Point-cofinite covers in the Laver model

POINT-COFINITE CO VERS IN THE LA VER MODEL ARNOLD W. MILLER AND BOAZ TSABAN Abstract. Let S 1 (Γ , Γ) b e the s tatemen t: F or each sequence of po in t-coﬁnite op en cov ers, o ne can pic k one elemen t from eac h cov er and o btain a p oint-coﬁnite cov er. b is the minimal cardinality of a set of reals not satisfying S 1 (Γ , Γ). W e prove the following assertions : (1) If there is an un bounded tower, then there are sets of reals of cardinality b , sa tisfying S 1 (Γ , Γ). (2) It is cons is ten t that all sets of rea ls satisfying S 1 (Γ , Γ) hav e cardinality smaller than b . These r esults can a lso be formulated as dealing with Arhangel’ski ˘ ı’s prop erty α 2 for spaces of contin uous real-v alued functions. The main technical res ult is that in Laver’s mo del, each s et of reals of cardinality b has an un bo unded Bo rel image in the Ba ir e space ω ω . 1. Back gr ound Let P b e a non trivial prop ert y of sets of r eals. The critic al c ar dinality of P , denoted non( P ), is the minimal cardinality of a set of reals not satisfying P . A natural question is whether there is a set of r eals of cardinalit y at least non( P ), whic h satisﬁe s P , i.e., a nontrivial example. W e consider t he following prop erty . Let X b e a set of reals. U is a p oint-c oﬁnite co v er o f X if U is inﬁnite, and for eac h x ∈ X , { U ∈ U : x ∈ U } is a coﬁnite subset of U . 1 Ha ving X ﬁxed in the bac kground, let Γ b e the family of all p o in t-coﬁnite op en co v ers of X . The follo wing prop erties w ere in tro duced by Hurewicz [8], Tsaban [19], and Sc heep ers [15], resp ectiv ely . U ﬁn (Γ , Γ) : F or all U 0 , U 1 , · · · ∈ Γ, none containing a ﬁnite sub- co v er, there are ﬁnite F 0 ⊆ U 0 , F 1 ⊆ U 1 , . . . suc h that { S F n : n ∈ ω } ∈ Γ. U 2 (Γ , Γ) : F or a ll U 0 , U 1 , · · · ∈ Γ, there are F 0 ⊆ U 0 , F 1 ⊆ U 1 , . . . suc h t ha t |F n | = 2 fo r all n , a nd { S F n : n ∈ ω } ∈ Γ. 1 Historically , p oint-coﬁnite covers w ere named γ -c overs , since they are related to a prop erty num ber ed γ in a list fro m α to ǫ in the seminal pap er [7] o f Ger lits and Nagy . 1 2 ARNOLD MILLER A ND BO AZ TSABAN S 1 (Γ , Γ) : F or all U 0 , U 1 , · · · ∈ Γ, there are U 0 ∈ U 0 , U 1 ∈ U 1 , . . . suc h t ha t { U n : n ∈ ω } ∈ Γ. Clearly , S 1 (Γ , Γ) implies U 2 (Γ , Γ), whic h in turn implies U ﬁn (Γ , Γ). None of these implications is rev ersible in ZF C [19]. The critical car- dinalit y of all three prop erties is b [9]. 2 Bartoszy ´ nski and Shelah [1] pro v ed that there are, pro v ably in ZF C, totally imp erfect sets of reals of cardinalit y b satisfying the Hurewicz prop ert y U ﬁn (Γ , Γ). Tsaban prov ed the same assertion for U 2 (Γ , Γ) [19]. These sets satisfy U ﬁn (Γ , Γ) in a ll ﬁnite p ow ers [2]. W e show that in order to o bta in similar results for S 1 (Γ , Γ), hy p othe- ses b ey ond ZFC are necessary . 