Thin-very tall compact scattered spaces which are hereditarily separable

THIN-VER Y T ALL COMP A CT SCA TTERED SP A CES W HICH ARE HERE DIT ARIL Y SEP ARABLE CHRISTINA BRECH AND PIOTR K OSZMIDER Abstract. W e strengthen t he property ∆ of a function f : [ ω 2 ] 2 → [ ω 2 ] ≤ ω considered b y Baumgartner and Shelah. This allows u s to consider new types of amalgamations in the for cing used b y Rabus, Juh´ asz and Soukup to construct thin-v ery tall compact scattered spaces. W e consistently obtain spaces K as abov e where K n is hereditarily separable for each n ∈ N . This serves as a coun terexample concerning cardinal functions on compact spaces as well as ha ving some applications in Banach spaces: the Banach space C ( K ) is an Asplund space of density ℵ 2 which has no F r´ ec het smo oth renorming, nor an uncoun table biorthogonal system. 1. Introduction Given a compact sc a ttered space K , we call the deriv ative of K (deno ted b y K ′ ) the subs e t of K fo r med b y its accumulation p oints and w e inductively define K ( α ) = ( K ( β ) ) ′ if α = β +1 and K ( α ) = T β <α K ( β ) if α is a limit ordina l. The height of K , ht ( K ), is the s mallest ordinal α such that K ( α ) is finite and nonempty , and the width of K , w d ( K ), is the supr emum of the cardinalities | K ( α ) \ K ( α +1) | for α < ht ( K ). W e c all K = S α ξ . Note that ζ ∈ h q ( ξ ) ∗ h q ( η ) ⊆ h q ( ξ ). Since ζ and ξ satisfying the hypothes is of the Ca se 4.2. ar e in D 2 \ D 1 , it follows from the definition of h q that ζ ∈ h 2 ( ξ ). T o finish, let us show the following: F act 2. ζ ∈ h 2 ( ξ ) ∗ h 2 ( θ ) . Pr o of of F act 2. First suppo se θ = δ 2 ( ξ ). If ζ 6∈ dom ( δ 2 ), then ζ / ∈ h 2 ( δ 2 ( ξ )). If ζ ∈ dom ( δ 2 ), fr o m F act 1 and the minimality of δ 2 ( ζ ), it follows that ζ / ∈ h 2 ( δ 2 ( ξ )). Since ξ ∈ h 2 ( δ 2 ( ξ )), we hav e that ζ ∈ h 2 ( ξ ) \ h 2 ( δ 2 ( ξ )) = h 2 ( ξ ) ∗ h 2 ( θ ). 1 This case i s si milar to Sub case 2.2 in the pro of of Claim 2.7.2 of [ 11]. THIN-VER Y T ALL COMP A CT SCA TTERED SP A CES WHICH ARE HS 9 Now suppo se θ = δ 2 ( ζ ). Analogous ly we prov e that ξ / ∈ h 2 ( δ 2 ( ζ )) and ζ ∈ h 2 ( ξ ) ∩ h 2 ( δ 2 ( ζ )) = h 2 ( ξ ) ∗ h 2 ( θ ), co ncluding the pro of o f F act 2 . Finally , since p 2 ∈ P f , there is γ ∈ i 2 ( { ξ , θ } ) such that ζ ∈ h 2 ( γ ) ⊆ h q ( γ ). By condition (B).(i), which can b e use d b y F ac t 1, we hav e that i 2 ( { ξ , θ } ) ⊆ f ( { ξ , θ } ) ∩ D 2 ⊆ f ( { ξ , η } ) ∩ D q = i q ( { ξ , η } ) . Hence, γ ∈ i q ( { ξ , η } ) and ζ ∈ h q ( γ ), concluding the pro of of Sub cas e 4 .2, Case 4 and thus concluding the pr o of of Claim 2. Now we know that q ∈ P f and let us chec k the other conclusio ns : it follows ea sily from the definition o f q and Lemma 2 .5 that q ≤ p 1 and analog ously it