Generalized Luzin sets

GENERA LI ZED LUZIN SETS R OBER T RA LOWSKI AND SZYMON ˙ ZEBERSKI Abstract. In this pap er we inv astigate the notio n of generalized ( I , J ) - Luzin set. This no tion genera lize the sta ndard notion of Luzin set and Sier pi´ nski set. W e find set theoretical condi- tions which imply the existence of generalized ( I , J ) - Luzin set. W e show ho w to construct large family of pair wise non-e quiv alent ( I , J ) - Luzin sets. W e find a class of f orcings whic h pr eserves the pr ope rt y of being ( I , J ) - Luzin s et. 1. Not a t ion and Terminology W e will use standar d set-theoretic notation follo wing [8]. In partic- ular for an y set X and an y cardinal κ , [ X ] <κ denotes the set of all subsets of X with size less than κ. Similarly , [ X ] κ denotes the family of subse ts of X of size κ. By P ( X ) w e denote the p o w er set of X . If A ⊆ X × Y t hen for x ∈ X and y ∈ Y we put A x = { y ∈ Y : ( x, y ) ∈ A } , A y = { x ∈ X : ( x, y ) ∈ A } . By A △ B w e denote the symmetric difference of sets A and B , i.e. A △ B = ( A \ B ) ∪ ( B \ A ) . In this pap er X denotes uncountable P olish space . By Op en( X ) w e denote the top ology o f X . By Borel( X ) w e denote the σ - field of all Borel sets. Let us recall that each Borel set can b e co ded by a function from ω ω . Precise definition of suc h co ding can b e found in [7]. If x ∈ ω ω is a Borel co de then by # x w e denote the Borel set co de d by x. I , J a re σ - ideals on X , i.e. I , J ⊆ P ( X ) are closed under coun table unions and subsets. Additionally w e assume that [ X ] ω ⊆ I , J . Moreov er I , J hav e Borel base i.e eac h set from the ideal can b e cov ered b y a Borel set from the ideal. Standard ex amples o f suc h ideals are the ideal L of Leb esgue measure zero sets and the ideal K of meager sets of P olish space. 1991 Mathematics Subje ct Classific ation. P rimary 0 3E20, 0 3E35; Secondary 03E17 , 0 3E15, 03E 55. Key wor ds and phr ases. Luzin set, Sie rpi´ nski set, definable forcing, mea surable function, mea ger set, null set. 1 2 ROBER T RA LOWSKI AND SZYMON ˙ ZEBERSKI Definition 1.1. L et M ⊆ N b e standar d tr ansitive mo dels of ZF. Co ding Bor el sets fr o m the id e al I is absolute iff ( ∀ x ∈ M ∩ ω ω )( M | = # x ∈ I ← → N | = # x ∈ I ) . W e sa y that I satisfies κ chain condition ( κ -c.c.) if eve ry family A of Borel subsets of X satisfying the following conditions: (1) ( ∀ A ∈ A )( A / ∈ I ) (2) ( ∀ A, B ∈ A )( A 6 = B → A ∩ B ∈ I ) has size smaller than κ. If I is ω 1 -c.c. then w e say that I is c.c.c. Let us recall that a function f : X → X is I -measurable if the preimage o f ev ery op en subset o f X is I -measurable i.e b elongs to the σ -field generated b y Borel sets and the ideal I . In other words f is I -measurable iff ( ∀ U ∈ Op en( X ))( ∃ B ∈ Borel( X ))( ∃ I ∈ I )( f − 1 [ U ] = B △ I ) . Let us recall the follow ing cardinal co efficien ts: Definition 1.2 (Cardinal co efficien ts) . non ( I ) = min {| A | : A ⊆ X ∧ A / ∈ I } add ( I ) = min {| A | : A ⊆ I ∧ S A / ∈ I } cov ( I ) = min {| A | : A ⊆ I ∧ S A = X } cov h ( I ) = min {| A | : A ⊆ I ∧ ( ∃ B ∈ Borel( X ) \ I )( B ⊆ S A ) } cof ( I ) = min {| A | : A ⊆ I ∧ A is a b ase o f I } wher e A is a b ase of I iff A ⊆ I ∧ ( ∀ I ∈ I )( ∃ A ∈ A )( I ⊆ A ) . Let us remark that