ab o v e ev ery no de. Next, w e consider the ideal of second coun table subsets of X : Definition 3.5 F or any sp ac e X , I X denotes the fa m ily of al l S ⊆ X such that S , with the subsp a c e top olo gy, is se c ond c ountable. I X need not b e an ideal. It is obv iously closed under subse ts, but need not b e closed under unio ns (consider ω ∪ { p } ⊂ β ω ). 3 NORMALIZING AR ONSZAJN COMP AC T A 6 Lemma 3.6 Assume that X ⊆ Q ω 1 is an HL Ar onsza jn c omp actum, as in L emma 3.2. Then I X is a σ –ide al, and, for al l S ⊆ X , the fol low ing ar e e quivalent: 1. S ∈ I X . 2. F or some α < ω 1 , σ ω 1 α ( S ) ∩ L α = ∅ . 3. Ther e is a G ⊇ S such that G ∈ I X and G is a G δ subset o f X . 4. Ther e is a n f ∈ C ( S, Q ) such that f is 1-1. 5. Ther e is a n f ∈ C ( S, Q ) such that f − 1 { y } is se c ond c o untable for al l y ∈ Q . Pro of. It is easy to verify (2) → (3 ) → (1) → (4 ) → (5). In particular, for (2) → (3): Fix α and let G = ( σ ω 1 α ) − 1 ( X α \L α ). Then G is a G δ set and σ ω 1 α : G ։ X α \L α is a 1-1 closed map, and hence a homeomorphism. F or (1) → ( 2 ): Fix an op en base for S of the form { V n ∩ S : n ∈ ω } , where each V n is op en in X . X is HL, so V n is an F σ . W e can thus fix ξ < ω 1 suc h that eac h V n = ( σ ω 1 ξ ) − 1 ( σ ω 1 ξ ( V n )). It follo ws that σ ω 1 ξ is 1-1 o n S . W e ma y then c ho ose α with ξ < α < ω 1 suc h t ha t σ ω 1 α ( S ) ∩ L α = ∅ . No w, I X is a σ –ideal b y (1) ↔ (2) . T o pro v e (5) → (2): Fix f as in (5). Let { U n : n ∈ ω } b e an op en base f or Q ; then f − 1 ( U n ) = S ∩ V n , where V n is open in X and hence an F σ set. W e can th us fix α < ω 1 suc h that V n = ( σ ω 1 α ) − 1 ( σ ω 1 α ( V n )). It follows that f is constan t on S ∩ ( σ ω 1 α ) − 1 { z } fo r all z ∈ X α . Th us, S ∩ ( σ ω 1 α ) − 1 { z } is second coun table for all z ∈ X α . But then S is con ta ined in the union of S { S ∩ ( σ ω 1 α ) − 1 { y } : y ∈ L α } and ( σ ω 1 α ) − 1 ( X α \L α ) ∼ = ( X α \L α ), so S ∈ I X b ecause I X is a σ – ideal. K This pro of sho ws t ha t ev ery Aronsza jn compactum is an ascending union of ω 1 P olish sp aces: namely , the ( σ ω 1 α ) − 1 ( X α \L α ). W e needed X to b e Aronsza jn in Lemma 3 .6; HS a nd HL are not enough to pro v e the equiv alence of (1)(3 )(4)(5). If S is the Sorg enfrey line con tained in the double arro w space X , then (4)(5) are tr ue but (1)(3) are false. Similar remarks hold for similar spaces whic h are b oth HS and HL. F or example, assuming CH, Filipp o v [2] constructed a locally connected con tinuu m whic h is HS and HL but not second coun ta ble. The space was obtained b y replacing a Luzin set o f p o in t s in [0 , 1] 2 b y circles. If S con tains one p oint from eac h o f the circles, then S satisfies (4) (5) but fails (1)(3). In b oth examples, t he space X itself satisfie s (5) but not (1)(3)(4). More generally , a ny space X that has an f ∈ C ( X , Q ) as in (5) cannot be an Aronsza jn compactum. Th us, a ZFC example of an HL Aro nsza jn compactum would settle in the nega t iv e the following we ll-kno wn question of F remlin ([3] 44Qc): is it cons isten t that for ev ery HL compactum X , there is an f ∈ C ( X , Q ) suc h that | f − 1 { y }| < ℵ 0 for all y ∈ Q ? In [5], Gruenhage giv es some of the history related to this question, and p oints out some related results suggesting that the answ er might b e “y es” under PF A. 4 CLOSURE PROPER TIES OF ARONSZAJN COMP A CT A 7 4 Closure Prop ertie s of Arons za jn Compact a Closure under subspaces was already men tio ned in Lemma 1.5. F or pro ducts, Lemma 2.1 implies: Lemma 4.1 Assume that X is an Ar onszajn c omp actum and Y is an arbitr ary s p ac e. Then X × Y is an Ar onsza jn c omp actum iff Y is c o m p act and c ountable. Regarding quotien ts, w e first pro v e: Lemma 4.2 Assume that X, Y ar e c om p act, ϕ : X ։ Y , and X, Y , ϕ ∈ M ≺ H ( θ ) . L et ∼ denote t he M e quivalenc e r el a tion on X a n d on Y . Then 1. If x 0 , x 1 ∈ X and x 0 ∼ x 1 , then ϕ ( x 0 ) ∼ ϕ ( x 1 ) ; so, the inverse image of an e quivalenc e class of Y is a union of e quivalenc e classes of X . 2. If y 0 , y 1 ∈ Y an d x 0 6∼ x 1 for al l x 0 ∈ ϕ − 1 { y 0 } and al l x 1 ∈ ϕ − 1 { y 1 } , then y 0 6∼ y 1 . Pro of. F or (1): If f ∈ C ( Y ) ∩ M separates ϕ ( x 0 ) from ϕ ( x 1 ) then f ◦ ϕ ∈ C ( X ) ∩ M separates x 0 from x 1 . F or (2): F or each x 0 ∈ ϕ − 1 { y 0 } and x 1 ∈ ϕ − 1 { y 1 } , there is an f ∈ C ( X , [0 , 1]) ∩ M suc h that f ( x 0 ) 6 = f ( x 1 ). By compactness of ϕ − 1 { y 0 } × ϕ − 1 { y 1 } , there are f 0 , . . . , f n − 1 ∈ C ( X , [0 , 1]) ∩ M for some n ∈ ω suc h that: for a ll x 0 ∈ ϕ − 1 { y 0 } and x 1 ∈ ϕ − 1 { y 1 } , there is some i < n suc h that f i ( x 0 ) 6 = f i ( x 1 ). The se yield an ~ f ∈ C ( X , [0 , 1] n ) ∩ M suc h that ~ f ( ϕ − 1 { y 0 } ) ∩ ~ f ( ϕ − 1 { y 1 } ) = ∅ . Since M con- tains a base f or [0 , 1] n , there are op en U 0 , U 1 ⊆ [0 , 1] n with each U i ∈ M such that U 0 ∩ U 1 = ∅ and each ~ f ( ϕ − 1 { y i } ) ⊆ U i , so that ϕ − 1 { y i } ⊆ ( ~ f ) − 1 ( U i ). Let V i = { y ∈ Y : ϕ − 1 { y } ⊆ ( ~ f ) − 1 ( U i ) } . Then the V i are op en in Y , eac h V i ∈ M , eac h y i ∈ V i , and V 0 ∩ V 1 = ∅ . There is thu s a g ∈ C ( Y ) ∩ M suc h that g ( V 0 ) ∩ g ( V 1 ) = ∅ , so that g ( y 0 ) 6 = g ( y 1 ). Th us, y 0 6∼ y 1 . K Theorem 4.3 Assume that X is an Ar o n szajn c omp actum, ϕ : X ։ Y , w ( Y ) = ℵ 1 , and χ ( Y ) = ℵ 0 . Then Y is an Ar onszajn c omp actum. Pro of. It is sufficien t to chec k that for a club of elemen tary submo dels M , all but coun ta bly man y M –classes of Y are singletons. Fix M as in Lemma 4.2; so all but coun ta bly man y M –classes of X are singletons. Then for all but coun tably many classes K = [ y ] o f Y : all M –classes of X inside of ϕ − 1 ( K ) are singletons, so tha t , b y the lemma, K is a singleton. K Note t hat we needed to assume that χ ( Y ) = ℵ 0 . Otherwise, when X is not HL, w e w ould get a trivial counterex ample of the form X/K , where K is a closed s et whic h is no t a G δ . 