follo ws from the definition of q and Lemma 2 .4 that q ≤ p 2 . Finally , we verify the condition we w ant q to satisfy , that is, ξ ∈ h 2 ( η ) ∪ e − 1 [ h 2 ( η )] if and only if e ( ξ ) ∈ h 2 ( η ): let ξ ∈ D 1 and η ∈ D 2 and w e consider ag ain the following cas e s: Case 1. ξ ∈ D 1 ∩ D 2 . It follows from the fa ct that in this cas e e ( ξ ) = ξ . Case 2. ξ ∈ D 1 \ D 2 . In this case, ξ ∈ h 2 ( η ) ∪ e − 1 [ h 2 ( η )] if a nd only if ξ ∈ e − 1 [ h 2 ( η )] if a nd only if e ( ξ ) ∈ h 2 ( η ), concluding the pro o f of the lemma.  3. The main resul ts T o a pply the key lemma prov ed in the previous s ection, the function f on which the forcing P f depe nds must s atisfy a stronger version of the pr op erty ∆: Definition 3.1. A function f : [ ω 2 ] 2 → [ ω 2 ] ≤ ω has the strong pro pe rty ∆ if f ( { ξ , η } ) ⊆ min { ξ , η } for all { ξ , η } ∈ [ ω 2 ] 2 and for any uncount able ∆-system A of finite subsets of ω 2 , there are distinct a, b ∈ A a nd an order -preserv ing bijection e : a → b which is the identit y on a ∩ b and s uch that ξ ≤ e ( ξ ) for all ξ ∈ a and for any τ ∈ a ∩ b , any ξ ∈ a \ b and any η ∈ b \ a we ha ve: 1) a ∩ min { ξ , η } ⊆ f ( { ξ , η } ); 2) τ < ξ ⇒ f ( { τ , η } ) ⊆ f ( { ξ , η } ); 3) τ < η ⇒ f ( { τ , ξ } ) ⊆ f ( { ξ , η } ). Finally we arrive at the main result of this pap er. Theorem 3.2. If f : [ ω 2 ] 2 → [ ω 2 ] ≤ ω has the str ong pr op erty ∆ , then V P f satisfies “for al l n ∈ N , K n f is her e d itarily sep ar able”. Pr o of. W e prov e this by induction on n ∈ N : in V P f , fix n ∈ N and supp ose that for a ll 0 ≤ i < n , K i f is here dita rily s e parable (take K 0 f = {∗} ) and let us show that K n f is her editarily sepa r able. W e will b e using a well-known fact that a regula r space is heredita rily separ a ble if and only if it has no uncountable left-separ ated sequence (see Theo rem 3.1 o f [24]). In V , supp ose ( ˙ x α ) α<ω 1 is a seq uence o f names such that P f forces that ( ˙ x α ) α<ω 1 is a left-separa ted sequence in K n f of car dinality ℵ 1 and for each α < ω 1 , we hav e that ˙ x α = ( ˙ x α 1 , . . . , ˙ x α n ), where e a ch ˙ x α i is a name for an e lemen t of K f . 10 CHRISTINA BRECH AND P IOTR KOSZMIDER Notice that if P f  ∃ 1 ≤ i ≤ n, ∃ X ⊆ ω 1 , | X | = ℵ 1 such that ∀ α, β ∈ X , ˙ x α i = ˙ x β i , then P f  ∃ 1 ≤ i ≤ n, ∃ X ⊆ ω 1 , | X | = ℵ 1 such that (( ˙ x α 1 , . . . , ˙ x α i − 1 , ˙ x α i +1 , . . . , ˙ x α n )) α ∈ X is a left-separated s e q uence in K n − 1 f , contradicting the inductiv e hypothesis. Therefore, w e can assume without los s of generality that P f forces that for a ll 1 ≤ i ≤ n a nd all α < β < ω 1 , ˙ x α i 6 = ˙ x i β and ˙ x α i ∈ L f = K f \ {∗} . By asser tion (+) following Definition 1 .4, for each α < ω 1 , there are names ˙ F α 1 , . . . , ˙ F α