2. Constr uctions W e sho w that certain w eak (but not prov able in ZF C) hypotheses suﬃce to ha v e nontrivial S 1 (Γ , Γ) sets, eve n ones whic h p ossess this prop ert y in all ﬁnite p ow e rs. Deﬁnition 2.1. A tower of cardinality κ is a set T ⊆ [ ω ] ω whic h can b e en umerated bijectiv ely as { x α : α < κ } , suc h that for all α < β < κ , x β ⊆ ∗ x α . A set T ⊆ [ ω ] ω is unb ounde d if the set of its en umeration functions are unbounded, i.e., for an y g ∈ ω ω there is an x ∈ T such that for inﬁnitely ma ny n , g ( n ) is less tha n the n -th elemen t of x . Sc heepers [1 6] pro v ed that if t = b , then there is a set of reals of cardinalit y b , satisfying S 1 (Γ , Γ). If t = b , then there is an un b ounded to w er of cardinality b , but the latter assumption is w eak er. Lemma 2.2 (folklore) . If b < d , then ther e is an unb ounde d tower of c ar dinality b . Pr o of. Let B = { b α : α < b } ⊆ ω ω b e a b -scale, tha t is, eac h b α is increasing, b α ≤ ∗ b β for all α < β < b , and B is unbounded. As | B | < d , B is not do minating. Let g ∈ ω ω exemplify that. F or eac h α < b , let x α = { n : b α ( n ) ≤ g ( n ) } . Then T = { x α : α < b } is an unbounded to w er: Clearly , x β ⊆ ∗ x α for α < β . Assume that T is b ounded, and let f ∈ ω ω exemplify that. F or eac h α , writing x α ( n ) for the n -t h elemen t of x α : b α ( n ) ≤ b α ( x α ( n )) ≤ g ( x α ( n )) ≤ g ( f ( n )) for all but ﬁnitely many n . Th us, g ◦ f shows that B is b ounded. A con tradiction.  2 Blass’s survey [4] is a go o d reference for the deﬁnitions a nd details ab out the sp ecial cardinals men tioned in this pap er. POINT-COFINITE COVERS IN THE LA VER MODEL 3 Theorem 2.3. If ther e is an unb ounde d tower (of any c ar di n ality), then ther e is a set of r e als X of c ar dinality b , which satisﬁes S 1 (Γ , Γ) . Theorem 2 .3 follows from t he following tw o pro p ositions. Prop osition 2.4. If ther e is an unb ounde d tower, then ther e is one o f c ar dinality b . Pr o of. By Lemma 2.2, it remains to consider t he case b = d . Let T b e an unbounded to w er of cardinality κ . L et { f α : α < b } ⊆ ω ω b e dominating. F or eac h α < b , pick x α ∈ T whic h is not b ounded by f α . { x α : α < b } is un b ounded, b eing unbounded in a dominating family .  Deﬁne a to p ology on P ( ω ) b y identifying P ( ω ) with the Can tor space 2 ω , via characteristic functions. Sc heep ers’s men tioned pro of actually establishes the fo llo wing result, to which w e give an alt ernat iv e pro of. Prop osition 2.5 (essen tially , Sc heepers [1 6]) . F or e ach unb ounde d tower T of c ar dinality b , T ∪ [ ω ] <ω satisﬁes S 1 (Γ , Γ) . Pr o of. Let T = { x α : α < b } b e an unbounded tow e r of car dinality b . F or each α , let X α = { x β : β < α } ∪ [ ω ] <ω . Let U 0 , U 1 , . . . b e p oin t- coﬁnite op en co