A classification of separable Rosenthal compacta and its applications

A CLASSIFICA TION OF SEP ARABLE ROSENTHAL COMP A CT A AND ITS APPLICA TIONS S. A. ARGYR OS, P . DODOS AND V. KANELLOPOULOS Contents 1. Int ro duction 2 2. Ramsey prop erties of p erfect s e ts and of subtrees of the Cantor tree 8 2.1. Notations 8 2.2. Partitions of trees 9 2.3. Partitions of p erfect sets 11 3. Increasing and de c r easing antic hains of a regular dyadic tree 11 4. Canonicalizing sequential compactness of trees of functions 14 4.1. Sequen tial compa ctness o f tr e es of functions 14 4.2. Equiv alence of families of functions 18 4.3. Sev en families of functions 20 4.4. Canonicalization 24 5. Analytic subspaces of separable Rosenthal co mpacta 27 5.1. Deﬁnitions and basic prop erties 27 5.2. Separable Rosenthal compacta in B 1 (2 N ) 29 6. Canonical em b eddings in analytic subspaces 32 6.1. Metrizable Rosenthal compacta 33 6.2. Non-metrizable separable Rosenthal compacta 34 7. Non- G δ po in ts in ana lytic subspaces 39 7.1. Kraw czyk trees 40 7.2. The em b edding o f ˆ A (2 N ) in ana lytic subspaces 42 8. Connections with Banach s pace Theory 48 8.1. Existence of unconditional families 48 8.2. Spreading and level unconditional tree bases 51 References 53 1 2000 Ma thematics Subje ct Classiﬁc ation : 03E15, 05C05, 05D10, 46B03, 46B26, 54C35, 54D30, 54D55. 1 2 S. A. AR GYR OS, P . DODOS AND V. KANELLOP OU LOS 1. Introduction The theory of Rose n thal compa cta, namely of co mpact subsets of the ﬁrst Baire class o n a Polish spac e X , w as initiated with the pioneering work o f H. P . Rosenthal [Ro2]. Signiﬁcant contribution of many resea rc hers coming from divergent area s has revealed the deep structural prop erties of this cla ss. Our aim is to study some as pects of separa ble Rosenthal c o mpacta, a s well as , to present some of their applications. The presen t w ork consists of three par ts. In the ﬁrst one we determine the prototypes of separable Ro s en thal co mpacta and we provide a classiﬁca tion theo- rem. The s e cond par t concer ns an extension of a theorem of S. T o dorˇ cevi´ c included in his profound study of Ro sen thal compacta [T o1]. The last one is devoted to applications. Our res ults, concerning the ﬁrst part, are ma inly included in Theorems 2 and 3 below. Roughly sp eaking, we asser t that there exist seven separ able Rosenthal compacta such that every K in the s ame clas s contains one of them in a very canonical w ay . W e start with the following. Deﬁnition 1. (a) L et I b e a c ountable set and X , Y b e Polish sp ac es. L et { f i } i ∈ I and { g i } i ∈ I b e two p ointwise b ounde d families of r e al-value d functions on X and Y r esp e ctively, indexe d by the set I . We say that { f i } i ∈ I and { g i } i ∈ I ar e e quivalent if the n atur al map f i 7→ g i is exten d e d to a top olo gic al home omorph ism b etwe en { f i } p i ∈ I and { g i } p i ∈ I . (b) L et X b e a Polish sp ac e and { f t } t ∈ 2 < N b e r elatively c omp act in B 1 ( X ) . We say that { f t } t ∈ 2 < N is minimal if for every dyadic subtr e e S = ( s t ) t ∈ 2 < N of t h e Ca ntor tr e e 2 < N , t he families { f t } t ∈ 2 < N and { f s t } t ∈ 2 < N ar e e quivale nt. Related to the ab o ve notions, the fo llo wing is proved. Theorem 2. (a) Up to e quivalenc e, t h er e ar e exactly seven minimal families. (b) F or every family { f t } t ∈ 2 < N r elatively c omp act in B 1 ( X ) , with X Polish, t her e ex- ists a r e gular dyadic su btr e e S = ( s t ) t ∈ 2 < N of 2 < N such t hat { f s t } t ∈ 2 < N is e quivalent to one of the seven minimal families. F o r any of the seven minimal families the corr e sponding p oin twise c lo sure is a separable Rose nthal compact con taining the family as a discrete set. W e denote them as follows A (2 < N ) , 2 6 N , ˆ S + (2 N ) , ˆ S − (2 N ) , ˆ A (2 N ) , ˆ D (2 N ) , a nd ˆ D  S (2 N )  . The pr ecise description of the families and the corres ponding co mpacta is given in § 4.3. The ﬁrs t tw o in the above list ar e metrizable space s. The next tw o are hered- itarily separ a ble, non-metrizable and mutually homeomorphic (thus, the ab o ve de- ﬁned notion of equiv alence of families is stro nger than saying that the c orresp onding closures a re homeomorphic). Th e space ˆ S + (2 N ), and so the space ˆ S − (2 N ) a s well, A CLASSIFICA TION OF S E P ARABLE ROSENTHAL COMP A CT A 3 can be realized as a clos ed subspac e of the s plit interv al S ( I ). F ollowing [E], we shall denote b y A (2 N ) the o ne po in t compactiﬁca tio n of the Can tor set 2 N . The space ˆ A (2 N ) is the sta nda rd s eparable extension of A (2 N ) (see [Po2], [Ma]). This is the only not ﬁrst countable space from the ab ov e list. The space ˆ D (2 N ) is the separable ex tens io n of the Alexa ndroﬀ duplicate of the Cantor set D (2 N ), as it was describ ed in [T o1]. Finally , the space ˆ D  S (2 N )  can b e re alized a s a closed sub- space of the Helly space. Its accumulation p oin ts is the closure of the standa rd