n for finite s ubsets of ω 2 such that P f forces that ∀ α < ω 1 ∀ 1 ≤ i ≤ n ˙ x α i ∈ h ( ˙ x α i ) \ [ ξ ∈ ˙ F α i h ( ξ ) and ∀ α < β < ω 1 ∃ 1 ≤ i ≤ n ˙ x α i / ∈ h ( ˙ x β i ) \ [ ξ ∈ ˙ F β i h ( ξ ) . F or each α < ω 1 , let p α = ( D α , h α , i α ) ∈ P f , x α 1 , . . . , x α n ∈ ω 2 and F α 1 , . . . , F α n ⊆ ω 2 be finite s uch tha t p α  ∀ 1 ≤ i ≤ n ˙ x α i = ˇ x α i and ˙ F α i = ˇ F α i . By Le mma 2.2 o f [11], w e ca n as s ume without loss of generality that for all α < ω 1 and all 1 ≤ i ≤ n , F α i ⊆ D α and x α i ∈ D α . By the ∆-system Lemma, we ca n as s ume as well that ( D α ) α<ω 1 forms a ∆- system with r o ot D . Since for ea ch pair { ξ , η } ⊆ D and each α < ω 1 , w e hav e that i α ( { ξ , η } ) ∈ [ f ( { ξ , η } )] <ω , we may as sume that fo r all α < β < ω 1 , if ξ , η ∈ D , ξ 6 = η , then i α ( { ξ , η } ) = i β ( { ξ , η } ). By thinning out, we can ass ume without loss of g enerality that ( D α ) α<ω 1 forms a ∆-system with ro ot D such that for every α < β < ω 1 : • p α is iso morphic to p β ; • p α is lower than p β ; • if e αβ : D α → D β is the order -preserv ing bijectiv e function, then e αβ ( x α i ) = x β i , for all 1 ≤ i ≤ n . Finally , we may as sume that for all 1 ≤ i ≤ n we hav e: either x α i = x β i for all α < β < ω 1 ; or x α i / ∈ D for a ll α < ω 1 and actually the second case holds by our initial assumption ab out the s e quence. Since f ha s the strong pro per ty ∆, there are α < β < ω 1 such that for all ζ ∈ D , all ξ ∈ D α \ D and all η ∈ D β \ D : (i) D α ∩ ξ ∩ η ⊆ f ( { ξ , η } ); (ii) if ζ < ξ , then f ( { ζ , η } ) ⊆ f ( { ξ , η } ); (iii) if ζ < η , then f ( { ζ , ξ } ) ⊆ f ( { ξ , η } ). Note that p α and p β satisfy the hypo thesis of Lemma 2 .7. Hence, there is q ≤ p α , p β in P f such that for all ξ ∈ D α and all η ∈ D β , ξ ∈ h q ( η ) if and only if e αβ ( ξ ) ∈ h p β ( η ) . THIN-VER Y T ALL COMP A CT SCA TTERED SP A CES WHICH ARE HS 11 Then, for all 1 ≤ i ≤ n and a ll ξ ∈ D β , we hav e that x α i ∈ h q ( ξ ) if and only if x β i ∈ h p β ( ξ ) . So we have that x α i ∈ h q ( x β i ) \ [ ξ ∈ F β i h q ( ξ ) . But q ≤ p α , p β and then q  ∀ 1 ≤ i ≤ n, ˙ x α i = ˇ x α i , ˙ x β i = ˇ x β i and ˙ F β i = ˇ F β i . Therefore, q  ∀ 1 ≤ i ≤ n, ˙ x α i = ˇ x α i ∈ h ( ˇ x β i ) \ [ ξ ∈ ˇ F β i h ( ξ ) = h ( ˙ x β i ) \ [ ξ ∈ ˙ F β i h ( ξ ) , contradicting the hypothesis a bo ut ˙ x α i , ˙ x β i and ˙ F β i .  Corollary 3.3. It is r elatively c onsistent with ZF C that ther e is a her e ditarily sep ar able c omp act s c atter e d sp ac e of height ω 2 . Pr o of. Since each level of the Cantor-Bendixso n de c omp o sition of K f is a discrete subset of K f , it follows that every le vel of it is countable. But | K f | = ℵ 2 and K f = S α (2 | P | ) + and a club set C ⊆ [ H ( θ )] ω such that whenever p ∈ M ∈ C and M ∩ ω 2 ∈ F then there is a ( P , M )- g eneric p 0 ≤ p , i.e., D ∩ M is pr edense b elow p 0 for every D ∈ M which is dense in P . F act 4. 