Are scales Fréchet?

ARE SCALES FR ´ ECHET? R. FIGUER O A-SIERRA, O. GUZM ´ AN, M. HR U ˇ S ´ AK, AND A. KWELA Abstract. W e con tin ue the study of Do w spaces of a b -scale, originally in tro duced by Dow in [14]. W e prov e that it is consistent that all such spaces are F r´ ec het, but it is also consistent that none of them is. W e use these spaces to exhibit (consisten tly) a △ 1 2 ideal that does not satisfy the Category Dic hotom y . Finally , w e pro v e that the Category Dic hotom y holds for all co-analytic ideals. 1. Introduction During the 2012 thematic program on F orcing and its Applications at the Fields Institute, Juh´ asz ask ed the follo wing question: Problem 1 (Juh´ asz) . Is ther e a c ountable, F r ´ echet 1 sp ac e with unc ountable π -weight? Although the consistency of existence of suc h spaces is easy to establish, no ZFC result was kno wn. The problem w as motiv ated b y the following result of Hru ˇ s´ ak and Ramos: Theorem 2 (H., Ramos [33]) . It is c onsistent that every c ountable, F r ´ echet top olo gic al gr oup is se c ond c ountable. Since for top ological groups the notions of weigh t and π -w eigh t coincide, Juh´ asz’s question asks to what extent the algebraic structure is needed for the consisten t result ab o v e. The question is particularly in teresting, as man y of the results in [33] do not use the algebraic structure at all. The problem remained open until Do w pro vided a solution: Theorem 3 (Do w [14]) . Ther e exists a c ountable, F r ´ echet, zer o dimensional sp ac e with π -weight at le ast b . Not only is this result impressive, but its proof is highly ingenious and illuminating. Giv en a b -scale B , Dow deﬁned a top ological space D ( B ) = ( ω <ω , τ B ) (whic h w e call the Dow sp ac e of B ) that is zero dimensional and Keywor ds: F r´ echet spaces, scales, Ideals on countable sets, Dow spaces, Kat ˇ etov order, cardinal in v arian ts. AMS Classiﬁc ation: 54A20, 54A35, 03E05 ,03E17 The second author w as supp orted b y the P APIIT grant IA 104124 and the CONAHCYT gran t CBF2023-2024-903. The third author was supported by a P API IT grant IN101323 and a SECIHTI gran t CBF-2025-I-898. 1 Undeﬁned concepts will be reviewed in the follo wing sections. 1 2 R. FIGUERO A-SIERRA, O. GUZM ´ AN, M. HR U ˇ S ´ AK, AND A. KWELA whose π -weigh t is exactly the b ounding num b er b . Do w then pro ceeds to tak e a sequential modiﬁcation to obtain a top ology σ B extending τ B suc h that D 1 ( B ) = ( ω <ω , σ B ) is F r´ ec het, zero dimensional and its π -w eight is at least b (the exact π -weigh t is currently unkno wn). Evidently , the adv antage of D 1 ( B ) o ver D ( B ) is that it is F r´ ec het. Nevertheless, the original space D ( B ) has certain adv antages ov er D 1 ( B ) . Mainly , the op en sets of D ( B ) admit a v ery nice com binatorial description and are intuitiv e to work with, but the same is no longer true for D 1 ( B ) . It is then natural to ask the follo wing question: Can the Dow space of a scale b e a F r´ echet space? In other words, w e wan ted to know if taking the sequential mo diﬁcation is really needed. W e will say that a b -scale is F r´ echet if its Do w space is F r´ echet. In this article, we will pro v e the follo wing result: Theorem 4. Both of the fol lowing statements ar e c onsistent (but not at the same time) with ZFC: (1) Every b -sc ale is F r ´ echet. (2) No b -sc ale is F r ´ echet. W e ﬁnish the paper with an application to the Kat ˇ eto v order on ideals and prov e that the Category Dic hotom y of [30] is true for all co-analytic ideals. The structure of the paper is as follo ws. Section 2 cov ers the nec essary notation and deﬁnitions that will b e used throughout the pap er. The next ﬁv e sections pro vide the required background on cardinal inv arian ts of the con tinuum, ﬁlters and ideals, top ology , and forcing that will b e needed in the article. The reader may skip these preliminary sections and return to them as needed. The deﬁnition of the Dow space of a b -scale from [14] is review ed in Section 8. In Section 9 we prov e that p = b implies that all b -scales are F r´ ec het. In Section 10 we pro duce a mo del where there is a non-F r´ echet b -scale and in Section 11 w e build a mo del where no F r ´ ec het b -scales exist. In Section 12 we use results from the previous sections to ﬁnd a consistent example of a △ 1 2 ideal that do es not satisfy the Category Dic hotomy and prov e that this is the least p ossible complexit y . Section 13 con tains op en questions that we do not kno w ho w to answ er. It is w orth p ointing out that coun table, F r ´ ec het spaces with uncountable π -weigh t hav e received a lot of atten tion recently , as can be seen from the recen tly published pap ers [17] and [15]. 2. Not a tion Our notation is basically standard and follows [43] and [38]. Giv en s, t ∈ ω <ω b y s ⌢ t w e denote the c onc atenation of s and t. W e write s ⌢ n instead of s ⌢ ( n ) . If F ⊆ ω <ω , deﬁne s ⌢ F = { s ⌢ z | z ∈ F } . W e sa y that T ⊆ ω <ω is a tr e e if it is closed under taking initial segmen ts. F or s ∈ T , deﬁne ARE SCALES FR ´ ECHET? 3 suc T ( s ) = { n | s ⌢ n ∈ T } . W e say s is the stem of T (denoted by s = st ( T )) if every no de of T is comparable with s and s is maximal with this prop ert y . The domain of a function f is denoted b y dom ( f ) and its image b y im ( f ) . By f ; X − → Y we mean that f is a partial function from X to Y (i.e., dom ( f ) ⊆ X ). The expression “for almost all” means “for all except ﬁnitely man y”. If κ is a cardinal, deﬁne H ( κ ) as the set consisting of all sets whose tran- sitiv e closure has size less than κ. W e will w ork with elementary submo dels of these sets. F or an in tro duction to this very imp ortant technique, we refer the reader to the surv ey [13]. W e will need the concepts of F σ , G δ , Borel, analytic, co-analytic and pro jectiv e subsets of a Polish space, whic h can b e consulted in [40] or [56]. 3. Preliminaries on cardinal inv ariants of the continuum Car dinal invariants of the c ontinuum play an imp ortan t role in this ar- ticle. W e no w review some basic deﬁnitions that will b e needed. T o learn more, see [7] and [58]. F or f , g ∈ ω ω , deﬁne f ≤ ∗ g if f ( n ) ≤ g ( n ) holds for almost all n ∈ ω . W e say a family B ⊆ ω ω is unb ounde d if B is un b ounded with respect to ≤ ∗ . A family D ⊆ ω ω is a dominating family if for ev ery f ∈ ω ω , there is g ∈ D such that f ≤ ∗ g . The b ounding numb er b is the size of the smallest unbounded family and the dominating numb er d is the smallest size of a dominating family . W e say that B = { f α | α ∈ b } ⊆ ω ω is a b -sc ale if ev ery f α is an increasing function, B is unbounded and f α < ∗ f β whenev er α < β . A sc ale is a dominating b -scale. It is not hard to see that the existence of a scale is equiv alen t to b = d . The order ≤ ∗ can b e extended to partial functions. Given f , g ; ω − → ω deﬁne f ≤ ∗ g if f ( n ) ≤ g ( n ) holds for almost all n in their common domain. It is well-kno wn that b -scales are not only unbounded with resp ect to total functions, but also with resp ect to inﬁnite partial functions. See [58, F act 3.4] for the pro of of the following lemma: Lemma 5. L et B ⊆ ω ω b e a b -sc ale and g ; ω − → ω an inﬁnite p artial function. Ther e is f ∈ B such that f ≰ ∗ g . Let A and B b e t wo subsets of ω . Deﬁne A ⊆ ∗ B ( A is an almost subset of B ) if A \ B is ﬁnite. F or H ⊆ [ ω ] ω and A, B ⊆ ω , w e say that A is a pseudo-interse ction of H if it is almost contained in ev ery elemen t of H . A family P ⊆ [ ω ] ω is c enter e d if the in tersection of ﬁnitely man y of its elemen ts is inﬁnite. The pseudo-in tersection num b er p is the least size of a c en tered family without an inﬁnite pseudo-intersection. W e say T = { A α | α < κ } is a pr e-tower if it is ⊆ ∗ -decreasing and it is a tower it has no inﬁnite pseudo- in tersection. The tower numb er t is the least length of a to wer. By c we denote the size of the real num b ers. It is easy to see that p ≤ t ≤ b ≤ d ≤ c . Moreo ver, an impressive theorem of Malliaris and Shelah establishes that p = t (see [48]). The following is essen tially [4, Theorem 4.2]. 4 R. FIGUERO A-SIERRA, O. GUZM ´ AN, M. HR U ˇ S ´ AK, AND A. KWELA Prop osition 6 (Baumgartner, Dordal [4]) . If ther e ar e no towers of length b , then every b -sc ale is a sc ale (so b = d ). Pr o of. Assume that there is a b -scale B = { f α | α ∈ b } that is not a dom- inating family . Let g ∈ ω ω b e not dominated by any function in B . It follo ws that the set A α = { n | f α ( n ) < g ( n ) } is inﬁnite for every α < b and T = { A α | α ∈ b } is a pretow er. Since there are no to wers of length b , there is X ∈ [ ω ] ω whic h is a pseudoin tersection of T . W e get f α ↾ X ≤ ∗ g ↾ X for ev ery α < b , whic h con tradicts Lemma 5. □ W e will need the following result in Section 9. Lemma 7. L et B = { f α | α ∈ b } b e a b -sc ale and M an elementary submo del of H ( κ ) (for some lar ge enough r e gular c ar dinal κ ) such that B ∈ M , the size of M is less than b and δ = M ∩ b ∈ b . If g ; ω − → ω is an inﬁnite p artial function in M , then f δ ≰ ∗ g . Pr o of. By Lemma 5, there is α < b such that f α ≰ ∗ g . Moreov er, b y elemen- tarit y , w e ma y assume that α < δ. Since f α ≤ ∗ f δ , the result follo ws. □ Let M be a mo del of ZF C and f ∈ ω ω . W e say that f is dominating (unb ounde d) over M if for every g ∈ M ∩ ω ω , it is the case that g ≤ ∗ f ( f ≰ ∗ g ). 