5 CONSTR UCTING AR ONSZAJN COMP A CT A 8 Examining whether an Aronsza jn compactum may b e b oth HS and HL reduces to considering zero dimensional spaces and connected spaces, b y the following lemma. Lemma 4.4 Assume that X is an HL Ar onsza jn c o mp actum, ϕ : X ։ Y . Th e n either Y is an Ar on s zajn c omp actum or some ϕ − 1 { y } is an Ar onsza jn c omp actum. Pro of. Y will b e an Aronsza jn compactum unless it is second countable. But if it is second coun table, then some ϕ − 1 { y } will b e not second coun table by Lemma 3.6, and then ϕ − 1 { y } will b e a n Aronsza jn compactum. K Corollary 4.5 Supp ose ther e is an Ar o n szajn c o mp actum X which is HS a nd HL. Then ther e is an Ar onsz ajn c omp actum Z which is HS and HL and which is either c onne cte d o r zer o dim e nsional. Pro of. Get ϕ : X ։ Y b y collapsing all connected comp onen ts to p oin ts. Then Z is either Y or some comp o nent. K Note that the cone ov er X is also connected, but is not an Aronsza jn compactum b y Lemma 4.1. 5 Constr u cting Aronsza jn Compacta W e b egin this section b y constructing a space X whic h prov es Theorem 1.6. W e construct X = X ω 1 as an in verse limit as a closed subspace of Q ω 1 . T o make X both HS and HL, we shall apply the follow ing lemma: Lemma 5.1 Assume that X is c om p act a n d for al l close d F ⊆ X , ther e is a c om p act metric Y and a map g : X ։ Y such that g ↾ g − 1 ( g ( F )) : g − 1 ( g ( F )) ։ g ( F ) i s irr e ducible. Then X is b oth HS and HL. Pro of. By irreducibilit y , g − 1 ( g ( F )) = F , so that F is a G δ and F is separable. Th us, X is a compact HL space in which all closed subsets are separable, so X is HS. K In applying the lem ma to X = X ω 1 , g will be some π ω 1 α ↾ X . W e shall use ♦ to capture all closed F ⊆ Q ω 1 so that all closed F ⊆ X will b e considered. This metho d w as also employ ed in [6], whic h constructed some compacta whic h were HS and HL but not Aronsza j n. As in standa r d inv erse limit constructions, w e inductiv ely construct X α ⊆ Q α , for α ≤ ω 1 . T o ensure t ha t X will be Aronsza jn, at eac h stage α < ω 1 , w e carefully select a coun ta ble set E α ⊆ X α of “expandable p oints”, and a t each stage β > α , w e cons truct X β ⊆ Q β so that | ( σ β α ) − 1 { x }| = 1 whenev er x / ∈ E α . Then the L α of Definition 3.1 will b e subsets of E α and hence coun ta ble. These preliminaries are included in the following conditions: 5 CONSTR UCTING AR ONSZAJN COMP A CT A 9 Conditions 5.2 X α , for α ≤ ω 1 , and P α , F α , E α , q α , for 0 < α < ω 1 , satisfy: 1. Each X α is a close d subset of Q α . 2. π β α ( X β ) = X α whenever α ≤ β ≤ ω 1 . 3. P α is a c ountable family of close d subsets of X α , and F α ∈ P α . 4. F or al l P ∈ P α : a. σ α +1 α ↾ (( σ α +1 α ) − 1 ( P )) : ( σ α +1 α ) − 1 ( P ) ։ P is irr e ducible, a n d b. ( σ β α ) − 1 ( P ) ∈ P β whenever α ≤ β < ω 1 . 5. F or al l c l o se d F ⊆ X , ther e is an α with 0 < α < ω 1 such that σ ω 1 α ( F ) = F α . 6. E α is a c ountable dense subset of X α , and q α ∈ E α . 7. E β ⊆ ( σ β α ) − 1 ( E α ) whene ver 0 < α ≤ β < ω 1 . 8. | ( σ α +1 α ) − 1 { x }| = 1 whenever 0 < α < ω 1 and x ∈ X α \{ q α } . 