a b o ve co efficien ts can b e defined for larger class of families (no t only ideals). Definition 1.3. We say that L ⊆ X is a ( I , J ) - Luzin set if • L / ∈ I , • ( ∀ B ∈ I )( B ∩ L ∈ J ) . Assume that κ is a c ar dinal n umb er. We say that L ⊆ X is a ( κ, I , J ) - Luzin set iff L is a ( I , J ) - Luzin set and | L | = κ . The ab ov e definition generalizes the standard notion o f Luzin and Sierpi ´ nski sets. Namely , L is Luzin set iff L is g eneralized ( L , [ R ] ≤ ω ) - Luzin set and S is Sierpi ´ nski set iff S is generalized ( K , [ R ] ≤ ω ) - Luzin set. The ab ov e notion generalizes also notio ns from [2]. Definition 1.4. We say that ide als I and J ar e ortho gonal if ∃ A ∈ P ( X ) A ∈ I ∧ A c ∈ J . In such c ase we write I ⊥ J . Definition 1.5. L et F ⊆ X X b e a family of functions. We say that A, B ⊆ X ar e e quival e nt with r esp e ct to F if ( ∃ f ∈ F ) ( B = f [ A ] ∨ A = f [ B ]) GENERALIZED LUZIN SETS 3 Definition 1.6. We say that A, B ⊆ X ar e Bor el e quivalent if A, B ar e e quivalent with r es p e ct to the famil y of al l Bor el functions. Definition 1.7. We say that I has F ubin i pr op erty iff for every Bor el set A ⊆ X × X { x ∈ X : A x / ∈ I } ∈ I = ⇒ { y ∈ X : A y / ∈ I } ∈ I Natural examples of ideals fulfilling F ubini prop erty are the ideal of n ull sets L (b y F ubini theorem) and the ideal of meager sets K (by Kuratow ski-Ulam theorem). By definition w e can o btain the follow ing pr o p erties: F act 1.1. Assume that I ⊥ J . (1) Th e r e exist a ( I , J ) - Luzin set. (2) I f L is a ( I , J ) - Luzin set then L is not ( J , I ) - Luzin set. Pr o of. (P art 1) By the definition of I ⊥ J w e can find tw o sets I ∈ I and J ∈ J suc h that I ∪ J = X . W e will sho w that J is ( I , J ) - Luzin set. J is not in I . Let us fix an y set A ∈ I . W e ha v e tha t A ∩ J ⊆ J ∈ J . (P art 2) By the definition of I ⊥ J w e can find t w o sets I ∈ I and J ∈ J such that I ∪ J = X . Assume tha t L is ( I , J ) - Luzin set and ( J , I ) - Luzin set. W e hav e that L ∩ J ⊆ J ∈ J and L ∩ I ⊂ I ∈ I By the prop erty of b eing ( J , I ) - Luzin set L ∩ J ∈ I . So L = ( L ∩ J ) ∪ ( L ∩ I ) ∈ I . what is a con tradiction with b eing ( I , J ) - Luzin set.  W e will tr y to find a wide class o f forcings whic h preserv es the prop- ert y of b ein g ( I , J ) - Luzin set. W e are mainly in terested in so called definable forcings (see [1 1]). Let us recall that P is definable forcing if P is of the f orm Bor el( X ) \ I , where X and I hav e absolute definition for standard tra nsitive mo dels of ZF of the same high t. 2. Existe nce of Luzin sets Let us start with a theorem whic h under suitable assumptions guar - an tees existence of uncoun tably many pairwise differen t ( I , J ) - Luzin sets. Theorem 2.1. Assume that κ = cov ( I ) = cof ( I ) ≤ non ( J ) . L et F b e a family of functions fr om X to X . Assume that | F | ≤ κ. Then we c an find a se quenc e ( L α ) α<κ such that (1) L α is ( κ, I , J ) - Luzin set, (2) for α 6 = β , L α is not e quivalent to L β with r e s p e ct to the fam ily F . 