4. Preliminaries on fil ters and ideals W e denote the p o wer set of a set X b y P ( X ). I ⊆ P ( X ) is an ide al on X if ∅ ∈ I and X / ∈ I , for ev ery A, B ⊆ X , if A ∈ I and B ⊆ A then B ∈ I and if A, B ∈ I then A ∪ B ∈ I . A family F ⊆ ℘ ( X ) is a called a ﬁlter on X if X ∈ F and ∅ / ∈ F , for every A, B ⊆ X , if A ∈ F and A ⊆ B then B ∈ F and if A, B ∈ F then A ∩ B ∈ F . Given a family B of subsets of X , we deﬁne B ∗ = { X \ B | B ∈ B } . Note that if F is a ﬁlter, then F ∗ is an ideal (called the dual ide al of F ) and if I is an ideal, then I ∗ is a ﬁlter (called the dual ﬁlter of I ). Let I b e an ideal on X . The collection of I -p ositive sets is I + = ℘ ( X ) ∖ I . If F is a ﬁlter, w e deﬁne F + = ( F ∗ ) + . It is easy to see that F + is the family of all sets that intersects every mem b er of F . If A ∈ I + then the r estriction of I to A , deﬁned as I ↾ A = ℘ ( A ) ∩ I , is an ideal on A. By I ⊥ w e denote the set of all sets that ha ve ﬁnite intersection with ev ery mem b er of I . It is easy to see that I ⊥ is an ideal. W e review some properties of ideals that will b e needed in this article. Deﬁnition 8. L et I b e an ide al on ω (or any c ountable set). (1) I is tall if for every X ∈ [ ω ] ω ther e is Y ∈ I such that Y ∩ X is inﬁnite. (2) I is a F r´ ec het ideal if for every A ∈ I + , ther e is B ∈ [ A ] ω ∩ I ⊥ . (3) I is ω -hitting if for every { X n | n ∈ ω } ⊆ [ ω ] ω ther e is Y ∈ I such that Y ∩ X n is inﬁnite for every n ∈ ω . ARE SCALES FR ´ ECHET? 5 (4) I is weakly selectiv e if for every X ∈ I + and P a p artition of X either P ⊆ I or P has a (p artial) sele ctor in I + (that is, a set that interse cts e ach element of P in no mor e than one p oint). Let F b e a ﬁlter on a set W . The ﬁlter F <ω is the ﬁlter on [ W ] <ω \ {∅} generated by  [ A ] <ω \ {∅} | A ∈ F  . Note that a set X is in ( F <ω ) + if and only if every A ∈ F con tains an elemen t of X . Deﬁnition 9. L et F b e a ﬁlter. We say that F is a FUF ﬁlter if for every X ∈ ( F <ω ) + ther e is Y ∈ [ X ] ω such that every F ∈ F c ontains almost al l elements of Y . In this wa y , F is a FUF ﬁlter if and only if ( F <ω ) ∗ is a F r ´ echet ideal. It is not hard to see that ev ery coun tably generated ﬁlter is a FUF ﬁlter. Gruenhage and Szeptyc ki ask ed if there w as a ZF C example of a FUF ﬁlter on ω that is not coun tably generated. In [8] it was prov ed that it is consistent that every FUF ﬁlter generated b y less than c many elemen ts is coun tably generated and the main theorem of [33] implies that all FUF ﬁlters are coun tably generated. W e now review the Katˇ etov order, whic h will be needed in Section 12. Deﬁnition 10. L et X, Y b e two sets, I an ide al on X , J an ide al on Y and f : X − → Y . (1) f is a Kat ˇ etov function fr om I to J if for every A ⊆ Y , we have that if A ∈ J , then f − 1 ( A ) ∈ I . (2) J ≤ K I me ans that ther e is a Katˇ etov function fr om I to J . The eventual ly diﬀer ent ideal E D is the ideal on ω 2 generated b y the set of columns {{ n } × ω | n ∈ ω } and the graphs of functions from ω to ω . Fix X a top ological space and N ⊆ X . W e say that N is nowher e dense if for ev ery non-empty op en set U ⊆ X , there is another op en set ∅  = V ⊆ U such that V ∩ N = ∅ . By nwd ( X ) we denote the ideal of no where dense subsets of X. By nwd w e mean the ideal of no where dense subsets of the rational n umbers. Readers interested in learning more ab out the Katˇ eto v order are encour- aged to see [30], [31], [28], [26], [10], [2], [16], [49], [53], [24], [1], [41], [44] or [19] among others. 5. Preliminaries on Topology W e now review some basic top ological concepts that will b e used in the pap er. Let X b e a top ological space 2 and b ∈ X . Recall that ( X , τ ) is zer o- dimensional if it has a basis of clop en sets. F or A ⊆ X a coun table set, w e say that A c onver ges to b (denoted by A − → b ) if every op en subset of b almost contains A. X is a F r´ echet sp ac e if for every a ∈ X and Y ⊆ X suc h that a ∈ Y \ Y , there is A ∈ [ Y ] ω that con verges to a. Let B b e a 2 All spaces under discussion are Hausdorﬀ. 6 R. FIGUERO A-SIERRA, O. GUZM ´ AN, M. HR U ˇ S ´ AK, AND A. KWELA collection of non-empty op en subsets of X. W e sa y that B is a π -b ase if ev ery non-empt y op en subset of X con tains an elemen t of B . The π -weight of X is the smallest size of a π -base of X . Moreov er, B is a lo c al π -b ase at b if every neighborho o d of b contains an elemen t of B . The π -char acter of b is the smallest size of a lo c al π -b ase at b. W e say that X has unc ountable π - char acter everywher e if ev ery p oint of X has uncountable π -c haracter. Let φ b e a top ological property . By X | = φ w e mean that “ X has property φ ”. W e mainly use this notation when we are w orking with several top ological spaces with the same underlying set. Let X be a top ological space and a ∈ X . Denote by N X ( a ) the neighbor- ho o d ﬁlter of a and I X ( a ) its dual ideal. Note that I X ( a ) =  A ⊆ X | a / ∈ A  . If there is no risk of confusion, w e simply write N ( a ) and I ( a ) . Many top ological prop erties at the point a can b e expressed as prop erties of its neigh b orhoo d ﬁlter and its dual ideal, as shown in T able 1. T op ological Com binatorial prop ert y translation a ∈ B B ∈ N ( a ) + A − → a A is a pseudoin tersection of N ( a ) equiv alently , A ∈ I ( a ) ⊥ a is a F r ´ ec het p oint I ( a ) is a F r´ ec het ideal a ∈ Y but no sequence Y ∈ I ( a ) + and I ( a ) ↾ Y is from Y con v erges to a a tall ideal T able 1. T op ological prop erties and their translations W e will need the follo wing result in Section 11, whic h is Prop osition 44 of [16]. Prop osition 11. L et X b e a c ountable F r ´ echet sp ac e with no isolate d p oints. The ide al nwd ( X ) is we akly sele ctive. 6. Preliminaries on For cing W e review some preliminaries on forcing that will b e needed. W e assume the reader is already familiar with the metho d of forcing as presented in [42]. Let F be a ﬁlter on ω . The L aver for cing of F (denoted by L ( F )) consists of all trees p ⊆ ω <ω that hav e a stem and if t ∈ T extends the stem , then suc T ( s ) ∈ F . Giv en p, q ∈ L ( F ) , denote p ≤ q if p ⊆ q . It is easy to see that L ( F ) is a c.c.c. partial order. If p ∈ L ( F ) and s ∈ p, deﬁne p s = { t ∈ p | t ⊆ s ∨ s ⊆ t } . It is clear that p s ∈ L ( F ) , it extends p and in case st ( p ) ⊆ s, we hav e that st ( p s ) = s. The L aver generic r e al will b e denoted b y l g en ∈ ω ω and is the only element that is a branch of every tree in the generic ﬁlter. T o learn more ab out this forcings, see [32]. ARE SCALES FR ´ ECHET? 7 By D we denote He chler for cing , whic h consists of all pairs ( s, f ) where s ∈ ω <ω and f ∈ ω ω . Deﬁne ( s, f ) ≤ ( t, g ) if t ⊆ s, g ( n ) ≤ f ( n ) for ev ery n ≥ | t | and s ( i ) ≥ g ( i ) for every i ∈ dom ( s ) \ dom ( t ) . Hechler forcing is the standard c.c.c. forcing for adding a dominating real. T o learn ab out Hec hler forcing, see [3], [9] or [51]. A family W ⊆ [ ω ] ω is ω -hitting if for ev ery { X n | n ∈ ω } ⊆ [ ω ] ω , there is W ∈ W that in tersects every X n . W e say P pr eserves ω -hitting families if ev ery ω -hitting family remain ω -hitting after forcing with P . It is not hard to see that a forcing preserving ω -hitting families can not ﬁll to w ers. A more reﬁned v ersion of the previous notion is the following: Deﬁnition 12. L et P b e a p artial or der. We say that P strongly preserves ω -hitting families if for every P -name ˙ B for an inﬁnite subset of ω , ther e is { B n | n ∈ ω } ⊆ [ ω ] ω such that for every X ∈ [ ω ] ω , if | X ∩ B n | = ω for every n ∈ ω , then P for c es that X ∩ ˙ B is inﬁnite. In [8], the reader can ﬁnd a characterization of the ﬁlters whose Lav er forcing strongly preserv es ω -hitting families. Moreov er, it is also prov ed that L ( F ) preserv es ω -hitting families if and only if L ( F ) strongly preserv es ω -hitting families. It is also known that Hec hler forcing strongly preserves ω -hitting families. The follo wing iteration theorem can b e found in [8]. Theorem 13 (Brendle, H. [8]) . The ﬁnite supp ort iter ation of c.c.c. for c- ings that str ongly pr eserve ω -hitting families, str ongly pr eserves ω -hitting families. Let P b e a partial order. W e say that C ⊆ P is c enter e d if any ﬁnitely man y elements of C hav e a lo wer bound in P . Recall that P is σ -c enter e d if it is the union of coun tably many of its centered sets. On the other hand, we sa y that P is σ -ﬁlter e d if it is the union of coun tably man y ﬁlters. Eviden tly , every σ -ﬁltered forcing is σ -cen tered. Juh´ asz and Kunen pro v ed that the con verse is not true (see [39]). Nevertheless, we hav e the following: Prop osition 14. L et P b e a p artial or der and B its Bo ole an c ompletion. The fol lowing ar e e quivalent: (1) B is σ -c enter e d. (2) B is σ -ﬁlter e d. (3) P is σ -c enter e d. W e will need the following w ell-known preserv ation result (see [57] and [21]). Prop osition 15. L et γ < c + and ⟨ P α , ˙ Q α | α < γ ⟩ b e a ﬁnite supp ort iter ation of σ -c enter e d for cings. P γ is σ -c enter e d. A remark able theorem of Bell is the following: Theorem 16 (Bell [5]) . L et κ b e a c ar dinal. The fol lowing ar e e quivalent: (1) κ < p . 