3. Supp ose F ⊆ [ ω 2 ] ω is a stationary set and P is an F -pr op er for ci ng notion, then P pr eserves ω 1 . Pr o of. The pr o of is a s traightforw ard version of Shelah’s par adigmatic pro of o f preserv ation of ω 1 by pro per forcings (see [26] o r [1 ]).  The following definition and lemmas are for mulations of well-kno wn techniques (originated in Shelah’s us e of elementary submo dels in for cing) a nd will simplify our further a rguments. Definition 4.4. Let P b e a notion of fo r cing, q ∈ P and let θ > (2 | P | ) + . Supp ose M ≺ H ( θ ) and P , π 1 , ..., π k ∈ M . W e say that a for mula φ ( x 0 , x 1 , ..., x k ) well reflects q in ( M ; π 1 , ..., π k ) whenever the following are satisfied: i) φ ( q , π 1 , ..., π k ) holds in H ( θ ); ii) whenever s ∈ M is such that φ ( s, π 1 , ..., π k ) holds in M , then q a nd s a re compatible. Definition 4.5. Supp ose F ⊆ [ ω 2 ] ω and supp ose P is a notion of forcing. W e say that P is simply F -pro per if there is θ such that whenever a) p ∈ P , b) M ≺ H ( θ ), M countable, c) p, P , F ∈ M , d) M ∩ ω 2 ∈ F , then there is p 0 ≤ p such that if q ≥ p 0 , then there are π 1 , ..., π k ∈ M and a formula φ ( x 0 , x 1 , ..., x k ) which w ell r eflects q in ( M , π 1 , ..., π k ). Lemma 4.6. If P is simply F -pr op er, then P is F - pr op er. 14 CHRISTINA BRECH AND P IOTR KOSZMIDER Pr o of. W e will prov e that whenever M , p ar e as in a) - d) of Definition 4.5, then p 0 is a ( P , M )-gener ic condition. Let D ∈ M b e dense, we will show tha t D ∩ M is predense b elow p 0 . L e t q ≤ p 0 , w e may w.l.o .g . assume that q ∈ D . Let π 1 , ..., π k ∈ M and φ ( x 0 , x 1 , ..., x k ) b e such that φ ( x 0 , x 1 , ..., x k ) well r eflects q in ( M , π 1 , ..., π k ). By i) o f Definition 4.4, we hav e φ ( q , π 1 , ...π k ) in H ( θ ). By its elementarit y , M satisfies the formula “ ∃ x ∈ P φ ( x, π 1 , ...π k ) & x ∈ D ”. So let s ∈ M witness this fact. Now b y Definition 4 .4.ii), s and q are co mpa tible, so D ∩ M contains a co ndition compatible with q whic h pr ov es that D ∩ M is predense below q which completes the pro of.  4.2. Adding a function wi th the s trong prop e rty ∆ . Fix a family F ⊆ [ ω 2 ] ω satisfying 1) - 6) of Prop o s ition 4.1. W e w ill a ssume familiarity of the rea der with elementary submo dels of structures H ( θ ). In particular w e will make use of facts such a s that co untable elemen ts of such mo dels are their subsets or that s uch mo dels contain ω . See [5] for more on this sub ject. W e consider the following forcing P whose conditions p are o f the form: p = ( a p , f p , A p ) where a) a p ∈ [ ω 2 ] <ω ; b) f