4 ROBER T RA LOWSKI AND SZYMON ˙ ZEBERSKI Pr o of. Let us enume rate the family F : F = { f α : α < κ } . No w, let us enume rate Borel base of ideal I : B I = { B α : α < κ } . No w without loss o f generalit y w e can assume that ( ∀ f ∈ F )( ∀ λ < κ )( κ ≤ | f [( [ ξ <λ B ξ ) c ] | ) Indeed, since cov ( I ) = κ a set ( S ξ <λ B ξ ) c is not in the ideal I . If the function f do es no t hav e the a b o v e prop erty and L is a ( I , J ) - Luzin set then f [ L ] = f [ L ∩ [ ξ <λ B ξ ] ∪ f [ L ∩ ( [ ξ <λ B ξ ) c ] and b oth sets has cardina lity less than κ. So f [ L ] is not ( I , J )- Luzin set. By induction w e will construct the family { x η α,ζ : η , ζ , α < κ } and { d η α,ζ : η , ζ , α < κ } suc h that d η α,ζ = f ζ ( x η α,ζ ) and for any differen t η , η ′ < κ { x η α,ζ : ζ , α < κ } ∩ { d η ′ α,ζ : ζ , α < κ } = ∅ and x η α,ζ ∈ X \ { d η ξ ,ζ : η , ξ , ζ < α } ∪ { x η ξ ,ζ : η , ξ , ζ < α } ∪ [ ξ <α B ξ ! for ev ery η , ζ < α. Assume that w e ar e in α -th step of construction. Fix η , ζ < α . It means that we ha v e constructed the followin g set O l d = { x λ β ,ξ , d λ β ,ξ : β , ξ , λ < α } ∪ { x λ α,ξ , d λ α,ξ : λ < η ∨ ( λ = η ∧ ξ < ζ ) } . Since | f ζ [( S ξ <α B ξ ) c ] | ≥ κ and | O l d | < κ we get that | f ζ [( [ ξ <α B ξ ∪ O ld ) c ] | ≥ κ. That’s wh y w e can find d η α,ζ ∈ f ζ [( [ ξ <α B ξ ∪ O l d ) c ] \ O l d. Let x η α,ζ b e such that d η α,ζ = f ζ ( x η α,ζ ). In this wa y we can finish the α -th step of construction. No w, let us define L α = { x α ξ ,ζ : ξ , ζ < κ } . Let us c hec k that L α is ( I , J ) - Luzin set. Indeed, if A ∈ I then there exists β < κ s.t. A ⊂ B β . Then w e hav e A ∩ L α ⊂ B β ∩ L α = B β ∩ { x α ξ ,ζ : ξ , ζ < β } ⊆ { x α ξ ,ζ : ξ , ζ < β } ∈ J GENERALIZED LUZIN SETS 5 b ecause | { x α ξ ,ζ : ξ , ζ < β }| ≤ | β | < κ ≤ non ( J ) . What is more, for ev ery function f = f α ∈ F a nd ev ery β 6 = γ w e ha v e that κ ≤ | f [ L γ ] \ L β | b ecause { d γ ξ ,α : α < ξ < κ } ⊆ f [ L γ ] \ L β . So L β 6 = f [ L γ ] .  In fact w e hav ed pro v ed a little stronger result. Remark 2.1. Assume that κ = cov ( I ) = cof ( I ) ≤ non ( J ) . L et F b e a family of functions fr om X to X . Assume that | F | ≤ κ. Then we c an find a se quenc e ( L α ) α<κ such that (1) L α is ( κ, I , J ) - Luzin set, (2) for α 6 = β and f ∈ F we have that κ ≤ | f [ L α ] △ L β | . Let us notice that f o r ev ery ideal I w e ha v e the inequalit y co v ( I ) ≤ cof ( I ) . This giv es the following corollary . Corollary 2.1. I f 2 ω = cov ( I ) = non ( J ) then ther e exists c on tin- uum many differ ent ( I , J ) - Luzin sets whic h ar en ’t Bor el e quivalent. In p articular, if CH hol d s then ther e ex i s ts c ontinuum man y differ ent ( ω 1 , I , J ) - Luzin sets which ar en ’t Bo r el e quivalent. W e can extend ab ov e corollary to a wilder class of f unctions - namely , I - measurable functions. Corollary 2.2. I f 2 ω = cov ( I ) = non ( J ) then ther e exists c on tin- uum man y differ ent ( I , J ) - Luzin sets which ar en ’t e quivalent with r esp e ct to al l I -me asur a ble functions. In p articular, if CH hol d s then ther e ex i s ts c ontinuum man y differ ent ( ω 1 , I , J ) - Luzin sets w hich a r en ’t e q uiva lent with r esp e ct to al l I - me asur able f unc tion s . Pr o of. First, let us notice that if a function f is I -measurable then there exists a set I ∈ I ∩ Bo r el ( X ) suc h that f ↾ ( X \ I ) is Borel. Indeed, it is enough to consider a countable base { U n } n ∈ ω of top ology of X . Then f − 1 [ U n ] = B n △ I n , where B n is Borel and I n is from the ideal I . Now, put I = S n ∈ ω I n . So w e can consider a family of partial Borel functions whic h domain is Borel set with complemen t in the ideal I . This family is naturally of size contin uum. So w e can use Corollary 2.1 and Remark 2.1 to finish the pro of.  