8 R. FIGUERO A-SIERRA, O. GUZM ´ AN, M. HR U ˇ S ´ AK, AND A. KWELA (2) F or every σ -c enter e d for cing P and { D α | α < κ } a c ol le ction of dense subsets of P , ther e is a ﬁlter G ⊆ P such that G ∩ D α  = ∅ for every α < κ. 7. Preliminaries on For cing and topology In this section w e review the metho d for destro ying the F r ´ echet prop ert y at a p oin t, as developed in [33] (which w as based on the techniques from [8]). Let X b e a top ological space and a ∈ X . According to T able 1, to destro y the F r´ ec het property at a by a forcing P , we must add a set ˙ A such that I ( a ) ↾ ˙ A is a tall ideal. But this is not enough, since the tallness of an ideal may not b e preserv ed under forcing iterations. The solution is to ensure that I ( a ) ↾ ˙ A is not only a tall ideal but an ω -hitting one, for which w e can pro ve preserv ation theorems under forcing. Deﬁnition 17. L et I b e an ide al on a c ountable set, P a for cing notion, and ˙ A a P -name. We say that P seals I with ˙ A if P for c es that ˙ A ∈ I + and I ↾ ˙ A is ω -hitting. Let X b e a coun table space. It is easy to see that L ( nwd( X ) ∗ ) forces ˙ A gen to be a dense subset of X . The following is Prop osition 5.2 of [33]: Prop osition 18 (H., Ramos [33]) . L et X b e a c ountable sp ac e with no isolate d p oints and a ∈ X : (1) If a has unc ountable π -weight, then L ( nwd( X ) ∗ ) se als I X ( a ) via ˙ A gen . (2) If X is F r ´ echet, then L ( nwd( X ) ∗ ) str ongly pr eserves ω -hitting fam- ilies. In this w ay , if X and a are as in the ab ov e prop osition, then L ( nwd( X ) ∗ ) forces ˙ A gen to b e a dense subset of X , yet it do es not contain sequences con verging to a. This prop ert y will b e preserved under any further forcing extension that preserves ω -hitting families. W e w ould lik e to p oin t out that the metho d of [33] has b een greatly reﬁned and expanded by Shibako v and the third author (see [35], [36] and [34]). 8. The Dow sp ace of a b -scale W e now review the construction of the Do w space from [14]. W e exp ect that the study of the top ological prop erties of Dow spaces will b e useful for inv estigating the combinatorial prop erties of b -scales, muc h as the study of the Mr´ owk a-Isb ell spaces resides in understanding the com binatorics of almost disjoin t families (see [58], [29] and [27]). Instead of w orking with b -scales on ω , we ﬁnd it more conv enient to work on b -scales consisting of functions from ω <ω to ω . T o this end, we ﬁrst adapt the relev ant deﬁnitions to our setting. F or conv enience, giv en m ∈ ω , we will denote △ m = m ≤ m . Let f , g : ω <ω − → ω . W e sa y that f is incr e asing if for every s, t ∈ ω <ω , if s is a ARE SCALES FR ´ ECHET? 9 prop er initial segmen t of t, then f ( s ) < f ( t ) and for ev ery n, m ∈ ω , if n < m, then f ( s ⌢ n ) < f ( s ⌢ m ) . As exp ected, deﬁne f < ∗ g if f ( s ) < g ( s ) holds for almost all s ∈ ω <ω . Moreo ver, deﬁne f < m g if f ( s ) < g ( s ) holds for all s / ∈ △ m . It follows that f < ∗ g if and only if there is m ∈ ω suc h that f < m g . Deﬁnition 19. L et B b e a family of functions fr om ω <ω to ω . (1) B is a weak b -scale if the fol lowing c onditions hold: (a) B c onsists of incr e asing functions. (b) B is unb ounde d (ther e is no function g such that f ≤ ∗ g for every f ∈ B ). (c) B is wel l-or der e d by ≤ ∗ . (d) F or every n ∈ ω , the set { f ∈ B | n < f ( ∅ ) } is c oﬁnal in B . (2) B is a b -scale if its or der typ e (with r esp e ct to ≤ ∗ ) is b . Although our main in terest is b -scales, it is conv enient to also consider w eak b -scales. W e will now deﬁne the sets that will b e part of a subbase in a Do w space. Deﬁnition 20. L et f : ω <ω − → ω . Deﬁne the tr e e U ( f ) ⊆ ω <ω r e cursively as fol lows: (1) ∅ ∈ U ( f ) . (2) If s ∈ U ( f ) , then suc U ( f ) ( s ) = ω \ { f ( s ) } . In this wa y , U ( f ) is a v ery wide tree, since ev ery no de branches in to all elemen ts of ω except one. The following is easy , but worth p ointing out. Lemma 21. L et f : ω <ω − → ω and s ∈ ω <ω . If s / ∈ U ( f ) , then ther e is i < | s | such that f ( s ↾ i ) = s ( i ) . W e now hav e the follo wing easy lemma: Lemma 22. L et B b e a we ak b -sc ale, s ∈ ω <ω and f ∈ B . (1) If s ∈ U ( f ) , then s ⌢ △ f ( s ) ⊆ U ( f ) . (2) The set { g ∈ B | s ∈ U ( g ) } is c oﬁnal in B . Pr o of. The ﬁrst point follo ws since f is an increasing function. F or the second p oint, choose n ∈ ω suc h that s ∈ △ n . Since { f ∈ B | n < f ( ∅ ) } is coﬁnal in B , we get the desired conclusion. □ Let s ∈ ω <ω . In this pap er, w e denote ⟨ s ⟩ = { t ∈ ω <ω | s ⊆ t } . This set should not be confused with { f ∈ ω ω | s ⊆ f } , which is also denoted b y ⟨ s ⟩ in the literature. Sets of the form ⟨ s ⟩ will b e often referred as c ones , while a c o c one is a set of the form ⟨ s ⟩ c = ω <ω \ ⟨ s ⟩ . A non-trivial c o c one is simply a non-empty co cone. Deﬁne 3 s ↑ = { t ∈ ω <ω | t ⊆ s } and for a set A ⊆ ω <ω , denote A ↑ = S  t ↑ | t ∈ A  . W e can now deﬁne the Do w space of a w eak b -scale. 3 In [14] our s ↑ is denoted by s ↓ . W e reverse this notation, as we picture our trees as gro wing do wn w ard. 10 R. FIGUERO A-SIERRA, O. GUZM ´ AN, M. HR U ˇ S ´ AK, AND A. KWELA Deﬁnition 23. L et B b e a we ak b -sc ale. The Do w space of B is the sp ac e D ( B ) = ( ω <ω , τ B ) wher e τ B is the top olo gy gener ate d by the subb ase c onsist- ing of the fol lowing sets: (1) ⟨ s ⟩ , ⟨ s ⟩ c for s ∈ ω <ω . (2) U ( f ) for f ∈ B . In [14] the top ology is not explicitly deﬁned from the b -scale (as in our presen tation), but it is instead constructed recursively . Nev ertheless, our presen tation falls under the scop e of [14] b ecause the family { U ( f ) | f ∈ B} satisﬁes Lemma 3.1 of that pap er. The follo wing notion was in tro duced in [14]: Deﬁnition 24. We say a top olo gic al sp ac e ( ω <ω , τ ) is ↑ -sequen tial if for every s ∈ ω <ω , the fol lowing c onditions hold: (1) ⟨ s ⟩ is an op en set . (2) ⟨ s ⌢ n ⟩ n ∈ ω c onver ges to s. (3) If A ⊆ ω <ω c onver ges to s, then A ↑ also c onver ges to s. Denote by A (1) the set of all con v ergence p oin ts of sequences contained in A. In [14] Do w pro ved the following: Theorem 25 (Do w) . L et B b e a b -sc ale. (1) D ( B ) is zer o dimensional. (2) D ( B ) is ↑ -se quential. (3) The π -char acter of every p oint in D ( B ) is b . (4) If A ⊆ ω <ω , then  A (1)  (1) = A (1) . (5) D ( B ) has no isolate d p oints. (6) Every ↑ -se quential top olo gy extending τ B has π -char acter at le ast b . Let U ⊆ ω <ω and n ∈ ω . Deﬁne U >n = { s ∈ U | s = ∅ ∨ s (0) > n } . W e ha ve the following: Lemma 26. L et B b e a we ak b -sc ale. The family:  ( U ( f 1 ) ∩ ... ∩ U ( f n )) >m | f 1 , ..., f n ∈ B ∧ m ∈ ω  is a lo c al b ase of ∅ . More constructions of countable F r´ echet spaces with uncoun table π -weigh t can be found in [17]. 9. All b -scales ma y be Fr ´ echet In this section we will pro ve that it is consistent that every b -scale is F r´ echet. In fact, we will prov e that the equalit y p = b implies that the Do w space of ev ery b -scale satisﬁes a strong form of the F r ´ ec het prop erty . W e recall the following deﬁnition: Deﬁnition 27. L et X b e a top olo gic al sp ac e. We say that X is F r´ ec het- Urysohn for ﬁnite sets if for every a ∈ X, its neighb orho o d ﬁlter N ( a ) is a FUF ﬁlter. ARE SCALES FR ´ ECHET? 