p : [ a p ] 2 → [ ω 2 ] <ω ; c) A p ∈ [ F ] <ω ; d) f p ( α, β ) ⊆ T { X : X ∈ A p , α, β ∈ X } ∩ min { α, β } for any distinct α, β ∈ a p . The order is just the inv erse inclusion, i.e., p ≤ q if a nd o nly if a p ⊇ a q , f p ⊇ f q , A p ⊇ A q . F act 4.7. P is simply F -pr op er. Pr o of. Let θ = ω 3 and let M and p b e as in a) - d) of Definition 4.5. The existence of such an M fo llows from the statio narity of F . Let X 0 = M ∩ ω 2 . Let p 0 = ( a p , f p , A p ∪ { X 0 } ). Finally let q ≤ p 0 . The pro of consists of using Lemma 4.6 and finding the parameters π 1 , ..., π k ∈ M and a formula φ ( x 0 , x 1 , ..., x k ) which well reflects q in ( M , π 1 , ..., π k ). Define q | M = ( a q ∩ M , f q | M , A q ∩ M ). Int ro duce notation δ = M ∩ ω 1 = rank ( M ), where the sec o nd equa lity follows from 4) of P rop osition 4 .1. Note that A q ∩ M = A q | M = { X ∈ A q : X ⊂ X 0 } . This follows fro m 5 ) of Prop ositio n 4 .1. The fact that [ M ] <ω ⊆ M implies tha t a q | M , A q | M ∈ M . Also as d) of the definition of the forcing is satisfied for q a nd α, β ∈ a q , w e hav e that f q ( α, β ) ⊆ X 0 = M ∩ ω 2 for α, β ∈ a q ∩ X 0 . So , we may conclude that f q | M ∈ M , in o ther words we hav e q | M ∈ M ∩ P . It is c le a r that q | M ≤ p . By 6) of Pro po sition 4 .1 a nd the fac t that [ M ] <ω ⊆ M , in M ther e is a Z ∈ F such that S { X ∩ M : rank ( X ) < δ, X ∈ A q } ⊆ Z . Let φ ( x 0 , x 1 , x 2 , x 3 , x 4 ) be the fo rmula which says that x 0 is a condition of the partial or der x 4 which extends in x 4 the condition x 3 and such tha t the difference b etw een the fir st co ordina te of x 0 and x 2 is disjoint from x 1 . Claim. φ ( x 0 , x 1 , x 2 , x 3 , x 4 ) wel l- re fle cts q in ( M , Z , a q | M , q | M , P ) . Pr o of of the Claim. It is c lear that φ ( q , Z , a q | M , q | M , P ) holds in H ( ω 3 ). Now let s ∈ M b e a condition satisfying φ ( s, Z, a q | M , q | M , P ) i.e., s extends in P the condition q | M and a s \ a q | M is disjoint fr om Z . Define the common extension r o f q and s as follows: a r = a s ∪ a q , f r = f s ∪ f q ∪ h , A r = A s ∪ A q , where h ( { α, β } ) = ∅ for { α, β } ∈ [ a s ∪ a q ] 2 − ([ a s ] 2 ∪ [ a q ] 2 ). Suc h an f r is a function on [ a r ] 2 since q | M ≥ q , s . Clear ly all clauses of the definition of the forcing P but d) a re trivially THIN-VER Y T ALL COMP A CT SCA TTERED SP A CES WHICH ARE HS 15 satisfied by r . So let us prov e d). Let α, β ∈ a r and X ∈ A r , w e will consider a few cases. Case 1. α, β ∈ a s , X ∈ A s It is trivial b ecaus e s ∈ P . Case 2. α, β ∈ a q , X ∈ A q It is trivial b ecaus e q ∈ P . Case 3. α, β ∈ a s , X ∈ A q . Since φ ( s, Z, a q | M , q | M , P ) holds in M w e have that either rank ( X ) ≥ δ = rank ( M ∩ ω 2 ) = r ank ( X 0 ) in which case d) is sa