No w, let us concentrate on ideal of null and meager sets. Corollary 2.3. (1) Assume that cov ( L ) = 2 ω . Ther e exists c ontin- uum many differ ent (2 ω , L , K ) - Luzin sets which ar en ’t e q uiv- alent w ith r esp e ct to the family of L eb esg ue - me asur a ble func- tions. 6 ROBER T RA LOWSKI AND SZYMON ˙ ZEBERSKI (2) Assume that cov ( K ) = 2 ω . Ther e exists c ontinuum many differ- ent (2 ω , K , L ) - Luzin sets which ar en ’t e quivalen t with r esp e ct to the family of Bair e - me asur able functions. Pr o of. Let us notice that the eq ualit y cov ( L ) = 2 ω implies that 2 ω = cov ( L ) = cof ( L ) = non ( K ) . Similarly , the equality cov ( K ) = 2 ω implies that 2 ω = cov ( K ) = cof ( K ) = non ( L ) (see [1]). Corollary 2.2 finishes the pro of.  3. Luzin se ts and for cing No w, let us fo cus on the class of forcings whic h preserv es b eing ( I , J )- L uzin set. Let us start with a t echnic al observ ation. Lemma 3.1. Assume that I has F ubini p r op erty. Supp ose that P I = Borel( X ) \ I is a pr op er defin able for cing. L et B ∈ I b e a set in V P I [ G ] . T h en B ∩ X V ∈ I . Pr o of. Let ˙ B – name for B , ˙ r – canonical name for generic real, C ⊆ X × X - Borel set from the ideal I . C is coded in g round mo del V and B = C ˙ r G . No w b y F ubini prop erty: { x : C x / ∈ I } ∈ I . Let x ∈ B ∩ X V then V [ G ] | = x ∈ B 0 < [ | x ∈ ˙ B | ] = [ | x ∈ C ˙ r | ] = [ | ( ˙ r , x ) ∈ C | ] = [ | ˙ r ∈ C x | ] = [ C x ] I Then we ha v e: B ∩ X V ⊆ { x : C x / ∈ I } ∈ I . But the last set is co ded in ground mo del b ecause the set C w as co ded in V .  Theorem 3.1. Assume that ω < κ and I , J a r e c.c.c. an d have F u- bini pr op e rty. S upp ose that P I = Borel( X ) \ I and P J = Borel( X ) \ J a r e definable for cings. Then P J pr eserves ( κ, I , J ) - Luzin set pr op erty . Pr o of. Let L b e a ( κ, I , J ) - Luzin set in V . In V [ G ] tak e an y B ∈ I then L ∩ B ∩ V = L ∩ B but L ∩ B ∈ I in V so L ∩ B ∈ J in V by definition of L. Finally , b y L emma 3.1 L ∩ B = L ∩ B ∩ V ∈ J in V [ G ] .  Theorem 3.2. L et ( P , ≤ ) b e a for cing notion such that { B : B ∈ I ∩ Borel( X ) , B is c o de d in V } is a b ase for I in V P [ G ] . Assume that Bor el c o des for sets fr om ide als I , J ar e absolute. Then ( P , ≤ ) pr eserve b eing ( I , J ) - Luzin sets. GENERALIZED LUZIN SETS 7 Pr o of. Let L b e a ( I , J ) - Luzin set in ground mo del V . W e will show that V P [ G ] | = L is ( I , J ) - Luzin set . Let us w ork in V P [ G ] . Fix I ∈ I . I has Borel base consis ting of sets co de d in V . So, there exists b ∈ ω ω ∩ V suc h that I ⊆ # b ∈ I . By absoluteness of Bo rel co des fro m I w e hav e tha t V | = # b ∈ I . L is a ( I , J ) - Luzin set in the mo del V . So, there is c ∈ ω ω ∩ V whic h co des Borel set from the ideal J suc h that V | = L ∩ # b ⊆ # c. By absoluteness of Borel co des from J w e get that V P [ G ] | = L ∩ B ⊆ L ∩ # b ⊆ # c ∈ J , what prov es that L is a ( I , J ) - Luzin set in generic extension.  