11 This class of spaces has b een studied in [18], [52], [22], [23], [8] and [20] among many others. It is not hard to see that ev ery space that is F r ´ ec het- Urysohn for ﬁnite sets is also F r ´ ec het. Theorem 28 ( p = b ) . The Dow sp ac e of every b -sc ale is F r ´ echet-Urysohn for ﬁnite sets (and in p articular, it is a F r ´ echet sp ac e). Pr o of. Let B = { f α | α < b } b e a b -scale. F or simplicit y , w e will pro ve that the neighborho o d ﬁlter of ∅ is a FUF ﬁlter. The argument for an arbitrary s ∈ ω <ω is essentially the same, only requiring more notation. F or ease of notation, let F = N ( ∅ ) and U α = U ( f α ) for α < b . Let X ∈ ( F <ω ) + . W e ﬁrst ﬁnd M an elementary submo del of H ( κ ) (for some large enough regular cardinal κ ) suc h that X , B ∈ M , the size of M is less than b and δ = M ∩ b ∈ b . Denote by U the set of all T α ∈ F U α for F ∈ [ δ ] <ω . Since ev ery non-trivial co-cone is a neighborho o d of ∅ , if follo ws that for ev ery U ∈ N ( ∅ ) and n ∈ ω , there is a ∈ X suc h that a ⊆ U and n < s (0) for ev ery s ∈ a \ {∅} . In this w a y , for ev ery U ∈ U , w e can deﬁne the function g U ∈ ω ω suc h that for every n ∈ ω , it is the case that g U ( n ) is the least natural n umber for whic h there is a ∈ X with the following prop erties: (1) a ⊆ U. (2) n < s (0) for ev ery s ∈ a \ {∅} . (3) If s ∈ a, then s ∈ △ g U ( n ) . Note that g U ∈ M for every U ∈ U . Lemma 7 implies that for every V ∈ U , there are inﬁnitely many n ∈ ω suc h that g U ( n ) < f δ (( n )) (of course, f δ (( n )) is the result of applying f δ to ( n ) ∈ ω <ω ). W e now deﬁne P as the set of all p = ( F , h, H ) with the following prop erties: (1) F ∈ [ ω ] <ω , h : F − → X and H ∈ [ δ ] <ω . (2) F or ev ery n ∈ F and s ∈ h ( n ) \ {∅} , the follo wing conditions hold: (a) n < s (0) . (b) s ∈ △ f δ (( n )) . Let p, q ∈ P . Deﬁne p ≤ q in case the following conditions are met: (1) F q ⊆ F p , h q ⊆ h p and H q ⊆ H p . (2) F or ev ery n ∈ F p \ F q and α ∈ H q , w e ha ve that h p ( n ) ⊆ U α . It is easy to see that P is a σ - cen tered forcing (an y ﬁnite set of conditions that share the same ﬁrst tw o coordinates are compatible). The following claim is also not hard to prov e: (1) F or ev ery n ∈ ω , the set D n = { p ∈ P | F p ⊈ n } is dense. (2) F or ev ery α < δ, the set E α = { p ∈ P | α ∈ H p } is dense. Since δ < b = p , b y Theorem 16, we can ﬁnd a ﬁlter G ⊆ P that intersects all of the dense sets described ab o ve. Deﬁne F G = S p ∈ G F p , h G = S p ∈ G h p (note that h G : F G − → ω ) and Y ⊆ X the image of h G . W e claim that Y is as desired. W e need to pro ve that ev ery U ∈ N ( ∅ ) contains almost ev ery elemen t of Y . Note that we ma y assume that U is a sub-basic set. Moreo ver, 12 R. FIGUERO A-SIERRA, O. GUZM ´ AN, M. HR U ˇ S ´ AK, AND A. KWELA if U is a non-trivial co-cone, the conclusion is straigh tforward, since the ﬁrst co ordinate of every element of h G ( n ) \ {∅} is larger than n. It remains to pro ve that if α < b , then a ⊆ U α for almost all a ∈ Y . This is clearly the case if α < δ (since G ∩ E α  = ∅ and G is a ﬁlter), so we ma y assume that δ ≤ α. Let n ∈ F G suc h that f α ( ∅ ) < n and f δ (( m )) ≤ f α (( m )) for ev ery m ≥ n (almost every elemen t of F G satisﬁes this conditions). W e need to pro ve that h G ( n ) ⊆ U α . W rite h G ( n ) \ {∅} = { s 0 , ..., s l } . W e kno w that n < s 0 (0) , ..., s l (0) and s 0 , ..., s l ∈ △ f δ (( n )) . Moreo ver, ( s 0 (0)) , ..., ( s l (0)) ∈ U α , since these v alues are ab o v e f α ( ∅ ) . F or each i ≤ l , we hav e that s i ∈ △ f δ (( n )) ⊆ △ f α (( n )) . It follows by Lemma 22 that each s i is in U α . □ In [17] Dow and P ecoraro prov ed that there is a countable, zero dimen- sional space that is not H -separable and has π -weigh t b . In [50] Nyik os pro ved that p = b implies that there is an uncoun tably generated FUF ﬁlter. Our result pro vides another proof of this result. In response to Theorem 3, Mo ore p osed the follo wing problem: Problem 29 (Moore) . Is ther e a c ountable, F r´ echet, zer o dimensional sp ac e with π -weight exactly b ? The Do w space of a F r ´ echet b -scale provides suc h an example. How ever, while Do w spaces ha ve π -weigh t b , the π -weigh t of their sequen tial mo diﬁca- tion remains unknown. Of course, the ab o ve problem w ould hav e a p ositive solution if there existed a F r´ ec het b -scale, but w e will see in a later section that this is consisten tly false. Nevertheless, we ha ve the follo wing result, whic h was indep endently prov ed by Dow and P ecoraro in [17]. Prop osition 30. If c ≤ ω 2 , then ther e is a c ountable, F r´ echet, zer o-dimensional sp ac e of π -weight exactly b . Pr o of. If p = b , then every b -scale is F r´ ec het b y Theorem 28. Otherwise, we p < b = c , so there is such space b y Theorem 3. □ In [15] Dow inv estigated the p ossible π -w eights of coun table, regular, F r´ echet spaces. He prov ed that in the Miller mo del every suc h space has π -weigh t at most ω 1 . On the other hand, after adding κ many random reals, for any inﬁnite cardinal λ ≤ κ, there exists a countable, regular, F r ´ ec het space with π -w eight exactly λ. An op en question remains whether there are (consisten tly) uncountable cardinals λ < µ < κ suc h that b oth λ and κ are realized as the π -weigh t of a countable, regular, F r´ ec het space, while µ is not. 10. There ma y be a non-Fr ´ echet b -scale In this section, we prov e that it is consistent that there is a b -scale whose Do w space is not F r´ ec het. This result will b e strengthened in the next sec- tion, where w e sho w that it is consisten t that no b -scale is F r´ ec het. Although the pro of in this section motiv ates some of the ideas used later, the argumen t here is not a sp ecial case of the one in the next section. More imp ortan tly , ARE SCALES FR ´ ECHET? 13 w e conjecture that there will b e F r ´ ec het b -scales in the model constructed in this section, which would establish the consistency of the simultaneous existence of both a F r ´ echet and a non-F r ´ ec het b -scale, which is unknown at the momen t. W e recommend that the reader consult Section 7 as this section and the next one, use deﬁnitions and results from there. F or ease of writing, if D is a w eak b -scale, we will write L ( D ) instead of L ( nwd( D ( D )) ∗ ). By Theorems 25, 28 and Prop osition 18, we get the follo wing: Lemma 31. (1) L et D b e a we ak b -sc ale such that D ( D ) is F r ´ echet. L ( D ) str ongly pr eserves ω -hitting families and for c es that ˙ A gen is a dense set, yet it do es not c ontain se quenc es c onver gent to ∅ . (2) ( CH ) If B is a b -sc ale, then L ( B ) str ongly pr eserves ω -hitting families and for c es that ˙ A gen is a dense set, yet it do es not c ontain se quenc es c onver ging to ∅ . W e can no w prov e the main result of this section. Theorem 32. It is c onsistent that ther e is a b -sc ale whose Dow sp ac e is not F r´ echet. Pr o of. W e start with a mo del of CH. Let D = { f α | α ∈ ω 1 } be a b -scale. Deﬁne P = L ( D ) ∗ ˙ D ω 2 , where D ω 2 denotes the ﬁnite-supp ort iteration of Hec hler forcing of length ω 2 . Let G ⊆ P b e a generic ﬁlter. W e will pro v e that in V [ G ] there exists a non-F r´ ec het b -scale. W e go to V [ G ] . Let A = A gen \ {∅} , where A gen is the range of the generic real of L ( D ) . By Lemma 31 and the strong preserv ation of ω -hitting of Hechler forcing, w e know that in D ( D ) the set A is dense and do es not con tain conv erging sequences to ∅ . Of course, D is no longer a b -scale (it is b ounded). Let B = { f α | α ∈ ω 2 } b e a b -scale extending D such that f ω 1 is dominating o ver V [ A ] . W e will prov e that B is not F r´ ec het. Claim 33. (1) D ( B ) | = ∅ ∈ A. (2) D ( B ) | = A do es not c ontain a c onver gent se quenc e to ∅ . The second p oint is easy . Since A do es not contain a conv ergen t sequence to ∅ in D ( D ) , then it cannot contain one in D ( B ) (every op en set in D ( D ) is op en in D ( B )) . W e no w prov e the ﬁrst p oin t. The argument is similar to the proof of Theorem 28. F or con venience, denote U α = U ( f α ) for α < ω 2 . Let U be the set of all T α ∈ F U α for F ∈ [ ω 1 ] <ω . Eviden tly , every element of U is an op en set in D ( D ) . Sub claim. Let W ∈ U . The set ⟨ ( n ) ⟩ ∩ W ∩ A is not empt y for almost all n ∈ ω . Indeed, if n is such that ⟨ ( n ) ⟩ ∩ W  = ∅ (whic h are almost all n ∈ ω ), since D ( D ) | = A is dense, it follows that ⟨ ( n ) ⟩ ∩ W ∩ A  = ∅ . This ﬁnishes the pro of of the sub claim. 