v ers of X b = T ∪ [ ω ] <ω . W e may assume that eac h U n is countable and that U i ∩ U j = ∅ whenev er i 6 = j . By the pro of of Lemma 1.2 of [6 ], for eac h k there are distinct U k 0 , U k 1 , · · · ∈ U k , and an increasing sequence m k 0 < m k 1 < . . . , suc h that for eac h n a nd k , { x ⊆ ω : x ∩ ( m k n , m k n +1 ) = ∅} ⊆ U k n . As T is unbounded, there is α < b suc h that for eac h k , I k = { n : x α ∩ ( m k n , m k n +1 ) = ∅} is inﬁnite. F or eac h k , { U k n : n ∈ ω } is an inﬁnite subset of U k , and thus a p oin t-coﬁnite co v er of X α . As | X α | < b , there is f ∈ ω ω suc h t ha t ∀ x ∈ X α ∃ k 0 ∀ k ≥ k 0 ∀ n > f ( k ) x ∈ U k n . F or eac h k , pic k n k ∈ I k suc h t ha t n k > f ( k ), W e claim that { U k n k : k ∈ ω } is a p oint-coﬁnite co v er of X b : If x ∈ X α , then x ∈ U k n k for all but ﬁnitely man y k , b ecause n k > f ( k ) for all k . If x = x β , β ≥ α , then x ⊆ ∗ x α . F or eac h large enough k , m k n k is large enough, so that x ∩ ( m k n k , m k n k +1 ) ⊆ x α ∩ ( m k n k , m k n k +1 ) = ∅ , and thus x ∈ U k n k .  R emark 2.6 . Zdomskyy p oin ts out that for the pro of to go through, it suﬃces that { x α : α < b } is such tha t there is an un b ounded { y α : α < b } ⊆ [ ω ] ω suc h that f o r each α , x α is a pseudoin tersection of 4 ARNOLD MILLER A ND BO AZ TSABAN { y β : β < α } . W e do not know whether t he assertion men tioned here is we ak er than the existence of a n un b ounded tow er. W e no w turn to non trivial examples o f sets satisfying S 1 (Γ , Γ) in all ﬁnite p ow ers. In general, S 1 (Γ , Γ) is not preserv ed by taking ﬁnite p ow- ers [9], and w e use a slightly stronger h yp othesis in our construction. Deﬁnition 2.7. Let b 0 b e the additivit y n um b er o f S 1 (Γ , Γ), t ha t is, the minimum cardinality of a family F of sets of reals, eac h satisfying S 1 (Γ , Γ), such that the union o f all mem b ers of F do es not satisfy S 1 (Γ , Γ). t ≤ h , and Sc heep ers pro v ed that h ≤ b 0 ≤ b [17]. It follows from Theorem 3 .6 that consisten tly , h < b 0 = b . It is op en whether b 0 = b is pro v able. If t = b or h = b < d , then there is an un b ounded tow er of cardinality b 0 . Theorem 2.8. F or e ach unb ounde d tower T of c ar dinality b 0 , al l ﬁnite p owers of T ∪ [ ω ] <ω satisfy S 1 (Γ , Γ) . Pr o of. W e say that U is an ω -c over of X if no mem ber of U con tains X as a subset, but eac h ﬁnite subset of X is con tained in some mem b er of U . W e need a mu ltidimensional v ersion of Lemma 1.2 of [6 ]. Lemma 2.9. Assume that [ ω ] <ω ⊆ X ⊆ P ( ω ) , and let e ∈ ω . F or e ach op en ω -c over U of X e , ther e a r e m 0 < m 1 < . . . and U 0 , U 1 , · · · ∈ U , s uch that for al l x 0 , . . . , x e − 1 ⊆ ω , ( x 0 , . . . , x e − 1 ) ∈ U n whenever x i ∩ ( m n , m n +1 ) = ∅ for al l i < e . Pr o of. As U is an op en ω - cov er of X e , there is an o