tisfied b ecause f r ( { α, β } ) = f s ( { α, β } ) ⊆ X 0 ∩ min { α, β } ⊆ X ∩ min { α, β } by d) for s and 3) of Pr op osition 4.1 or rank ( X ) < δ and then by the definition of φ and Z we g e t that α, β ∈ a s ∩ a q , so we are again in Case 2. Case 4. α, β ∈ a q , X ∈ A s . This means that α, β ∈ M , b ecause s ∈ M , i.e., α, β ∈ a s ∩ a q so we are a gain in Case 1. Case 5. α ∈ a s \ a q and β ∈ a q \ a s . Then h ( { α, β } ) = ∅ . The pro o f of the claim completes the pro of o f F act 4.7.  Definition 4.8. F o r p ∈ P , call the set a p ∪ f [[ a p ] 2 ] ∪ S A p the supp ort of p and denote it by supp ( p ). Definition 4. 9 . W e s ay that tw o conditions p, q of P are isomor phic (via π : supp ( p ) → supp ( q )) if π : supp ( p ) → supp ( q ) is an or der preser ving bijection constant on supp ( p ) ∩ sup p ( q ) and i) π [ a p ] = a q ; ii) { π [ X ] : X ∈ A p } = A q ; iii) f q ( { π ( α ) , π ( β ) } ) = π [ f p ( { α, β } )] for all α, β ∈ a p . Lemma 4. 10. Supp ose p, q ∈ P ar e isomorphic via π : supp ( p ) → supp ( q ) . Then they ar e c omp atible. Pr o of. Define the commo n e xtension r of p and q as follows: a r = a p ∪ a q , f r = f p ∪ f q ∪ h , A r = A p ∪ A q , where h ( { α, β } ) = ∅ fo r { α, β } ∈ [ a p ∪ a q ] 2 − ([ a p ] 2 ∪ [ a q ] 2 ). The only non-automatic co ndition of the definition of P which needs to b e chec ked is d). Case 1. α, β ∈ a r . If X ∈ A r , we ar e trivially done. If X ∈ A q and α, β ∈ X , then α, β ∈ supp ( p ) ∩ supp ( q ), hence α, β ∈ a p ∩ a q and hence a gain use d) for q . Case 2. α, β ∈ a q . Similar to the prev io us case. Case 3. α ∈ a r \ a q , β ∈ a q \ a r . In this case h is empty .  16 CHRISTINA BRECH AND P IOTR KOSZMIDER F act 4.1 1. Assuming CH t he for cing P is ω 2 -c.c. Thus by F a ct 4.7, L emma 4.6 and F act 4.3, P pr eserve s c ar dinals. Pr o of. By the previous lemma the pro of is a standard application of the ∆-system lemma to the seq uence of supp or ts { su pp ( p ξ ) : ξ < ω 2 } o f so me conditio ns p ξ ∈ P under our ca rdinal arithmetic a ssumption.  Theorem 4.12. In V P ther e is a function f : [ ω 2 ] 2 → [ ω 2 ] <ω with the str ong pr op erty ∆ . Pr o of. Clearly , we claim that f = S { f p : p ∈ G } defines s uch a function, wher e G is a P - generic ov er V . Let ˙ f be a name for it. Fix a set A = { ˙ a α : α < ω 1 } of P -na mes for elements o f an uncountable ∆-system of n -tuples ˙ a = { ˙ a i : i < n } of elements of ω 2 for which there are bijections e as in Definition 3.1 (any uncountable ∆- s ystem has a n uncountable s uch a subsys tem). Fix a c o ndition p ∈ P . T ake a mo del M ≺ H ( ω 3 ) such that M ∩ ω 2 = X 0 ∈ F and p ∈ P ∩ M ; F ∈ M and { ˙ a α : α < ω 1 } ∈ M . W e will show that there are α 1 < α 2 < ω 1 and r ≤ p such that r for ces 1), 2), 3) of Definition 