9. | ( σ α +1 α ) − 1 { q α }| > 1 . W e discus s b elo w how to satisfy these conditions. Conditions (1) and (2) sim- ply determine our X = X ω 1 ⊆ Q ω 1 with eac h X α = π ω 1 α ( X ). ♦ is used for (5). Constructing an X that satisfies Conditions (1 - 9) is enough to prov e Theorem 1.6: Lemma 5.3 Conditions (1 − 9) imply that X = X ω 1 is an A r onszajn c omp actum an d is b oth HS and HL. Pro of. By (4) and induction on β , σ β α ↾ (( σ β α ) − 1 ( P )) : ( σ β α ) − 1 ( P ) ։ P is ir reducible whenev er α ≤ β ≤ ω 1 and P ∈ P α . Then X is HS and HL by Lemma 5.1 and (5)(3). By (6)(7)(8) and induction, | ( σ β α ) − 1 { x }| = 1 whenev er 0 < α ≤ β ≤ ω 1 and x ∈ X α \E α . So, L α := { x ∈ X α : | ( σ ω 1 α ) − 1 { x }| > 1 } ⊆ E α , whic h is countable b y (6). Finally , w ( X ) = ℵ 1 b y (9), and χ ( X ) = ℵ 0 b ecause X is HL. K T o obtain Conditions (1 − 9), w e m ust add some further conditions so that the natural construction av oids contradictions. F or examp le, satisfying Conditions (6 ) and ( 7) at stage β requires T α<β ( σ β α ) − 1 ( E α ) 6 = ∅ . So w e add Conditions ( 1 0 - 12) b elo w making the E α in to the leve ls of a tree; the selection of the E α will resem ble the standard inductiv e construction of an Aronsza jn tree. The sets F α ma y b e scattered or ev en singletons. This cannot b e a v oided, b ecause w e are using the F α to ensure that al l closed sets are G δ sets, so that X is HL; making just the perfect sets G δ could pro duce a F edorc huk space (as in [7]) , whic h is not ev en first coun table. If x ∈ P ∈ P α and x is isolated in P , then the irreducibilit y condition in (4) requires tha t | ( σ α +1 α ) − 1 { x }| = 1, but that con tra dicts (9) if x = q α . No w, if ev ery p oint of E α is isolat ed in some P ∈ P α , then w e cannot c ho ose q α ∈ E α , as required b y (6). W e shall a v oid thes e problems b y requiring that if x ∈ E α and P ∈ P α , then either x / ∈ P or x is in the p erfect k ernel of P . This can b e ensured by c ho osing F α first (as give n b y ♦ ), and then choosing E α ; for limit α , our Aronsza jn 5 CONSTR UCTING AR ONSZAJN COMP A CT A 10 tree construction will give us plen t y of options for c ho osing the p oints of E α , and we shall mak e F α trivial for successor α . The additional conditions that handle this will emplo y the notation in the follow ing: Definition 5.4 If F is c om p act a n d not sc atter e d, let k er( F ) denote the p erfe ct k ernel of F ; otherwise, ker( F ) = ∅ . T o satisfy Condition (8 ), w e construct X α +1 from X α b y c ho osing an appropriate h α ∈ C ( X α \{ q α } , Q ), and letting X α +1 = cl ( h α ). Iden tifying Q α +1 with Q α × Q and h α with its graph, h α ( x ) is the y ∈ Q suc h that x ⌢ y ∈ X α +1 . Note that h α is indeed con tinuous b ecause its g raph is close d. Th us, to cons truct X so that Conditions (1 - 9) are met, w e add the follo wing: Conditions 5.5 h α and r n α , for 0 < α < ω 1 and n < ω , s atisfy: 10. ( σ β α )( E β ) = E α whenever 0 < α ≤ β < ω 1 . 11. |E α +1 ∩ ( σ α +1 α ) − 1 { q α }| > 1 . 12. If x ∈ E α , t hen ( σ α + n α )( q α + n ) = x for some n ∈ ω . 13. X α has no isolate d p o i nts whenev er α > 0 . 14. F α = ∅ whene v er α is a suc c essor o r di n al. 15. P β = { F β } ∪ { ( σ β α ) − 1 ( P ) : 0 < α < β & P ∈ P α } . 16. E α ∩ ( P \ k er( P )) = ∅ whenever P ∈ P α . 17. r n α ∈ X α \{ q α } and the se quen c e h r n α : n ∈ ω i c on ver ges to q α . 18. h α ∈ C ( X α \{ q α } , Q ) , and X α +1 = cl( h α ) . 