14 R. FIGUERO A-SIERRA, O. GUZM ´ AN, M. HR U ˇ S ´ AK, AND A. KWELA Giv en W ∈ U , we can deﬁne g W ∈ ω ω suc h that for ev ery n ∈ ω , the follo wing holds: (1) g W ( n ) = 0 if ⟨ ( n ) ⟩ ∩ W ∩ A = ∅ . (2) If ⟨ ( n ) ⟩ ∩ W ∩ A  = ∅ , let g W ( n ) b e the least natural n umber such that ( n ) ⌢ △ g W ( n ) ∩ W ∩ A  = ∅ . Note that if W ∈ U , then g W ∈ V [ A ] . It follo ws that g W ( n ) < f ω 1 (( n )) for almost all n ∈ ω . W e are in position to prov e that D ( B ) | = ∅ ∈ A no w. It is enough to pro ve that A intersects every set of the form ( W ∩ U α 1 ∩ ... ∩ U α m ) >k where W ∈ U , ω 1 ≤ α 1 , ..., α m < ω 2 and k ∈ ω . Find n ∈ ω with the follo wing prop erties: (1) g W ( n )  = 0 . (2) g W ( n ) < f ω 1 (( n )) ≤ f α 1 (( n )) , ..., f α m (( n )) (3) n  = f α 1 ( ∅ ) , ..., f α m ( ∅ ) . (4) k < n. Since g W ( n )  = 0 , there is s ∈ ( n ) ⌢ △ g W ( n ) ∩ W ∩ A. It follo ws by Lemma 22 that s ∈ U α i for all i ≤ m. Since s (0) = n > k , w e hav e that s is in A ∩ ( W ∩ U α 1 ∩ ... ∩ U α m ) >k . □ Let us review the proof of the previous theorem. W e started with a b -scale D and forced with L ( D ) . The pro of actually sho ws that if w e complete D to a scale B in any ω -hitting preserving extension suc h that the ﬁrst element of B \ D is dominating ov er V [ A gen ] , then B will not b e F r ´ echet. Of course, this do es not need to b e the case for every b -scale, so a more careful approach is required in order to construct a mo del in whic h no b -scale is F r ´ ec het. W e do not kno w whether there are F r ´ ec het scales in the mo del of Theorem 32. In fact, we conjecture that the scale induced by the Hec hler reals is F r´ echet, but w e do not know how to pro ve it. 11. There ma y be no Fr ´ echet b -scale In this section, w e will pro ve that it is consisten t that no b -scale is F r´ echet. W e follo w the approac h used in the model constructed in [33]. W e will p erform a ﬁnite support iteration of forcings of the t yp e L ( D ) for D a weak b -scale. As in [33], we will need a diamond sequence in order to guess an initial segment of a b -scale “at the right time”. By Lemma 31, its Lav er forcing will add A gen , a dense set containing no sequences that conv erge to ∅ . The main diﬃcult y lies in proving that A gen still accumulate to ∅ , ev en after extending the b -scale. This is where our approach diverges from the one tak en in [33], where the algebraic structure is used, whic h is not a v ailable in our setting. A completely diﬀeren t argument is required. Deﬁnition 34. L et F b e a ﬁlter on the c ountable set X , ˙ P an L ( F ) -name for a p artial or der, and ⟨ ˙ F n | n ∈ ω ⟩ a se quenc e of L ( F ) -names of ﬁlters of P such that L ( F ) ⊩ “ P = S n ∈ ω ˙ F n ”. L et ( p, ˙ u ) ∈ L ( F ) ∗ ˙ P . ARE SCALES FR ´ ECHET? 15 (1) We say that ( p, ˙ u ) is suitable if ther e is n ∈ ω such that ( p, ˙ u ) ⊩ “ ˙ u ∈ ˙ F n ”. (2) A type is a p air of the form ( a, m ) wher e a ∈ X <ω and m ∈ ω . (3) We say that ( p, ˙ u ) is of typ e ( a, m ) if st ( p ) = a and p ⊩ “ ˙ u ∈ ˙ F n ”. (4) L et φ b e a formula. We say that ( a, m ) prefers “ φ ” if ther e is no ( q , ˙ v ) of typ e ( a, m ) that for c es the ne gation of φ. it follows that a condition is suitable if it has a type. Note that the set of suitable conditions is dense. The follo wing lemma is easy to v erify . Lemma 35. L et F b e a ﬁlter on a c ountable set, ˙ P an L ( F ) -name for a σ -ﬁlter e d for cing , ( a, m ) a typ e and φ a formula. (1) If ( p 1 , ˙ u 1 ) , ..., ( p n , ˙ u n ) ar e of typ e ( a, m ) , then ther e is ( q , ˙ v ) of typ e ( a, m ) such that ( q , ˙ v ) ≤ ( p 1 , ˙ u 1 ) , ..., ( p n , ˙ u n ) . (2) If ( a, m ) pr efers “ φ ” and ( p, ˙ u ) is of typ e ( a, m ) , then ther e is ( q , ˙ v ) ≤ ( p, ˙ u ) such that ( q , ˙ v ) ⊩ “ φ ”. (3) If ( p, ˙ u ) is of typ e ( a, m ) and ( a, m ) do es not pr efer “ φ ”, then ther e is ( q , ˙ v ) ≤ ( p, ˙ u ) of typ e ( a, m ) that for c es the ne gation of φ. Note that in the second p oint ab ov e, ( q , ˙ v ) ma y not b e of type ( a, m ). Another v ery imp ortan t prop erty is the follo wing: Lemma 36 (Pure Preference Property) . L et F b e a ﬁlter on a c ountable set, ˙ P an L ( F ) -name for a σ -ﬁlter e d for cing, X a ﬁnite set and ˙ y an L ( F ) ∗ ˙ P - name for an element of X . F or every typ e ( a, m ) , ther e is z ∈ X such that ( a, m ) pr efers “ ˙ y = z ”. Pr o of. Assume this is not the case, so for every z ∈ X there is ( p z , ˙ u z ) of t yp e ( a, m ) suc h that ( p z , ˙ u z ) ⊩ “ ˙ y  = z ”. Find ( q , ˙ v ) that extends ( p z , ˙ u z ) for ev ery z ∈ X . It follo ws that ( q , ˙ v ) ⊩ “ ˙ y / ∈ X ”, which is a con tradiction. □ W e will b e working with forcings of the t yp e L ( B ) ∗ ˙ P , where B is a weak b -scale. The elements of L ( B ) are subtrees of ( ω <ω ) <ω that branch in to subsets of ω <ω . F or the conv enience of the reader, w e adopt the following notational con ven tions: (1) Elements of ω <ω will be denoted b y s, t and z . (2) Elements of ( ω <ω ) <ω will be denoted b y a, b and c. (3) Elements of L ( B ) will be denoted b y p, q and r . (4) Names for elemen ts of ˙ P will b e denoted b y ˙ u, ˙ v and ˙ w . Accordingly , if p ∈ L ( B ) , then a typical elemen t of p will b e denoted by a , b or c and a typical element of suc p ( a ) will b e denoted b y s, t or z . Prop osition 37. L et B a we ak b -sc ale that is dominating and F r´ echet, ˙ P an L ( B ) -name for a σ -ﬁlter e d for cing and ˙ g 1 , ..., ˙ g m b e L ( B ) ∗ ˙ P -names for functions fr om ω <ω to ω that ar e dominating. F or every U ∈ N D ( B ) ( ∅ ) , we have that: L ( B ) ∗ ˙ P ⊩ “ U ∩ U ( ˙ g 1 ) ∩ ... ∩ U ( ˙ g m ) ∩ ˙ A gen \ {∅}  = ∅ ”. 16 R. FIGUERO A-SIERRA, O. GUZM ´ AN, M. HR U ˇ S ´ AK, AND A. KWELA Pr o of. W e pro ceed by contradiction. Assume there is a condition ( p, u ) forcing that the intersection is empty . F or conv enience, w e ma y assume that if a ∈ p extends the stem, then ∅ / ∈ suc p ( a ) and the range of a and suc p ( a ) are disjoin t. Let ˙ g b e the name of the function from ω <ω to ω such that ˙ g ( s ) = min { ˙ g 1 ( s ) , ..., ˙ g m ( s ) } . It is easy to see that ˙ g is forced to be dominating o ver V . W e may assume there is n ∈ ω suc h that: (1) ( p, u ) ⊩ “ ˙ g 1 ( ∅ ) , ..., ˙ g m ( ∅ ) < n ”. (2) If k ≥ n, then ( k ) ∈ U. Let a ∈ p and n ∈ ω . W e deﬁne the following: (1) l ( a, n ) = min { k > n | a ⊆ △ k } + n. (2) M ( a, n ) is the set of all s ∈ ω <ω for whic h there is k ∈ ω such that ( a, n ) prefers “ ˙ g ( s ) < k ”. (3) W e sa y ( a, n ) is ugly if M ( a, n ) \ △ l ( a,n ) . (4) Let ( p, ˙ u ) ≤ ( p, u ) . W e say that ( a, n ) c an b e r e alize d b elow ( p, ˙ u ) if there is ( q , ˙ v ) ≤ ( p, ˙ u ) of t yp e ( a, n ) . In tuitively , M ( a, n ) is the collection of all s ∈ ω <ω suc h that ( a, n ) can b ound ˙ g ( s ) (in term of preference) and a type is ugly if it can b ound some- thing that is “v ery far aw ay”. W e no w ha ve the followin g: Claim 38. Ther e is ( p, ˙ u ) ≤ ( p, u ) such that no ugly typ e is r e alize d b elow ( p, ˙ u ) . F or every s ∈ ω <ω , deﬁne X ( s ) as the set of all t yp es ( a, n ) such that s ∈ M ( a, n ) \ △ l ( a,n ) . In other words, X ( s ) is the set of all types ( a, n ) such that s testiﬁes that ( a, n ) is ugly . Note that X ( s ) is a ﬁnite set. This is simply b ecause s is in △ l ( a,n ) for almost all types ( a, n ) . W e can then deﬁne h : ω <ω − → ω such that if s ∈ ω <ω and ( a, n ) ∈ X ( s ) , then ( a, n ) prefers “ ˙ g ( s ) < h ( s )” (in case X ( s ) = ∅ , w e can simply take h ( s ) = 0). Since ˙ g is forced to b e a dominating real, there are k ∈ ω and ( p, ˙ u ) ≤ ( p, u ) suc h that ( p, ˙ u ) ⊩ “ h ≤ k ˙ g ” and st ( p ) ⊈ △ k . W e claim that ( p, ˙ u ) is as desired. If this w as not true, then there is an ugly type ( a, n ) that can b e realized b elow ( p, ˙ u ) . Let ( q , ˙ v ) ≤ ( p, ˙ u ) that is of type ( a, n ) . Since st ( q ) ⊈ △ k (b ecause st ( p ) ⊆ st ( q )), it follows that l ( a, n ) > k . Since ( a, n ) is an ugly t yp e, w e know that there is s ∈ M ( a, n ) \ △ l ( a,n ) . Since ( a, n ) ∈ X ( s ) , it follows that ( a, n ) prefers “ ˙ g ( s ) < h ( s )”. In this wa y , w e can ﬁnd ( r , ˙ w ) ≤ ( q , ˙ v ) such that ( r , ˙ w ) ⊩ “ ˙ g ( s ) < h ( s )”. But this is a con tradiction since s / ∈ △ k (recall that l ( a, n ) > k and s is not even in △ l ( a,n ) ) and ( p, ˙ u ) ⊩ “ h ≤ k ˙ g ”. This ﬁnishes the pro of of the claim. Fix ( p, ˙ u ) ≤ ( p, u ) suc h that no ugly t yp e is realized b elow it. F or sim- plicit y , we will assume that st ( p ) = ∅ . The argument for the general case is essen tially the same, only requiring m uch more notation. Let T b e the set of all types that can b e realized b elow ( p, ˙ u ) . F or every type ( a, n ) ∈ T , ﬁx a condition ( p ( a, n ) , ˙ u ( a, n )) ≤ ( p, ˙ u ) of type ( a, n ). Note that the stem of p ( a, n ) is a. W e record some prop erties of T . F or every ( a, n ) ∈ T , the following holds: ARE SCALES FR ´ ECHET? 