p en ω -co v er V of X suc h t ha t { V e : V ∈ V } reﬁnes U [9]. Let m 0 = 0. F or each n ≥ 0: Assume that V 0 , . . . , V n − 1 ∈ V are giv en, and U 0 , . . . , U n − 1 ∈ U are s uc h t ha t V e i ⊆ U i for all i < n . Fix a ﬁnite F ⊆ X suc h that F e is not con tained in any of the sets U 0 , . . . , U n − 1 . As V is an ω -co v er of X , there is V n ∈ V suc h that F ∪ P ( { 0 , . . . , m n } ) ⊆ V n . T ak e U n ∈ U suc h that V e n ⊆ U n . Then U n / ∈ { U 0 , . . . , U n − 1 } . As V n is op en, for each s ⊆ { 0 , . . . , m n } there is k s suc h that for each x ∈ P ( ω ) with x ∩ { 0 , . . . , k s − 1 } = s , x ∈ V n . Let m n +1 = max { k s : s ⊆ { 0 , . . . , m n }} . If x i ∩ ( m n , m n +1 ) = ∅ for all i < e , then ( x 0 , . . . , x e − 1 ) ∈ V e n ⊆ U n .  The assumption in the theorem that there is an unbounded tow er of cardinalit y b 0 implies that b 0 = b . The pro o f is by induction on the p o w er e of T ∪ [ ω ] <ω . The case e = 1 follow s from Theorem 2.5. POINT-COFINITE COVERS IN THE LA VER MODEL 5 Let U 0 , U 1 , · · · ∈ Γ( ( T ∪ [ ω ] <ω ) e ). W e ma y assume that these cov ers are countable. As in the pro of of Theorem 2 .5 (this time using Lemma 2.9), there are for eac h k m k 0 < m k 1 < . . . and U k 0 , U k 1 , · · · ∈ U k (so that { U k n : n ∈ ω } ∈ Γ( ( T ∪ [ ω ] <ω ) e )), suc h that for all y 0 , . . . , y e − 1 ⊆ ω , ( y 0 , . . . , y e − 1 ) ∈ U k n whenev er y i ∩ ( m k n , m k n +1 ) = ∅ for all i < e . Let α 0 b e suc h that X e α 0 is not contained in a ny mem b er of S n U n . As T is unbounded, there is α suc h that α 0 ≤ α < b , and for eac h k , I k = { n : x α ∩ ( m k n , m k n +1 ) = ∅} is inﬁnite. Let Y = { x β : β ≥ α } . ( T ∪ [ ω ] <ω ) e \ Y e is a union of few er t ha n b 0 homeomorphic copies of ( T ∪ [ ω ] <ω ) e − 1 . By the induction h ypothesis, ( T ∪ [ ω ] <ω ) e − 1 satisﬁes S 1 (Γ , Γ), and therefore so do es ( T ∪ [ ω ] <ω ) e \ Y e . F or eac h k , { U k n : n ∈ I k } is a p oint-coﬁnite co v er of ( T ∪ [ ω ] <ω ) e \ Y e , and thu s there are inﬁnite J 0 ⊆ I 0 , J 1 ⊆ I 1 , . . . , suc h that { T n ∈ J k U k n : k ∈ ω } is a p oint-coﬁnite cov er o f ( T ∪ [ ω ] <ω ) e \ Y e . 3 F or each k , pic k n k ∈ J k suc h tha t : m k n k > m k − 1 n k − 1 +1 , x α ∩ ( m k n k , m k n k +1 ) = ∅ , and U k n k / ∈ { U 0 n 0 , . . . , U k − 1 n k − 1 } . { U k n k : k ∈ ω } ∈ Γ( T ∪ [ ω ] <ω ): If x ∈ ( T ∪ [ ω ] <ω ) e \ Y e , then x ∈ U k n k for all but ﬁnitely man y k . If x = ( x β 0 , . . . , x β e − 1 ) ∈ Y , then β 0 , . . . , β e − 1 ≥ α , and thus x β 0 , . . . , x β e − 1 ⊆ ∗ x α . F or eac h large enough k , m k n k is larg e enough, so that x β i ∩ ( m k n k , m k n k +1 ) ⊆ x α ∩ ( m k n k , m k n k +1 ) = ∅ for all i < e , and t hus x ∈ U k n k .  