19. If q α ∈ P ∈ P α , then r n α ∈ k er( P ) for infinitely many n , and every y ∈ Q with q ⌢ α y ∈ X α +1 is a limit p oint of the se quen c e h h α ( r n α ) : n ∈ ω & r n α ∈ ker( P ) i . Observ e that (10)(11 ) (12) will giv e us the follow ing: Lemma 5.6 L α = E α whenever 0 < α < ω 1 . In the tree T ( X ), altho ugh only the no de q α ∈ L α has more than o ne successor in L α +1 , (12) ensures that a t limit lev els γ , there ar e 2 ℵ 0 c hoices for the elemen ts of E γ , so that we may a v o id t he p oin ts in F γ \ k er( F γ ), as required by (16 ) . By (14)(1 5), ∅ ∈ P α for all α > 0, and non-empt y sets are a dded in to the P α only at limit α . The following pro of giv es the bare-b ones construction; refinemen ts of it pro duce the spaces of Theorem 5.8. Pro of of Theorem 1.6. Before w e s tart, use ♦ to c ho ose a closed e F α ⊆ Q α for eac h α < ω 1 , so that { α < ω 1 : π ω 1 α ( F ) = e F α } is stationary for all closed F ⊆ Q ω 1 . 5 CONSTR UCTING AR ONSZAJN COMP A CT A 11 T o b egin the induction: X 0 m ust b e {∅} = Q 0 , and P α , F α , . . . . . . are only defined f o r α > 0 . No w, fix β with 0 < β < ω 1 , and assume that all conditions hav e b een met b elow β . W e define in o r der X β , F β , P β , E β , q β , r n β , h β . If β is a limit, then X β is determined by (1)(2 ) and the X α for α < β . X 1 can b e any p erfect subset of Q 1 . If β = α + 1 ≥ 2, then X β = cl( h α ), as required by (18). No w let F β = e F β if e F β ⊆ X β and β is a limit; otherwise, let F β = ∅ . P β is no w determined b y (15). E 1 can b e an y coun ta ble dense subset of X 1 . If β = α + 1 ≥ 2, let E β = ( σ β α ) − 1 ( E α \{ q α } ) ∪ D β , where D β is an y subset of ( σ β α ) − 1 { q α } suc h that 2 ≤ |D β | ≤ ℵ 0 . Observ e that E β is dense in X β (without using D β ), so (6 ) is preserv ed, and D β guar- an tees t ha t (11) is preserv ed. T o v erify (16) at β , note that by (15 ) at α , ev ery non-empt y set in P β is of the form b P := ( σ β α ) − 1 ( P ) fo r some P ∈ P α . So, if ( 1 6) f a ils at β , fix P ∈ P α and x ∈ E β ∩ ( b P \ ke r( b P )). Then x ∈ ( σ β α ) − 1 { q α } , so q α ∈ P , and hence q α ∈ ker( P ); but then by (1 9), x is a limit o f a sequence of elemen ts of ke r( b P ), so that x ∈ k er ( b P ). F or limit β , let E β = { x ∗ : x ∈ S α<β E α } , where, x ∗ , for x ∈ E α , is some y ∈ X β suc h tha t π β α ( y ) = x and π β ξ ( y ) ∈ E ξ for all ξ < β . An y such c hoice o f the x ∗ will satisfy ( 10). But in fact, using (11 )(12), for eac h suc h x there are 2 ℵ 0 p ossible c hoices of x ∗ , so w e can satisfy (16) b y av oiding the countable sets P \ k er( P ) for P ∈ P β . T o facilitat e (12), list eac h E α as { e j α : j ∈ ω } ; let e j 0 = ∅ ∈ X 0 . Then, if β is a success or ordinal of the form γ + 2 i 3 j , where γ is a limit or 0, c ho ose q β ∈ E β so that σ β γ + i ( q β ) = e j γ + i . F or other β , q β ∈ E β can b e c ho sen arbitrarily . Next, we ma y choose the r n β to satisfy (19) b ecause if q β ∈ P ∈ P β , then q β ∈ ker( P ) b y (16 ), so that q β is a lso a limit of points in k er ( P ). Finally , w e m ust c ho ose h β ∈ C ( X β \{ q β } , Q ). Conditions (18 )(19) only require that h β ha v e a discon tinuit y at q β with the property that ev ery limit p oin t of the function at q β is also a limit of eac h o f the sequence s h h β ( r n β ) : n ∈ ω & r n β ∈ ker( P ) i . Since X β is a compact metric spac e with no isolated p oin ts, w e may accomplish this b y making eve ry p o int of Q a limit p oin t o f eac h h h β ( r n β ) : n ∈ ω & r n β ∈ ker( P ) i . K If we c ho ose each h β as ab o v e and also set X 1 = Q , then our X will b e connected, and it is fairly easy to choose the h β so that X fails to be locally conne cted. The next theorem sho ws how to mak e X connected and locally connected. W e construct X so t hat each X α is homeomorphic to the Menger sp onge , MS , and all the maps σ β α are monotone. The Menger sp onge [10] is a o ne dimensional lo cally connected metric con tin uum; the prop erties of MS used in inductiv e constructions suc h as these are summarized in [7], whic h con tains further references to the literature. A map is monotone iff a ll p o in t in vers es ar e connected. Monotonicit y o f the σ β α will imply that X is lo cally connected. 5 CONSTR UCTING AR ONSZAJN COMP A CT A 12 A t successor stages, to construct X α +1 ∼ = MS , w e a ssume that X α ∼ = MS and apply the f ollo wing sp ecial case of Lemmas 2.7 and 2.8 of [7]: Lemma 5.7 Assume that q ∈ X ∼ = MS and that for e ach j ∈ ω , the se quenc e h r n j : n ∈ ω i c onver ges to q , w i th e a ch r n j 6 = q . L et π : X × [0 , 1] ։ X b e the natur al pr oje ction. Then ther e is a Y ⊆ X × [0 , 1] such that: 1. Y ∼ = MS and π ( Y ) = X . 2. | Y ∩ π − 1 { x }| = 1 for al l x 6 = q . 3. π − 1 { q } = { q } × [0 , 1] . 4. L et Y ∩ π − 1 { r n j } = { ( r n j , u n j ) } . Then, for e ach j , every p oint in [0 , 1] is a limit p oint of h u n j : n ∈ ω i . Constructing X as such an inv erse limit of Menger sp onges will mak e X one dimen- sional. The results quoted from [7] ab o ut MS were patterned on an earlier construction of v an Mill [11 ], whic h in v olv ed an inv erse limit of Hilb ert cub es; replacing MS by Q here w ould yield an infinite dimensional v ersion of this Aronsza jn compactum. The follo wing summarizes sev eral possibilities for X and it s a sso ciated tree: Theorem 5.8 Assume ♦ . F or e ach o f the fol l o wing 2 · 3 = 6 p ossibilities, ther e is an A r onszajn c o m p actum X with asso ciate d Ar onszajn tr e e T such that X is HS and HL. Pos s i b ilities for T : a. T is S uslin . b. T is sp e cial. Possibilities for X : α . dim( X ) = 0 . β . dim( X ) = 1 and X is c onne cte d and lo c al ly c o n ne cte d. γ . dim( X ) = ∞ and X is c on n e cte d and lo c al ly c onne cte d. Pro of. W e refine the pro of of Theorem 1.6, T o obtain ( a ) or ( b ), the refinemen t is in the c hoice of the E β for limit β . T o obtain ( α ) or ( β ) or ( γ ), the refinemen t is in the c hoice of X 1 and the functions h α . Since these refinemen ts are indep enden t of eac h other, the discussion of ( a )( b ) is unrelated to the discussion o f ( α )( β )( γ ). F or ( a ): W e use ♦ to kill all p oten tia l uncoun table maximal antic hains A ⊂ T . Fix a sequence h A α : α < ω 1 i suc