17 (1) a ∈ p. (2) If b ∈ p ( a,n ) and a ⊆ b, then ( b, n ) ∈ T . In particular, if z ∈ suc p ( a,n ) ( a ) , then ( a ⌢ z , n ) ∈ T . (3) If q ≤ p ( a, n ) , then ( st ( q ) , n ) ∈ T . W e now hav e the follo wing: Claim 39. F or every ( a, n ) ∈ T and z ∈ suc p ( a,n ) ( a ) ∩ U >l ( a,n ) ther e is i ( z , a, n ) such that: (1) i ( z , a, n ) < | z | . (2) Ther e is j ≤ m such that ( a ⌢ z , n ) pr efers “ ˙ g j ( z ↾ i ( z , a, n )) = z ( i ( z , a, n )) ”. In p articular, ( a ⌢ z , n ) pr efers “ ˙ g ( z ↾ i ( z , a, n )) ≤ z ( i ( z , a, n )) ”. Let q = p ( a, n ) a ⌢ ( z ) (in other words, q is the tree obtained by adding z to the stem of p ( a, n )). Note that q ⊩ “ z ∈ ˙ A gen ∩ U ”. Since U ∩ U ( ˙ g 1 ) ∩ ... ∩ U ( ˙ g m ) ∩ ˙ A gen \ {∅} is forced to b e empty , Lemma 21 implies that: ( q , ˙ u ( a, n )) ⊩ “ ∃ j, k ( ˙ g j ( z ↾ k ) = z ( k ))”. W e can no w use the Pure Preference Prop ert y to ﬁnd the exact j and k . This ﬁnishes the pro of of the claim. Let ( a, n ) ∈ T . Deﬁne the following items: (1) W ( a, n ) = { z ↾ i ( z , a, n ) | z ∈ suc p ( a,n ) ( a ) ∩ U >l ( a,n ) } . (2) F or ev ery s ∈ W ( a, n ) , denote E xt s ( a, n ) = { z ∈ suc p ( a,n ) ( a ) ∩ U >l ( a,n ) | s = z ↾ i ( z , a, n ) } . W e will no w prov e the follo wing claim: Claim 40. L et ( a, n ) ∈ T and s ∈ W ( a, n ) . The fol lowing holds: (1) s  = ∅ and s (0) > l ( a, n ) . (2) W ( a, n ) is inﬁnite. (3) D ( B ) | = E xt s ( a, n ) is nowher e dense. W e prov e the ﬁrst p oint. Pic k any z ∈ E xt s ( a, n ) . It follo ws by deﬁnition that z (0) > l ( a, n ) . In order to prov e that s  = ∅ and s (0) > l ( a, n ) , it is enough to show that i ( z , a, n )  = 0 . If it was the case that i ( z , a, n ) = 0 , then we would hav e that ( a ⌢ z , n ) prefers “ ˙ g j ( ∅ ) = z (0)” for some j ≤ m. It follows that ( a ⌢ z , n ) prefers “ ˙ g j ( ∅ ) > n ”, which is imp ossible (see the prop erties of n at the beginning of the pro of ). W e no w prov e the second p oint. Cho ose any k > l ( a, n ) . Since suc p ( a ) is dense, w e c an ﬁnd z ∈ suc p ( a ) ∩ U >l ( a,n ) with z (0) = k. The conclusion follo ws by the ﬁrst p oin t of the claim. It is time to pro ve the third p oint. Let W ⊆ ω <ω b e a non-empt y op en set. W e need to ﬁnd a non-empty open set U ⊆ W that is disjoin t with E xt s ( a, n ) . If W ∩ ⟨ s ⟩ c  = ∅ , then this is a non-empty set disjoin t with E xt s ( a, n ), since this set is contained in ⟨ s ⟩ . Moreov er, w e may assume there is t ∈ ω <ω suc h that s ⊊ t and W ⊆ ⟨ t ⟩ (since W ⊆ ⟨ s ⟩ , there is t ⊋ s suc h that t ∈ W , w e then c hange W for W ∩ ⟨ t ⟩ ). 18 R. FIGUERO A-SIERRA, O. GUZM ´ AN, M. HR U ˇ S ´ AK, AND A. KWELA By the ﬁrst p oin t of the claim, we know that s / ∈ △ l ( a,n ) . Since ( a, n ) is not ugly , it follo ws that ( a, n ) do es not prefer “ ˙ g ( s ) ≤ t ( | s | )”. By Lemma 35, w e can ﬁnd ( q , ˙ v ) ≤ ( p ( a, n ) , ˙ u ( a, n )) of t yp e ( a, n ) suc h that ( q , ˙ v ) ⊩ “ ˙ g ( s ) > t ( | s | )”. W e ma y assume that suc q ( a ) is an op en dense subset of D ( B ) . W e claim that U = W ∩ suc q ( a ) (which is non-empt y) is disjoin t with E xt s ( a, n ) . Assume there is z ∈ U ∩ E xt s ( a, n ) . W e hav e the follo wing: (1) s ⊊ t ⊆ z (recall that W ⊆ ⟨ t ⟩ ). (2) z ↾ i ( z , a, n ) = s ( z ∈ E xt s ( a, n )). In this wa y , there is j ≤ m suc h that ( a ⌢ z , n ) prefers “ ˙ g j ( s ) = z ( | s | ) = t ( | s | )”. In particular, ( a ⌢ z , n ) prefers “ ˙ g ( s ) ≤ t ( | s | )”. Let ( r , ˙ x ) ≤ ( q , ˙ v ) such that ( r , ˙ x ) ⊩ “ ˙ g ( s ) ≤ t ( | s | )”. But this is imp oss ible since ( q , ˙ v ) ⊩ “ ˙ g ( s ) > t ( | s | )”. This ﬁnishes the pro of of the claim. W e need another claim: Claim 41. L et ( a, n ) ∈ T . Ther e is h ( a,n ) : W ( a, n ) − → ω <ω with the fol lowing pr op erties: (1) h ( a,n ) ( s ) ∈ E xt s ( a, n ) (so s ⊊ h ( a,n ) ( s ) ). (2) im  h ( a,n )  ∈ nwd( D ( B ) ) + . T ake an enumeration W ( a,n ) = { s i | i ∈ ω } (recall that this set is inﬁnite). Deﬁne N 0 = E xt s 0 ( a, n ) and N i +1 = E xt s i ( a, n ) \ N 0 ∪ ... ∪ N i . It follo ws that { N i | i ∈ ω } is a partition of suc p ( a,n ) ( a ) ∩ U >l ( a,n ) in to no where dense sets. Since D ( B ) is F r ´ echet, Proposition 11 implies that there is Z ⊆ suc p ( a,n ) ( a ) ∩ U >l ( a,n ) that is not no where dense suc h that | Z ∩ N k | ≤ 1 for every k ∈ ω (w e could mak e sure that | Z ∩ N k | = 1 whenev er N k  = ∅ if w e w anted). W e can no w deﬁne h ( a,n ) : W ( a, n ) − → ω <ω as follo ws: F or s i ∈ W ( a, n ) , if Z ∩ N i  = ∅ , let h ( a,n ) ( s i ) b e the only p oint in Z ∩ N i . In case Z ∩ N i = ∅ , let h ( a,n ) ( s i ) b e any elemen t of E xt s ( a, n ) . It follows that im  h ( a,n )  con tains Z, so it is not no where dense. This ﬁnishes the pro of of the claim. Let ( a, n ) ∈ T . W e no w deﬁne the function h ( a,n ) : W ( a, n ) − → ω <ω giv en b y h ( a,n ) ( s ) = h ( a,n ) ( s ) ( | s | ) (this is p ossible b ecause s ⊊ h ( a,n ) ( s )). Note that if s ∈ W ( a, n ) , then we hav e the following: (1) h ( a,n ) ( s ) ∈ suc p ( a,n ) ( a ) ∩ U >l ( a,n ) and s ⊊ h ( a,n ) ( s ) . (2)  a ⌢ h ( a,n ) ( s ) , n  prefers “ ˙ g ( s ) ≤ h ( a,n ) ( s )” (this is b ecause h ( a,n ) ( s ) ↾ | s | = s ). Since { h ( a,n ) | ( a, n ) ∈ T } is a countable set and B is a dominating family , there is f ∈ B such that h ( a,n ) ≤ ∗ f for ev ery ( a, n ) ∈ T . Recall that ˙ g is forced to b e a dominating real. W e can ﬁnd a suitable ( q , ˙ v ) ≤ ( p, ˙ u ) and k ∈ ω suc h that ( q , ˙ v ) ⊩ “ f < k ˙ g ”. Let ( a, n ) b e the type of ( q , ˙ v ) . W e may assume that k < l ( a, n ) . Claim 42. Ther e is s ∈ W ( a, n ) with the fol lowing pr op erties: (1) h ( a,n ) ( s ) ∈ suc q ( a ) . ARE SCALES FR ´ ECHET? 19 (2) ( q , ˙ v ) ⊩ “ f ( s ) < ˙ g ( s ) ”. (3) h ( a,n ) ( s ) < f ( s ) . Since im  h ( a,n )  ∈ nwd( D ( B ) ) + , it follo ws that im  h ( a,n )  ∩ suc q ( a ) is in- ﬁnite. Moreo ver, we know that h ( a,n ) ≤ ∗ f , so we can ﬁnd s ∈ W ( a, n ) suc h that h ( a,n ) ( s ) ∈ suc q ( a ) and h ( a,n ) ( s ) < f ( s ) . Since s (0) > l ( a, n ) , w e con- clude that s / ∈ △ k (recall that k < l ( a, n )) . It follo ws that ( q , ˙ v ) ⊩ “ f ( s ) < ˙ g ( s )”. This ﬁnishes the pro of of the claim. W e can no w ﬁnish the proof. Let s ∈ W ( a, n ) as ab ov e and z = h ( a,n ) . Since ( a ⌢ z , n ) prefers “ ˙ g ( s ) ≤ h ( a,n ) ( s )”, there is ( r, ˙ w ) < ( q z , ˙ v ) such that ( r , ˙ w ) ⊩ “ ˙ g ( s ) ≤ h ( a,n ) ( s )”. It follows that ( r, ˙ w ) ⊩ “ ˙ g ( s ) < f ( s )”. But this is a contradiction b ecause ( q , ˙ v ) ⊩ “ f < k ˙ g ”. □ Let S ω 1 ( ω 2 ) = { α < ω 2 | cof ( α ) = ω 1 } . Recall the follo wing principle: ♢ ( S ω 1 ( ω 2 )) There is { D α | α ∈ S ω 1 ( ω 2 ) } suc h that D α ⊆ α for all α ∈ S ω 1 ( ω 2 ) with the prop erty that for every X ∈ ω 2 , the set { α | X ∩ α = D α } is stationary . A sequence as ab ov e is called a ♢ ( S ω 1 ( ω 2 ))- se quenc e. It is w ell kno wn that ♢ ( S ω 1 ( ω 2 )) holds in the constructible univ erse of G¨ odel (see [38] or [11]). W e can now prov e the main theorem of the section, whic h was inspired in the proof of the main theorem of [33]. Theorem 43. It is c onsistent that no b -sc ale is F r ´ echet. Pr o of. W e start with a mo del of CH + ♢ ( S ω 1 ( ω 2 )) . Let { D α | α ∈ S ω 1 ( ω 2 ) } b e a ♢ ( S ω 1 ( ω 2 ))-sequence. Let us construct a ﬁnite support iteration ⟨ P α , ˙ Q α | α < ω 2 ⟩ suc h that for every α < ω 2 , the following holds: (1) If α ∈ S ω 1 ( ω 2 ) and D α co des a P α -name for a b -scale B α that is dominating and F r ´ echet, we let ˙ Q α b e a P α -name for L ( B α ) . (2) If α is not as ab ov e, let ˙ Q α b e a P α -name for Hechler forcing. It follo ws that P ω 2 is a c.c.c. forcing that preserv es ω -hitting families. Since at ev ery step we add a dominating real, it follo ws that P ω 2 ⊩ “ b = d = c = ω 2 ” and by the preserv ation of ω -hitting families, P ω 2 forces that there are no tow ers of length ω 2 (see [4] for more details). Prop osition 6 implies that P ω 2 forces that every b -scale is a dominating family . Let G ⊆ P ω 2 b e a generic ﬁlter. W e shall prov e that there are no F r´ ec het b -scales in V [ G ] . Aiming tow ards a con tradiction, assume that there is a F r´ echet b -scale B ∈ V [ G ]. By a standard closing oﬀ argument, there is a set C ⊆ S ω 1 ( ω 2 ) which is a club relativ e to S ω 1 ( ω 2 ) such that if α ∈ C , then V [ G α ] | = B α is a dominating F r´ ec het w eak b -scale , where G α = G ∩ P α and B α = B ∩ V [ G α ] . It follo ws that there is α ∈ C such that D α co des B α . In this w ay , we ha ve that P α +1 = P α ∗ L ( ˙ B α ) . Let A gen ∈ V [ G α +1 ] b e the image of the generic real added by L ( ˙ B α ) . Claim 44. D ( B ) | = A gen ac cumulates to ∅ (in V [ G ] ). 20 R. FIGUERO A-SIERRA, O. GUZM ´ AN, M. HR U ˇ S ´ AK, AND A. KWELA It is enough to pro ve that for every U ∈ V [ G α ] basic op en set of ∅ and g 1 , ..., g m ∈ B \ B α , it is the case that U ∩ U ( ˙ g 1 ) ∩ ... ∩ U ( ˙ g m ) ∩ ˙ A gen \ {∅}  = ∅ . Note that since V [ G α ] | = B α is a dominating family and g 1 , ..., g m ∈ B \ B α , then they are dominating ov er V [ G α ] . Let β > α such that g 1 , ..., g m ∈ V [ G β ] . W e go to V [ G α ] . Let B b e the Bo olean completion of the quotien t P β ⧸ G α . Prop ositions 14 and 15 imply that B is σ -ﬁltered. Prop osition 37 implies that U ∩ U ( ˙ g 1 ) ∩ ... ∩ U ( ˙ g m ) ∩ ˙ A gen \ {∅}  = ∅ . Claim 45. D ( B ) | = A gen do es not c ontain a se quenc e c onver ging to ∅ (in V [ G ] ). Let Y b e an inﬁnite subset of A gen . Since I D ( B α ) ( ∅ ) ↾ A gen is an ω -hitting ideal, there is B ∈ I D ( B α ) ( ∅ ) such that Y ∩ B is inﬁnite. Let U be a basic neigh b orhoo d of ∅ (in D ( B α )) such that B ∩ U = ∅ . It follo ws that U \ Y is inﬁnite. Since U is still an op en set in D ( B ), the result follo ws. □ 12. The Ca tegor y Dichotomy for definable ideals In [30] the third author pro ved that if I is a Borel ideal, then either I ≤ K nwd or there is X ∈ I + suc h that E D ≤ K I ↾ X . This statement is known as the Cate gory Dichotomy. 