The ab ov e theorem give s us a series of corollaries. Corollary 3.1. L et ( P , ≤ ) b e any for cing notion which do es not change the r e als i. e. ( ω ω ) V = ( ω ω ) V P [ G ] . Assume that Bor el c o de s for sets fr om ide als I , J ar e a b s olute. Then ( P , ≤ ) pr eserve b ein g ( I , J ) - Luzin sets. Corollary 3.2. Assume that ( P , ≤ ) is a σ -close d for cing and Bo r el c o d es for sets fr om ide als I , J ar e absolute. Then ( P , ≤ ) pr eserve ( I , J ) - Luzin sets. Corollary 3.3. L et λ ∈ O n b e an or dinal numb e r. L et P λ = h ( P α , ˙ Q α ) : α < λ i b e iter ate d for cing with c ountable s upp ort. Sp ouse that (1) for any α < λ P α  ˙ Q α − σ close d , (2) B o r el c o des for sets fr om id e als I , J ar e absolute, then P λ pr eserve ( I , J ) - Luzin sets. Pr o of. Our forcing P λ is σ - closed b ecause it is coun table supp ort itera- tion of σ -closed forcings. So, w e can apply Coro lla ry 3.2 to finish t he pro of.  No w, let us consider some prop erties of countable supp ort iteration connected with preserv ation o f some relation. W e will follow notation giv en by Goldstern (see [4 ]). First, let us consider measure case. Let Ω is a family o f clop en sets of Cantor space 2 ω and C r andom = { f ∈ Ω ω : ( ∀ n ∈ ω ) µ ( f ( n )) < 2 − n } with discrite top o logy . If f ∈ C r andom then let us define t he follow ing set A f = T n ∈ ω S k ≥ n f ( k ). No w, we are r eady to define the following relation ⊑ = S n ∈ ω ⊑ n where ( ∀ f ∈ C r andom )( ∀ g ∈ 2 ω )( f ⊑ n g ← → ( ∀ k ≥ n ) g / ∈ f ( k )) . Definition of the notion of preserv ation of relation ⊑ r andom b y forcing notion ( P , ≤ ) can b e found in pap er [4]. Let us fo cus on the f ollo wing consequenc e of that definition. 8 ROBER T RA LOWSKI AND SZYMON ˙ ZEBERSKI F act 3.1 (Goldstern) . If ( P , ≤ ) pr e s e rves ⊑ r andom then P  µ ∗ (2 ω ∩ V ) = 1 . No w, w e sa y that forcing notion P pres erv es o uter measure iff P preserv es ⊑ r andom . It is w ell kno wn that Lav er forcing preserv es some stronger prop ert y than ⊑ r andom (see [5]). So, L a v er f orcing preserv es outer measure. In [4] w e can find the f ollo wing theorem: Theorem 3.3 (Goldstern) . L et P λ = (( P α , Q α ) : α < γ ) b e any c ountable supp ort iter ation such that ( ∀ α < γ ) P α  Q α pr eserves ⊑ r andom then P γ pr eserves the r elation ⊑ r andom . Theorem 3.4. Assume that P is a for cing notion which pr eserv e s ⊑ r andom . Th e n P pr eserves b eing ( L , K ) -Luzin set. Pr o of. Assume tha t V | = L is ( L , K )-Luzin set. L et us w ork in V P [ G ] . T ak e an y n ull set A ∈ L . Then there is a null set B in ground mo del suc h that A ∩ V ⊆ B . Indeed, let us a ssume that there is no such B ∈ V . Then without loss of generality (2 ω \ A ) ∩ V ∈ L . But A ∈ L then we ha v e that 2 ω ∩ V ⊂ A ∪ ((2 ω \ A ) ∩ V ) which is a null set. But by F a ct 3.1 µ ∗ (2 ω ∩ V ) = 1 . So w e ha v e a contradiction. Then inte rsection A ∩ L ⊆ B ∩ L ∈ K is a meager set in ground mo del. Then by absolutnes of b orel co des of meager sets the set A ∩ L is a meager set what finishes the pro of .  Remark 3.1. In c onstructible univers e L le t us c onsider the c ountable for cing iter ation P ω 2 = (( P α , Q α ) : α < ω 2 ) o f the length ω 2 as fol lo w s, for a ny α < ω 2 • if α is even then P α  ′′ Q α is r an dom for cing ′′ , • in o dd c ase P α  “ Q α is L av er for cing ′′ . Pr eviously we notic e d that b oth r andom and L aver for cing, pr eserves ⊑ r andom and then by The or em 3.3 P ω 2 pr eserves r elation ⊑ r andom . B y The or em 3 . 