There is an additional w ay to obta in non trivial S 1 (Γ , Γ) sets: The h yp othesis b = cov( N ) = cof ( N ) prov ides b -Sierpi ´ nski sets, a nd b - Sierpi ´ nski sets satisfy S 1 (Γ , Γ), ev en for Bor el p o int-coﬁnite co v ers . Details are av ailable in [1 8]. W e record the f ollo wing consequence o f Theorem 2.3 f or later use. Corollary 2.10. F or e ach unb ounde d tower T of c ar dinality b , T ∪ [ ω ] <ω satisﬁes S 1 (Γ , Γ) f o r op en c overs, but not for Bor el c overs. Pr o of. The latter prop erty is hereditary for subs ets [18]. By a theorem of Hurewicz, a set of reals satisﬁes U ﬁn (Γ , Γ) if, and o nly if, eac h con tin- uous image o f X in ω ω is b ounded. It follows that the set T ⊆ T ∪ [ ω ] <ω do es not ev en satisfy U ﬁn (Γ , Γ).  3. A consiste nc y resul t By the results of the previous section, we hav e the following. Lemma 3.1. Assume that every set of r e als with pr op erty S 1 (Γ , Γ) has c ar dinality < b , and c = ℵ 2 . T hen ℵ 1 = t = co v( N ) < b = ℵ 2 . 3 Cho osing inﬁnitely man y elemen ts from each c o ver, instea d of one, can be done by adding to the given sequence of c overs all coﬁnite subsets of the given cov ers. 6 ARNOLD MILLER A ND BO AZ TSABAN Pr o of. As there is no un b ounded to w er, we hav e that t < b = d . As c = ℵ 2 , ℵ 1 = t < b = ℵ 2 . Since there are no b -Sirepi ´ nski sets and b = cof ( N ) = c , cov( N ) < b .  In L a v er’s mo del [11], ℵ 1 = t = cov( N ) < b = ℵ 2 . W e will show that indeed, S 1 (Γ , Γ) is trivial there. Lav er’s mo del w as constructed to realize Bo rel’s Conjecture, asserting that “strong measure zero” is trivial. In some sense, S 1 (Γ , Γ) is a dual of strong measure zero. F or example, the canonical examples of S 1 (Γ , Γ) sets are Sierpi ´ nski sets, a measure theoretic ob ject, whereas the canonical examples of strong measure zero se ts are Luzin s ets, a Baire category theoretic ob ject. More ab out that can b e seen in [1 8 ]. The main technic al r esult of this pap er is the following. Theorem 3.2. In the L aver mo del, if X ⊆ 2 ω has c ar dinality b , then ther e is a Bor el map f : 2 ω → ω ω such that f [ X ] is unb ounde d. Pr o of. The notation in this pro of is as in Lav er [1 1]. W e will use the follo wing slightly simpliﬁed ve rsion of Lemma 14 o f [1 1]. Lemma 3.3 (La v er) . L et P ω 2 b e the c ountable supp ort iter ation of L a ver for cing, p ∈ P ω 2 , a n d ˚ a b e a P ω 2 -name such that p  ˚ a ∈ 2 ω . Then ther e ar e a c ondition q str onger than p , a n d ﬁnite U s ⊆ 2 ω for e ach s ∈ q (0 ) extendi n g the r o ot of q (0) , such that for al l such s a n d al l n : q (0) t ˆ q ↾ [1 , ω 2 )  “ ∃ u ∈ ˇ U s u ↾ n = ˚ a ↾ n ” for al l but ﬁnitely many imme diate suc c esso rs t o f s in q (0) . Assume that X ⊆ 2 ω has no un b ounded Borel image in M [ G ω 2 ], La v er’s mo del. F o r ev ery co de u ∈ 2 ω for a Borel function f : 2 ω → ω ω there exists g ∈ ω ω suc h t ha t for ev ery x ∈ X we ha v e that f ( x ) ≤ ∗ g . By a standard L¨ ow