3.1 for ˙ a α 1 and ˙ a α 2 . First take a condition p 0 ≤ p as in F act 4.7, i.e., a p 0 = a p , f p 0 = f p , A p 0 = A p ∪ X 0 . T ak e q ≤ p 0 and α 1 ∈ ω 1 such that there is b suc h that b \ M 6 = ∅ , q  ˙ a α 1 = ˇ b and b ⊆ a q . This can be done as { ˙ a α : α < ω 1 } is a seq ue nce of names for an uncountable ∆-system of se ts and | M | = ω . Pro ceed as in the pr o of of F act 4.7, i.e., c ho ose Z and φ as in F a ct 4 .7. So, we have φ ( q , Z , a q | M , q | M , P ) in H ( ω 3 ) and so by the elementarit y o f M , we can find a n s and α 2 such that φ ( s, Z , a q | M , q | M , P ) holds in M and moreover there is a such that a \ ( b ∩ M ) ∈ [ M \ Z ] <ω and suc h that s  ˙ a α 2 = ˇ a and a ⊆ a s . Now we will o btain another amalgamation r o f s and q which will force 1 ), 2) and 3 ) of Definition 3.1. Let a r = a s ∪ a q , f r = f s ∪ f q ∪ h . F or ξ ∈ a s \ a q and η ∈ a q \ a s : ∗∗ ) h ( { ξ , η } ) = [ A ∪ B ∪ C ] ∩ D where A = a ∩ min { ξ , η } B = [ { f s ( { τ , ξ } ) : τ ∈ a ∩ b, τ < η } C = [ { f q ( { τ , η } ) : τ ∈ a ∩ b, τ < ξ } D = min { ξ , η } ∩ \ { X ∈ A q : ξ , η ∈ X , r ank ( X ) ≥ δ } First let us chec k that r is a common extensio n of q and s . The pro o f also follows the cases as in the Claim in the pro o f o f F act 4.7. All ar e chec ked in the same manner except for Case 5 wher e one may assume that X ∈ A q as β 6∈ M . This time the inclusion in the set D guar antees that d) holds in Case 5 . Now we will chec k 1), 2) and 3) o f Definition 3 .1 for a, b a s ab ove and f r . This will b e eno ugh since r  ˙ a α 1 = ˇ b, ˙ a α 2 = ˇ a and r  f r ⊆ ˙ f . Supp ose ξ ∈ a \ b and η ∈ b \ a . By the form o f the definition o f f r ( { ξ , η } ) = h ( { ξ , η } ) it will b e eno ugh to prov e that the sets A , B a nd C are actually included in min { ξ , η } ∩ X for any X ∈ A q such that ran k ( X ) ≥ δ and ξ , η ∈ X . So, let X ∈ A q be any such element that ran k ( X ) ≥ δ and ξ , η ∈ X . THIN-VER Y T ALL COMP A CT SCA TTERED SP A CES WHICH ARE HS 17 F or 1) of Definition 3 .1, note that since X 0 = M ∩ ω 2 and r ank ( X 0 ) = δ we have that M ∩ min { ξ , η } is included in X by 3) of P rop osition 4 .1. Hence, as a ⊆ M , w e hav e a ∩ min { ξ , η } ⊆ min { ξ , η } ∩ X , that is we obtain 1). T o get 2) of Definition 3.1 ass ume that τ ∈ a ∩ b a nd τ < ξ , hence min { τ , η } ≤ min { ξ , η } . Note ag ain, b y 3) of Pr o p o sition 4 .1, that M ∩ ξ ⊆ X ∩ ξ which implies in this ca s e that τ ∈ X . Hence, since τ , η ∈ a q , by d) o f the definition of the for cing, we have that f q ( { τ , η } ) ⊆ X ∩ min { τ , η } ⊆ X ∩ min { ξ , η } , so we o btain 2). T o g et 3) of Definition 3.1 as sume that τ ∈ a ∩ b and τ < η , hence min { τ , ξ } ≤ min { ξ , η } . W e have that τ , ξ ∈ M ∩ ξ , a nd ag ain M ∩ ξ ⊆ X . Hence, since ξ , τ ∈ a s , by d) of the definition of the forcing, and the fact that s ∈ M , w e hav e that f s ( { τ , ξ } ) ∩ min { τ , ξ } ⊆ X ∩ min { τ , ξ } ⊆ X ∩ min { ξ , η } , so we obtain 3).  