4 the ( L , K ) -Luzin se ts ar e pr eserve d by our iter ation P ω 2 . Mor e ov e r, in generic extens i on we have cov ( L ) = ω 2 and 2 ω = ω 2 (for details se e [4] ). Asuume that in the gr ound m o del A is ( L , K ) -Luzin se t with outer me asur e e qual to one. Then in generic e x tension it has outer me asur e one and | A | = ω 1 . So, i t do es not c on tain any L eb esgue p ositive Bor el set. Thus A is c ompletely L -nonm e asur able set. The analogous mac hinery can b e used for ideal of meager sets K . Let us recall the necessary definitions (see [4]). Let C C ohen b e set of all functions from ω <ω in to itself. Then ⊑ C ohen = S n ∈ ω ⊑ C ohen n and for any n ∈ ω let ( ∀ f ∈ C C ohen )( ∀ g ∈ ω ω )( f ⊑ C ohen n g iff ( ∀ k < n )( g ↾ k ⌢ f ( g ↾ k ) ⊆ g )) . GENERALIZED LUZIN SETS 9 Then finally we hav e t he following theorem: Theorem 3.5. Assume that P is a for cing notion which pr eserv e s ⊑ C ohen . Then P pr eserves b eing ( K , L ) -Luzin set. The another preserv ation theorem whic h is due to Shelah (see [9] and also [10]) is as fo llows Theorem 3.6 (Shelah) . L et P λ = (( P α , ˙ Q α ) : α < λ ) b e any c ountable supp ort iter ation such that ( ∀ α < γ ) P α  Q α is p r op er and P α  Q α  every new op en dense set c ontains old o p en de n se set then P λ  every new op en dense c ontains old op en dense set. W e can easily deriv e Corollary 3.4. L et P λ = (( P α , ˙ Q α ) : α < λ ) b e any c ountable supp ort iter ation such that ( ∀ α < λ ) P α  Q α is p r op er and P α  Q α  every new op en dense set c ontains old o p en de n se set Then P λ pr eserves b eing ( K , L ) -Luzin s e t. Reference s [1] T. Ba rtoszynski, H. Judah, S. Shelah, The Cichon D iagr am, J. Symbo lic Logic v ol. 58 (2 ) (19 93), pp.401-4 23, [2] J. Cicho ´ n, On two-c ar dinal pr op erties of ide als , T r ans. Am. Math. So c. v ol 314 , no . 2 (1989), pp 693- 708, [3] J. Cicho´ n, J. Pa wlik o wski, On ide als of subsets of the plane and on Cohen r e als , J. Symbolic Logic vol. 5 1 , no. 1 (19 86), pp 560-56 9 [4] M. Goldstern, T o ols for your for cing c onst ru ction, Isra el Mathematical Con- ference P ro ceedings, vol. 06 (1992), pp.307 -362, [5] H. Judah, S. Shelah, The Kunnen Mil ler chart, Jo urnal of Sy m bolic Lo gic, (1990). [6] A. Kanamori, The Higher Infinite: L ar ge Car dinals in S et The ory fr om Their Be ginnings Springer- V erlag (20 09), [7] K ec hris, Classic al Descriptive Set The ory (Gradua te T exts in Mathematics) (v. 1 56) Springer (1995), [8] T. Jech, Set The ory millenium edition, (2003), [9] S. Shelah, Pr op er and impr op er for cing , (199 8). [10] Ch. Schlindw ein, Understanding pr eservation the or ems, II , to app ear in Math. Log. Q uart. arXiv:1001 .0922 v1 [11] J. Zapletal, F or cing Ide al ize d (Cambridge T racts in Mathematics) (2008). Institute of Ma thema tics and Computer Science, W r oc la w Un iver- sity of Technol ogy, Wybrze ˙ ze Wyspia ´ nskiego 27, 50-370 Wroc la w, Poland. E-mail addr ess , Rob ert Ra lowski: robert .ralow ski@pwr.wroc.pl E-mail addr ess , Szymon ˙ Zeb erski: s zymon.z ebersk i@pwr.wroc.pl