enheim-Sk olem argument, see The orem 4.5 on page 2 81 of [3], or section 4 on page 580 of [12], w e ma y ﬁnd α < ω 2 suc h that for eve ry co de u ∈ M [ G α ] there is an upp er b ound g ∈ M [ G α ]. By the argumen ts emplo y ed b y Lav er [11, Lemmata 10 and 11], w e ma y assume that M [ G α ] is the g round mo del M . Since the con tin uum h ypothesis holds in M and | X | = b = ℵ 2 , there are p ∈ G ω 2 and ˚ a suc h that p  ˚ a ∈ ˚ X and ˚ a / ∈ M . W ork in the gro und mo del M . Let q ≤ p b e a s in Lemma 3.3. D eﬁne Q = { s ∈ q (0) : ro ot( q (0)) ⊆ s } POINT-COFINITE COVERS IN THE LA VER MODEL 7 and let U s , s ∈ Q , b e the ﬁnite sets from the Lemma. Let U = S s ∈ Q U s . Deﬁne a Borel map f : 2 ω → ω Q so that for ev ery x ∈ 2 ω \ U and for eac h s ∈ Q : If f ( x )( s ) = n , then x ↾ n 6 = u ↾ n for eac h u ∈ U s . F or x ∈ U , f ( x ) ma y b e arbitrary . There m ust b e a g ∈ ω Q ∩ M and r ≤ q suc h t ha t r  f ( ˚ a ) ≤ ∗ ˇ g . Since p forced that a is not in the ground mo del, it cannot b e tha t a is in U . W e ma y extend r (0) if necessary so that if s = ro ot( r (0)), then r  f ( ˚ a )( s ) ≤ ˇ g ( s ) . But this is a contradiction to Lemma 3.3, since for a ll but ﬁnitely man y t ∈ r (0) whic h a re immediate extensions of s : r (0) t ˆ q ↾ [1 , ω 2 )  f ( ˚ a )( s ) > ˇ g ( s ) .  In [20], Tsaban and Z do mskyy pro v e that S 1 (Γ , Γ) for Borel co v ers is equiv alent to the Koˇ cinac pr o p ert y S cof (Γ , Γ) [10], asserting that for all U 0 , U 1 , · · · ∈ Γ , there are coﬁnite subsets V 0 ⊆ U 0 , V 1 ⊆ U 1 , . . . suc h that S n V n ∈ Γ. The main result of [5 ] can b e reform ulated as follow s. Theorem 3.4 (Dow [5]) . In L a ver’s mo del, S 1 (Γ , Γ) implies S cof (Γ , Γ) . F or the reader’s conv e nience, we giv e D o w’s pro of, adapted to the presen t no t a tion. Pr o of. A family H ⊆ [ ω ] ω is ω -splitting if for eac h coun table A ⊆ [ ω ] ω , there is H ∈ H whic h splits eac h elemen t of A , i.e., | A ∩ H | = | A \ H | = ω for all A ∈ A . The main tec hnical result in [5] is the following. Lemma 3.5 (Dow) . I n L aver’s mo del, e ach ω -splitting fam i l y c o n tains an ω -splitting family of c ar dinality < b . Assume that X satisﬁes S 1 (Γ , Γ). Let U 0 , U 1 , . . . b e op en p oint-coﬁnite coun table cov ers of X . W e may a ssume 4 that U i ∩ U j = ∅ whenev er i 6 = j . Put U = S n<ω U n . W e iden tify U with ω , its cardinality . Deﬁne H ⊆ [ U ] ω as follows. F or H ∈ [ U ] ω , put H ∈ H if and only if there exists V ∈ [ U ] ω , a p oin t-coﬁnite co v er of X , suc h that H ∩ U n ⊆ ∗ V f or all n . W e claim that H is an ω -splitting family . As H is closed under taking inﬁnite subsets, it suﬃces to show that it is ω -hitting , i.e., for an y coun table A ⊆ [ U ] ω there exists H ∈ H whic h 4 T o see why , repla ce ea c h U n by U n \ S i

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