Remark 4.13. F o r any k ≤ ω and any un c ountable ∆ - s ystem one c an have k sets satisfying 1), 2) and 3) of Defin ition 3.1. This fol lows fr om t he Dushnik-Mil ler the or em se e [9] The or em 9.7. 4.3. CH and the strong prop e rty ∆ . In this section we prove that CH implies that there is no function f s uch as in the previous section, even if we allow f to take countable sets as v a lues. This a lso proves that the strong pro pe r ty ∆ ca nnot be obtained as in Baumgartner and Shelah [2], that is , by a forcing which pr eserves CH. Prop ositio n 4.14. ( CH) Ther e is no f : [ ω 2 ] 2 → [ ω 2 ] ω such that for every ∆ - system A of finite subsets of ω 2 of c ar dinality ℵ 1 , ther e exist distinct a, b ∈ A such that ∗ ) ∀ ξ ∈ a \ b ∀ η ∈ b \ a a ∩ ξ ∩ η ⊆ f ( { ξ , η } ) . Pr o of. Suppose that f : [ ω 2 ] 2 → [ ω 2 ] ω . F or an A ⊆ ω 2 and ξ ∈ ω 2 \ A define f A,ξ : A → [ A ] ω , f A,ξ ( η ) = f ( { η , ξ } ) ∩ A ∀ η ∈ A. Let M ≺ H ( ω 3 ) be closed under its countable subsets (here we use CH) | M | = ω 1 , ω 1 ⊆ M ; ω 1 , ω 2 , f ∈ M and such that sup( M ∩ ω 2 ) = γ has an uncountable cofinality . By r ecursion construct a sequence ( α ξ , β ξ ) ξ<ω 1 which satisfies: 1) ω 1 < α ξ , β ξ ∈ M ∩ γ ; 2) α ξ < β ξ < α η for all ξ < η ; 3) f A η ,β η = f A η ,γ where A η = { α ξ , β ξ : ξ < η } ; 4) α η 6∈ f ( { β η , γ } ). T o justify that this construc tio n ca n b e c arried out assume tha t we hav e A η satisfying 1)-4) and let us show how to obtain α η , β η . As A η ⊆ M and M is closed under its co un table sets we hav e A η ∈ M . Also f A η ,γ ∈ M as M is clos ed under countable se ts. Hence, b y the elementarit y there is β η ∈ M \ sup( A η ) s uch that f A η ,β η = f A η ,γ and cf ( β η ) = ω 1 . Now f ( { β η , γ } ) ∩ M is in M again, so using the fact that cf ( β η ) = ω 1 we c an find α η ∈ M satisfying A η < α η < β η and α η 6∈ f ( { β η , γ } ) whic h completes the co nstruction. Now define γ η < ω 1 such that for ξ < η < ω 1 we ha ve γ ξ < γ η 6∈ [ { f ( { β ξ , β η } ) : ξ < η } . 18 CHRISTINA BRECH AND P IOTR KOSZMIDER Finally define A = {{ γ ξ , α ξ , β ξ } : ξ < ω 1 } . Supp ose ξ < η . Note that, a s by 4), α ξ 6∈ f ( { β ξ , γ } ) and by 3), f ( { β ξ , γ } ) = f ( { β ξ , β η } ), we hav e α ξ ∈ ( β ξ ∩ β η ) \ f ( { β ξ , β η } ) . 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IMECC-UNICAMP, Caix a Post al 6065, 13083-97 0, Ca mpinas, SP, Brazil E-mail addr ess : christina .brech@gmail .com Instytut Matema tyki; Politechnika L ´ odzka, ul. W ´ olcza ´ nska 215; 90-924 L ´ od ´ z, Po land E-mail addr ess : pkoszmide r.politechni ka@gmail.co m