4 It is natural to ask for which other classes of ideals the Category Dichotom y can b e extended. The article [16] con tains man y results in this direction. Here w e are interested in ho w far the dic hotomy can hold within the pro jectiv e hierarch y . Using results from the previous sections, we will pro vide (consistently) a new example of an ideal that do es not satisfy the Category Dic hotomy with the least p ossible complexit y . In [25] the second author and Na v arro prov ed that the Category Di- c hotomy holds for analytic ideals, as w ell as for all ideals in the Solov a y mo del. W e will now prov e that the dic hotomy also holds for co-analytic ideals. Let I b e an ideal on a coun table set. The game G Cat ( I ) (which was ﬁrst pla yed by Laﬂamme, see [46] and [47]) is play ed in the following wa y: I A 0 A 1 ... A 0 ... I I b 0 b 1 b n The game lasts ω rounds. In round n, Play er I plays A n ∈ I and Pla y er I I resp onds with b n / ∈ A n . Player I I wins the game if the { b n | n ∈ ω } ∈ I + . In this game and the one b elow, Play er I will b e a man and Pla yer I I a woman. In the pro of of [30, Theorem 3.1], the third author obtained the follo wing result. Prop osition 46 (H. [30]) . L et I b e an ide al on ω . 4 Note that the tw o alternatives of the Category Dichotom y are not mutually exclusive, so it is not a dic hotomy in the traditional sense. ARE SCALES FR ´ ECHET? 21 (1) If for every X ∈ I + , the Player I I has a winning str ate gy in G Cat ( I ↾ X ) , then I ≤ K nwd. (2) If Player I has a winning str ate gy in G Cat ( I ) , then ther e is X ∈ I + such that E D ≤ K I ↾ X . The follo wing is a trivial consequence of this last prop osition: Corollary 47. L et Γ b e a class of ide als such that if I ∈ Γ and X ∈ I + , then I ↾ X ∈ Γ . If for every I ∈ Γ the game G Cat ( I ) is determine d, then every ide al in Γ satisﬁes the Cate gory Dichotomy. In [45] the fourth author and Sab ok deﬁned an “unfolded” version of the previous game, whic h w e will now review. W e need to introduce some notation: (1) Let R b e the set of all functions f : ω − → ω ∪ {− 1 } such that f ( n ) < n for all n ∈ ω and the set { n | f ( n )  = − 1 } is inﬁnite. (2) Let f ∈ R. Deﬁne e f : ω − → ω as the function obtained in the follow- ing w ay: enumerate { n | f ( n )  = − 1 } = { n i | i ∈ ω } in an increasing w ay and let e f ( i ) = f ( n i ) . In tuitively , if we imagine f ∈ R as an inﬁnite sequence of elemen ts from ω ∪ {− 1 } , then e f is the sequence obtained b y deleting all the o ccurrences of − 1 and then “compressing” to get rid of the empt y spaces. Let I b e an ideal on a countable set and F : ω ω − → I + an y function. The game H Cat ( I , F ) is pla y ed in the follo wing w ay: I A 0 A 1 ... A n ... I I ( b 0 , m 0 ) ( b 1 , m 1 ) ( b n , m n ) The game lasts ω rounds. In round n, Pla y er I plays A n ∈ I and Play er I I resp onds with ( b n , m n ) such that b n / ∈ A n and m n ∈ n ∪ {− 1 } (intuitiv ely , m n = − 1 can b e interpreted as Pla yer I I refraining from making a mo v e). Player I I wins the game if the following conditions are met: (1) The set { n | m n  = − 1 } is inﬁnite. (2) F ( e g ) = { b n | n ∈ ω } , where g : ω − → ω ∪ {− 1 } is the function suc h that g ( n ) = m n . Note that if Pla y er I I was the winner of the game, then { b n | n ∈ ω } ∈ I + . The ﬁrst p oint of the following prop osition was pro ved by the fourth author and Sab ok as part of the pro of of Theorem 1.6 of [45]. W e provide a pro of for completeness, but ﬁrst we in tro duce some more notation. Let n ∈ ω . Deﬁne T n as the set of all t : n − → n ∪ {− 1 } such that t ( i ) < i for all i < n. Note that eac h T n is a ﬁnite set. Let s ∈ ω n and t ∈ T n . Deﬁne s ∗ t : n − → ω × ( ω ∪ {− 1 } ) given b y ( s ∗ t ) ( i ) = ( s ( i ) , t ( i )) . Prop osition 48 (K., Sab ok [45]) . L et I b e a c o-analytic ide al on ω and F : ω ω − → I + a c ontinuous surje ction 5 . 5 Note that the existence of suc h function is guaran teed by the fact that I is co-analytic. 22 R. FIGUERO A-SIERRA, O. GUZM ´ AN, M. HR U ˇ S ´ AK, AND A. KWELA (1) If Player I has a winning str ate gy in H Cat ( I , F ) , then he has a win- ning str ate gy in G Cat ( I ) . (2) If Player I I has a winning str ate gy in H Cat ( I , F ) , then she has a winning str ate gy in G Cat ( I ) . Pr o of. Let σ : ( ω × ( ω ∪ {− 1 } )) <ω − → I b e a a winning strategy for the Pla yer I in the game H Cat ( I , F ) . W e will use σ to deﬁne a strategy π : ω <ω − → I for him in the game G Cat ( I ) . Both games start with π ( ∅ ) = σ ( ∅ ) . Assume we are at round n + 1 and Play er I has to make a mov e (in G Cat ( I )) after Play er I I pla yed s ∈ ω n +1 . Play er I plays (in G Cat ( I )) the set π ( s ) = S t ∈ T n +1 σ ( s ∗ t ) ∈ I . W e claim that this is a winning strategy for Pla yer I in G Cat ( I ) . Assume this is not the case, so there is a run of the game G Cat ( I ) where Pla yer I follo wed π , but Play er I I w as declared the winner. Let ( b i ) i ∈ ω b e the sequence play ed by Pla yer I I . Since she was the winner, we kno w that Y = { b i | i ∈ ω } ∈ I + . Recall F is onto, so Y is in the range of F . Moreo ver, we can ﬁnd g ∈ R suc h that F ( e g ) = Y . Consider the run of the game H Cat ( I , F ) where Play er I follow ed σ and Play er I I pla yed ( b n , g ( n )) at round n. This is a v alid run of the game since for every n ∈ ω , it is the case that g ( n ) ∈ n ∪ {− 1 } and σ (( b 0 , g (0)) , ..., ( b n, g ( n ))) is con tained in π ( b 0 , ..., b n ) , so b n +1 is not in σ (( b 0 , g (0)) , ..., ( b n, g ( n ))) (it is not even in π ( b 0 , ..., b n ) , which is bigger). It follo ws that Pla y er II w on the game, but this w as imp ossible since σ was a winning strategy . W e no w prov e the second part of the prop osition. Play er I I can easily win G Cat ( I ) b y pla ying the ﬁrst co ordinates of her winning strategy of H Cat ( I , F ) . □ W e now prov e the determinacy of these games for co-analytic ideals. Prop osition 49. L et I b e a c o-analytic ide al on ω and F : ω ω − → I + a c ontinuous surje ction. The games H Cat ( I , F ) and G Cat ( I ) ar e determine d. Pr o of. By Prop osition 48, the determinacy of H Cat ( I , F ) implies the deter- minacy of G Cat ( I ) . W e no w sho w that H Cat ( I , F ) is determined. By Martin’s Determinacy Theorem for Borel games (see [40]), it is enough to prov e that { ( B , g ) | g ∈ R ∧ F ( e g ) = B } ⊆ P ( ω ) × ( ω ∪ {− 1 } ) ω (the winning set for Pla yer I I ) is Borel. This is easy: mem b ership in R is G δ while the condition F ( e g ) = B is closed b y the con tinuit y of F . □ Since the restriction of a co-analytic ideal is also co-analytic, by Prop osi- tion 49 and 47, w e obtain the following: Theorem 50. Every c o-analytic ide al satisﬁes the Cate gory Dichotomy. Ho w far can the Category Dic hotomy b e extended? If we assume suitable determinacy axioms or instances of the Close d-sets Covering Pr op erty (see [55], [12] and [25]), we can extend the Category Dic hotomy throughout the en tire Pro jectiv e Hierarch y . Ho wev er, just in ZFC, the class of analytic and ARE SCALES FR ´ ECHET? 23 co-analytic ideals is the highest we can go, since it is consisten t that the Category Dic hotomy fails for △ 1 2 ideals. W e w ere able to ﬁnd t wo consisten t examples of △ 1 2 ideals in the literature for whic h the Category Dichotom y fails: (1) The ide al gener ate d by a c o-analytic tight MAD family. In [6], Bergfalk, Fisc her and Switzer pro ved that V = L implies that there is a co- analytic tigh t MAD family . The ideal generated by such family is a △ 1 2 ideal and can not satisfy the Category Dic hotomy (see Prop osi- tion 24 of [16]). (2) The dual of a △ 1 2 R amsey ultr aﬁlter. In [54, Theorem 5.1], Sc hilhan pro ved that it is consisten t that there is a △ 1 2 Ramsey ultraﬁlter. The dual of such ultraﬁlter cannot satisfy the Category Dichotom y (see [16, Prop osition 23]). W e will now provide a new example of a △ 1 2 ideal for which the Category Dic hotomy fails using Do w spaces. The follo wing is [16, Theorem 45]: Theorem 51 (Dow, F., G., H. [16]) . If X is a top olo gic al sp ac e with the fol lowing pr op erties: (1) X is c ountable. (2) X is zer o dimensional. (3) X is F r ´ echet. (4) X has unc ountable π -char acter everywher e. Then nwd ( X ) do es not satisfy the Cate gory Dichotomy. It follows that if B is a F r ´ ec het b -scale, then the ideal nwd( D ( B ) ) do es not satisfy the Category Dic hotom y . W e no w pro ve the follo wing: Theorem 52. If V = L , then ther e exists a F r ´ echet b -sc ale B such that nwd  D ( B )  is a ∆ 1 2 ide al. The follo wing lemma is used in the pro of of the preceding theorem. Lemma 53. L et ( X , τ ) b e a c ountable top olo gic al sp ac e, and supp ose that B is a c o-analytic b asis for τ . Then the top olo gy τ is Σ 1 2 , and the ide al nwd ( X , τ ) is ∆ 1 2 . Pr o of. As B is co-analytic, for each s ∈ X the set X s =  A ⊆ X : s ∈ A → ∃ U ∈ B  s ∈ U ∧ U ⊆ A  is Σ 1 2 . Moreov er, for an y A ⊆ X A ∈ τ ⇐ ⇒ ∀ s ∈ X ( A ∈ X s ) . Therefore