h that eac h A α is a coun table subset of Q <α and suc h that for all A ⊆ Q <ω 1 : if each A ∩ Q <α is countable, then { α < ω 1 : A ∩ Q <α = A α } is statio nary . Let T β = S {L α : α < β } = S {E α : α < β } (se e Lemma 5.6), and use ⊳ fo r the tree order. F or eac h limit β < ω 1 , mo dify the construction of E β in the pro of of Theorem 1.6 as follows : W e still ha ve E β = { x ∗ : x ∈ T β } , where, x ∗ , for x ∈ T β , is 5 CONSTR UCTING AR ONSZAJN COMP A CT A 13 c hosen so that x ⊳ x ∗ and x ∗ defines a path through T β . But no w, if A β ⊆ T β and A β is a maximal an tic ha in in T β , then make sure that eac h x ∗ is a b ov e some elemen t of A β . T o do this, use maximalit y of A β first to c ho ose x † ∈ T β so that x ⊳ x † and x † is ab o v e some eleme n t of A β , and then c ho ose x ∗ so tha t x ⊳ x † ⊳ x ∗ . There are still 2 ℵ 0 p ossible c hoices for x ∗ , so w e can satisfy (16) by av oiding the coun table se ts P \ k er( P ) as b efore. No w, the usual argumen t sho ws that T is Suslin. F or ( b ): Let Lim denote the set o f coun table limit ordinals, and let T Lim = S {L α : α ∈ Lim } = S {E α : α ∈ Lim } . T o make T sp ecial, inductiv ely define an order pre- serving map ϕ : T Lim → Q . T o make the induction work, we also assume inductiv ely: ∀ γ , β ∈ Lim ∀ x ∈ L γ ∀ q ∈ Q [ γ < β & q > ϕ ( x ) → ∃ y ∈ L β [ x ⊳ y & ϕ ( y ) = q ]] ( ∗ ) T o start the induction, ϕ ↾ L ω : L ω → Q can b e arbitra ry . F or β = α + ω , where α is a limit o rdinal: First, determine the x ∗ exactly as in the pro of of Theorem 1.6. Then, no t e that f or eac h x ∈ L α , the set S x := { y ∈ E β : x ⊳ y } has size ℵ 0 , so w e can let ϕ ↾ S x map S x onto Q ∩ ( ϕ ( x ) , ∞ ). F or β < ω 1 whic h is a limit of limit ordinals: Let E β = { x ∗ q : x ∈ T β & q ∈ Q ∩ ( ϕ ( x ) , ∞ ) } , where eac h x ∗ q is c hosen so tha t x ⊳ x ∗ q and x ∗ q defines a pat h thr o ugh T β and the x ∗ q are all differen t as q v aries. W e let ϕ ( x ∗ q ) = q , whic h will clearly preserv e ( ∗ ), but we m ust mak e sure that ϕ remains order preserving. F or this, choose x ∗ q so tha t q > sup { ϕ ( z ) : z ∈ T Lim & z ⊳ x ∗ q } . Suc h a c hoice is p ossible using ( ∗ ) on T β . As b efore, there are 2 ℵ 0 p ossible c hoices of x ∗ q , so w e can still a v oid the countable sets P \ k er( P ). F or ( α ), just mak e sure that X α is homeomorphic to the Can tor s et 2 ω whenev er 0 < α < ω 1 . In view of (1 3), this is equiv alen t to making X α zero dimensional. F or α = 1, w e simply c ho o se X 1 so tha t X 1 ∼ = 2 ω . Then, for larger α , just mak e sure that in (9) , w e alw a ys hav e | ( σ α +1 α ) − 1 { q α }| = 1, whic h will hold if in (18), w e c ho ose h α ∈ C ( X α \{ q α } , 2) (identifying 2 = { 0 , 1 } as a subset of Q ). T o mak e this c hoice, and satisfy (1 9): First, let A j , for j ∈ ω , b e disjoin t infinite subsets o f ω suc h that for eac h P ∈ P α , if q α ∈ P then for some j , r n α ∈ ker( P ) for a ll n ∈ A j . Next, let X α = K 0 ⊃ K 1 ⊃ K 2 ⊃ · · · , where each K i is clop en, T i K i = { q α } , and, for eac h j , there are infinitely many ev en i and infinite ly man y o dd i suc h that K i \ K i +1 ∩ { r n α : n ∈ A j } 6 = ∅ . Now , let h α b e 0 on K i \ K i +1 when i is ev en a nd 1 on K i \ K i +1 when i is o dd. F or ( β ), construct X so t ha t eac h X α is homeomorphic to the Menger sp onge , MS , and all the maps σ β α are monotone. Then dim( X ) = 1 will follow fr om the fact that X is a n in vers e limit of one dimensional spaces. F or monotonicit y of the σ β α , it suffi ces to ensure that eac h σ α +1 α is monotone. By Condition (8), that will follow if w e mak e ( σ α +1 α ) − 1 { q α } connected; in fact we shall 6 CHAINS OF CLOPEN SETS 14 mak e ( σ α +1 α ) − 1 { q α } homeomorphic to [0 , 1 ], as in the pro of of Theorem 1.6. But w e also need to v erify inductiv ely that X α ∼ = MS . A t limits, this follow s fr o m Lemma 2.5 of [7]. A t successor stages, w e assume that X α ∼ = MS and iden tify [0 , 1] as a subspace of Q , so that X α +1 ma y b e the Y of Lemma 5.7. ( γ ) is pro v ed analogously to ( β ). Construct X α ∼ = Q rather than MS , applying the results ab out Q in [11 ] § 3. As in [1 1] § 2, a ll the σ β α are cell-lik e Z ∗ -maps. K 6 Chains of C lop en Sets The double ar r o w space has an uncoun table c hain (under ⊂ ) of clop en sets of real t yp e. This cannot happ en in an Aronsza jn compactum: Lemma 6.1 If X is an A r onszajn c omp actum and E is an unc ountable c h ain of clop en subsets of X , then E c annot b e of r e al typ e. Pro of. Supp o se that E is suc h a c hain. D eleting some elemen ts of E , w e ma y assume that ( E , ⊂ ) is a dense total order. Let D b e a coun table dense subset of E . Since X is an Aronsza jn compactum, there is a map ϕ : X ։ Z , where Z is a compact metric space, A = ϕ − 1 ( ϕ ( A )) f o r all A ∈ D , and { y ∈ Z : | ϕ − 1 { y }| > 1 } is countable. Since D is dense in E , the sets ϕ ( B ) for B ∈ E are all different. Eac h ϕ ( B ) is closed, and only coun ta bly man y of the ϕ ( B ) can b e clop en. Whenev er ϕ ( B ) is not clopen, c ho ose y B ∈ ϕ ( B ) ∩ ϕ ( X \ B ). Since D is dense in E , these y B are all different p oin ts, so t here are uncoun tably many suc h y B . But ϕ − 1 { y B } meets b o th B and X \ B , so eac h | ϕ − 1 { y B }| ≥ 2 , a con tradiction. K Note tha t if this argumen t is applied with a c hain of clop en sets in the double arro w space, then the | ϕ − 1 { y B }| will be exactly 2. Lemma 6.2 If X is any sep ar able sp ac e, and E is an unc ountable chain of clop en subsets of X , then E must b e of r e a l typ e. Pro of. If D ⊆ X is dens e, then ( E , ⊂ ) is isomorphic to a chain in ( P ( D ) , ⊂ ). K Corollary 6.3 If X is a sep ar able A r onszajn c omp actum and E is a chain of clop en subsets of X , then E is c ountable. Note t hat if X is a zero dimensional compacted Aronsza jn line whic h is also Suslin (see Lemma 2.4), then X has uncountable c hain of clop en sets, but X is not separable. REFERENCES 15 References [1] V. V. F edorc h uk, F ully close d mappings and their applications (Russian), F un- dam. Prikl. Mat. 9 (2003) no. 4, 105-235; English translation in J. Math. Sci. (N.Y.) 136 (2006), no. 5 , 42 0 1-4292 . [2] V. V. Filipp o v, Perfectly normal bicompacta (Russian), Dokl. A kad. 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