τ b elongs to Σ 1 2 , as it is a countable intersection of Σ 1 2 sets. Recall that △ 1 2 denotes the class Σ 1 2 ∩ Π 1 2 . W e ﬁrst sho w that nwd ( X , τ ) is a Π 1 2 subset of P ( X ). T o that end, it suﬃces to v erify that its complement nwd ( X , τ ) + = P ( X ) \ nwd ( X , τ ) 24 R. FIGUERO A-SIERRA, O. GUZM ´ AN, M. HR U ˇ S ´ AK, AND A. KWELA is Σ 1 2 . This follo ws from the following equiv alence: A ∈ nwd ( X, τ ) + ⇐ ⇒ ∃ U ∈ B \ {∅} ∀ V ∈ B \ {∅} ( V ⊆ U → V ∩ A  = ∅ ) whic h deﬁnes a Σ 1 2 condition under the assumption that B is co-analytic. Since the complement of a Σ 1 2 set is a Π 1 2 set, the result follo ws. On the other hand, we show that nwd ( X , τ ) is also a Σ 1 2 subset of P ( X ). W e b egin with the case of closed no where dense sets. Let nwd ( X , τ ) denote the family of closed no where dense subsets of X . W e claim that nwd ( X , τ ) is Σ 1 2 . Indeed, a set F ⊆ X is closed and no where dense if and only if its complemen t X \ F is a dense op en set. Equiv alently , F ∈ nwd ( X , τ ) ⇐ ⇒ X \ F ∈ τ ∧ ∀ U ∈ B \ {∅}  U ∩ ( X \ F )  = ∅  . Since the top ology τ is Σ 1 2 and B is a co-analytic, the universal quantiﬁcation o ver B \ {∅} deﬁnes a co-analytic set. Therefore, the conjunction ab o ve is Σ 1 2 , and it follo ws that nwd ( X , τ ) is Σ 1 2 . No w, we consider the general case. A subset A ⊆ X is nowhere dense if and only if it is contained in some closed nowhere dense set. That is, A ∈ nwd ( X, τ ) ⇐ ⇒ ∃ F ∈ nwd ( X, τ )  A ⊆ F  from whic h follo ws that nwd ( X, τ ) is also a Σ 1 2 subset of P ( X ). □ Pr o of of The or em 52. By Lemma 53, it suﬃces to construct a F r ´ ec het b - scale such that the top ology τ B admits a co-analytic basis. Note that b y Theorem 28, V = L implies that all b -scales are F r´ ec het. Claim 54. V = L implies that ther e is a c o-analytic b -sc ale. Pr o of. Consider F ⊆  ω ω <ω  ≤ ω × ω ω <ω × ω ω <ω deﬁned b y declaring that ( A, g , f ) ∈ F if and only if the follo wing conditions hold: (1) f is increasing from ω <ω to ω , that is, for s, t ∈ ω <ω and i, j ∈ ω : • If s ⊂ t , then f ( s ) < f ( t ), and • If i < j , then f ( s ⌢ i ) < f ( s ⌢ j ). (2) F or ev ery h ∈ ran( A ) we hav e h < ∗ f . (3) g < ∗ f . (4) f ( ∅ ) > g ( ∅ ). Note that F is Borel, since it is deﬁned as a ﬁnite conjunction of Borel conditions. Moreov er, for ev ery ( A, g ) ∈  ω ω <ω  ≤ ω × ω ω <ω , the section F ( A,g ) = n f ∈ ω ω <ω : ( A, g , f ) ∈ F o is coﬁnal in the T uring degrees 6 . 6 W e refer to elemen ts of 2 ω , ω ω , or ω ω <ω collectiv ely as r e als . F or x and y reals, we sa y x is T uring r e ducible to y , denoted by x ≤ T y , if there exists an oracle T uring machine ARE SCALES FR ´ ECHET? 25 W e will prov e this for A countably inﬁnite. The argumen t for A ﬁnite is essen tially the same. Fix { h i : i ∈ ω } an enumeration of A , and an arbitrary real y ∈ 2 ω . W e construct a function f ∈ F ( A,g ) suc h that y ≤ T f . First, deﬁne a function f 0 ∈ F ( A,g ) b y recursion on ω <ω . F or eac h s ∈ ω <ω , c ho ose f 0 ( s ) so large that: (1) If t ⊂ s , then f 0 ( t ) < f 0 ( s ), (2) If i < j , then f 0 ( s ⌢ i ) < f 0 ( s ⌢ j ) , and (3) f 0 ( s ) > max { h i ( s ) : i ≤ | s |} + g ( s ), Since only ﬁnitely many inequalities m ust b e satisﬁed at each stage, this construction is p ossible. No w, we co de y along the branch ⟨ 0 ⟩ , ⟨ 0 , 0 ⟩ , . . . . Deﬁne f by f ( s ) = ( 2 f 0 ( s ) + 1 if s is not of the form ⟨ 0 n ⟩ for any n, 2 f 0 ( ⟨ 0 n ⟩ ) + y ( n ) if s = ⟨ 0 n ⟩ for some n, where ⟨ 0 n ⟩ denotes the sequence of n man y zeros. Note that f ∈ F ( A,g ) and y is computable from f since for eac h n , y ( n ) = f ( ⟨ 0 n ⟩ ) mo d 2 , and the parity of f at these no des can b e obtained by an oracle T uring mac hine with input f . Th us y ≤ T f , establishing that F ( A,g ) is coﬁnal in the T uring degrees. Therefore, it follows from [59, Theorem 1.3.] that there exists a co- analytic set that is compatible with F , that is, there is a co-analytic b - scale □ No w, let B = { f α : α < ω 1 } b e a co-analytic b -scale. Note that b y construction, B is well-ordered by < ∗ , that is, for any distinct f , g ∈ B , either f < ∗ g or g < ∗ f . W e shall prov e that the top ology τ B admits a co-analytic basis B . W e verify the following statemen ts. (1) The family U = { U ( f ) : f ∈ B } is co-analytic. Let Φ : ω ω <ω → 2 ω <ω b e the mapping deﬁned b y f 7→ U ( f ). W e claim that Φ is contin uous. W e need to show that the preimage of an y subbasic op en set is op en. F or s ∈ ω <ω , consider V s = { A ⊆ ω <ω : s ∈ A } a subbasic op en set. Based on the deﬁnition of U ( f ), w e ha ve: f ∈ Φ − 1 ( V s ) ⇐ ⇒ s ∈ U ( f ) ⇐ ⇒ ∀ t ⊊ s ( f ( t )  = s ( | t | )) . This preimage is a ﬁnite intersection of op en sets: Φ − 1 ( V s ) = \ t ⊊ s { f ∈ ω ω <ω : f ( t )  = s ( | t | ) } . that computes x given y as an oracle. A set of reals A is c oﬁnal in the T uring degrees if for ev ery x ∈ 2 ω there exists some y ∈ A such that x ≤ T y . 26 R. FIGUERO A-SIERRA, O. GUZM ´ AN, M. HR U ˇ S ´ AK, AND A. KWELA Therefore, Φ − 1 ( V s ) is op en. Similarly , for the complemen t of the subbasic open set: f ∈ Φ − 1 (2 ω <ω \ V s ) ⇐ ⇒ s / ∈ U ( f ) ⇐ ⇒ ∃ t ⊊ s ( f ( t ) = s ( | t | )) . This preimage is a ﬁnite union of op en sets: Φ − 1 (2 ω <ω \ V s ) = [ t ⊊ s { f ∈ ω ω <ω : f ( t ) = s ( | t | ) } , whic h is also op en. Since the preimages of subbasic sets are open, w e conclude that Φ is contin uous. Moreo ver, Φ is injective. Indeed, let f , g ∈ ω ω <ω with f  = g . F or the minimal t such that f ( t )  = g ( t ), deﬁne s = t ⌢ f ( t ). Then s ∈ U ( g ) \ U ( f ), so Φ( f )  = Φ( g ). By Theorem 2.6 in [37], the injective contin uous image of a co-analytic set is co-analytic. Therefore, w e conclude that U = { U ( f ) : f ∈ B } is co-analytic. (2) F or eac h n ∈ ω , the following family is co-analytic U n =    \ f ∈ C U ( f ) : C ∈ B n    . Let X = ω ω <ω and consider the mapping Φ n : X n → 2 ω <ω deﬁned b y ( f 0 , . . . , f n − 1 ) 7→ T i N , the v alue g ( s ) is distinct from f ( s ) for every f ∈ C 2 . This implies that for any such s , if s ∈ U 2 then the successor no de s ∗ = s ⌢ g ( s ) belongs to U 2 \ U 1 . Giv en that U 2 ∩ K A,B is non-empty and the trees in U + p ossess an inﬁnite branc hing structure, there are inﬁnitely many no des s ∈ U 2 ∩ K A,B with | s | > N . F or eac h such s , the successor s ∗ remains within the cones generated by A . F urthermore, the ﬁniteness of B ensures that w e can choose at least one s such that s ∗ = s ⌢ g ( s ) ∈ K A,B . Thus, Ψ A,B ( U 1 )  = Ψ A,B ( U 2 ). By Theorem 2.6 in [37], w e conclude that Ψ A,B ( U + ) is co-analytic. Since Ψ A,B ( U ∅ ) = {∅} is a closed set, it follows that X A,B = Ψ A,B ( U ∅ ) ∪ Ψ A,B ( U + ) is co-analytic. 28 R. FIGUERO A-SIERRA, O. GUZM ´ AN, M. HR U ˇ S ´ AK, AND A. KWELA (5) The family B generated by the cones ⟨ s ⟩ and co-cones ⟨ s ⟩ c for s ∈ ω <ω , together with the trees U ( f ) for f ∈ B , is co-analytic. Note that for an y U ∈ 2 ω <ω U ∈ B ⇐ ⇒ ∃ A, B ∈ [ ω <ω ] <ω ( U ∈ X A,B ) . As each X A,B is co-analytic and the class of co-analytic sets is closed under coun table unions, it follows that B is co-analytic. This completes the pro of that τ B admits a co-analytic basis. □ 13. Open Questions In this ﬁnal section, we present several op en questions that we hav e b een unable to solve. W e ﬁrst restate Mo ore’s problem: Problem 55 (Moore) . Is ther e a c ountable, F r´ echet, zer o dimensional sp ac e with π -weight exactly b ? In the pap er, w e pro ved that it is consistent that ev ery b -scale is F r´ ec het and also that none is. Ho wev er, the follo wing remains unsolv ed: Problem 56. Is it c onsistent that ther e ar e F r ´ echet b -sc ales and a non- F r ´ echet b -sc ales at the same time? W e conjecture that this happ ens in the model of Theorem 32. Problem 57. Is ther e a F r´ echet b -sc ale in the mo del of The or em 32? W e do not know muc h ab out the ideal nwd( D ( D )) in case D is not a F r´ echet b -scale. Problem 58. (1) Is it c onsistent that ther e is a b -sc ale D such that nwd( D ( D ) ) satisﬁes the Cate gory Dichotomy? (2) Is it c onsistent that for every b -sc ale D , the ide al nwd( D ( D ) ) satisﬁes the Cate gory Dichotomy? Finally , the follo wing problem ma y also b e of interest. Problem 59. How do the c ombinatorial pr op erties of a sc ale r eﬂe ct on the top olo gic al pr op erties of its Dow sp ac e and vic e versa? References [1] M. Arciga-Alejandre, M. 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Ra ´ ul Figueroa-Sierra Departamen to de Matem´ aticas, Universidad de Los Andes (Bogot´ a) r.ﬁgueroa@uniandes.edu.co Osv aldo Guzm´ an Cen tro de Ciencias Matem´ aticas, UNAM. oguzman@matmor.unam.mx Mic hael Hru ˇ s´ ak Cen tro de Ciencias Matem´ aticas, UNAM. mic hael@matmor.unam.mx Adam Kw ela F aculty of Mathematics, Ph ysics and Informatics, Universit y of Gda´ nsk Adam.Kw ela@ug.edu.pl

Are scales Fréchet?

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