uncountable discrete subset of the Helly space. Theorem 2 is essentially a success of the inﬁnite-dimensional Ramsey Theor y for trees and pe rfect sets. There is a long histor y on the int era ction b et ween Ramsey Theory and Rosenthal co mpacta, which can be tr aced back to the cla ssical J. F ara- hat’s pro of [F] o f H. P . Ros e n thal’s ℓ 1 Theorem [Ro1] a nd its tree e x tension due to J. Ster n [Ste]. This interaction was further expanded by S. T o dorˇ cevi´ c in [T o1] with the use o f the para meter ized Ramse y Theory for p erfect sets. The new Ramsey theoretic ingredient in the pr oof of Theo r em 2 is a result concerning partitions of tw o cla s ses o f antic hains o f the Ca n tor tree, which w e call incr e asing and de cr e asing . W e will brieﬂy co mmen t on the pro of of Theorem 2 and the critical role of this res ult. O ne starts with a family { f t } t ∈ 2 < N relatively compact in B 1 ( X ). A ﬁrst top ological re ductio n shows that in order to unders ta nd the closur e of { f t } t ∈ 2 < N in R X it is enough to de ter mine all subsets of the Cantor tree for which the corresp onding subsequence of { f t } t ∈ 2 < N is p oin twise co nvergen t. A sec o nd reduction shows that it is enough to deter mine only a coﬁna l subset of conv ergent subsequences. One is then led to analyze which classes o f subsets of the Cantor tree are Ramsey and coﬁnal. First, w e observe tha t every inﬁnite subs et of 2 < N either contains an inﬁnite chain, o r an inﬁnite a n tic hain. It is well-known, and g oes bac k to Stern, that chains are Ramsey . On the other hand, the set of all antic hains is not. Ho wev er, the classes of increas ing and decreasing a n tic hains are Ramsey and, more over, they are coﬁnal in the set of a ll antic hains. Using the ab o ve prop erties of chains and of increas ing and decreas ing antic hains we are a ble to hav e a satisfa ctory control ov er the convergen t subsequences of { f t } t ∈ 2 < N . Fina lly , rep eated a pplications of F. Galvin’s theorem on partitions o f doubletons of p erfect sets of re a ls p ermit us to fully cano nic a lize the to pologica l b e ha vior of { f t } t ∈ 2 < N yielding the pro of of Theorem 2. A dir ect conseq uence of Theorem 2 (b) is that for every sepa r able Rosenthal compact and for e v ery countable dense subset { f t } t ∈ 2 < N o f it, there exists a r egular dyadic subtree S = ( s t ) t ∈ 2 < N such that the p oin twise closure of { f s t } t ∈ 2 < N is home- omorphic to one of the ab o ve describ ed c o mpacta. In ge ner al, for a given c oun table dense s ubs e t { f n } n of a separable Rosenthal compa c t K , w e say that one of the minimal families canonically e m beds into K with resp ect to { f n } n if there e x ists an inc r easing injection φ : 2 < N → N such tha t the family { f φ ( t ) } t ∈ 2 < N is equiv alent 4 S. A. ARGYR OS, P . DODOS AND V. KANELLOPOULOS to it. The next theorem is a supplement of Theo rem 2, showing that the minimal families can b e chosen to characterize certa in top ological prop erties of K . Theorem 3. L et K b e a sep ar abl e Rose nthal c omp act and { f n } n a c ountable dense subset of K . (a) If K c onsists of b ounde d functions in B 1 ( X ) , is metrizable and non-sep ar able in the supr emum norm, then 2 6 N c anonic al ly emb e ds into K with r esp e ct to { f n } n such that its image is norm non-sep ar able. (b) If K is n on-metrizable and her e ditarily sep ar abl e, then either ˆ S + (2 N ) , or ˆ S − (2 N ) c anonic al ly emb e ds into K with r esp e ct to { f n } n . (c) If K is not her e ditarily sep ar abl e and ﬁrst c ountable, then either ˆ D (2 N ) , or ˆ D  S (2 N )  c anonic al ly emb e ds into K with r esp e ct to { f n } n . (d) If K is not ﬁrst c ountable, then ˆ A (2 N ) c anonic al ly emb e ds into K with resp e ct to { f n } n . In p articular, if K is non-metrizable, then one of the non- metrizabl e pr ototyp es c anonic al ly emb e ds into K with r esp e ct to any dense su bset of K . Part (a) is a n extensio n o f the classical Ch. Stegall’s result [St], whic h led to the characterization of the Radon-Niko dym prop ert y in dual Ba nac h spaces. W e men tion that T o dorˇ cevi ´ c [T o 1 ] has shown that in case (b) ab o ve the split in terv al S ( I ) em beds in to K . It is an immediate consequenc e of the abov e theor em that every not hereditarily separa ble K contains a n uncountable discrete s ubs pace of the size of the co n tin uum, a r esult due to R. Pol [Po1]. The pro ofs of par ts (a), (b) and (c) use v ar ian ts o f Stegall’s fundamen tal constr uc tio n, similar in spirit as in the work of G. Go defroy and M. T alag rand [GT]. Part (d) is a co nsequence o f a more general structural re s ult concerning non- G δ po in ts which we are ab out to describ e. T o this end, we star t with the following. Deﬁnition 4. L et K b e a sep ar able R osenthal c omp act on a Polish sp ac e X and C a close d subsp ac e of K . We say that C is an analytic subsp ac e if ther e ex ist a c ountable dense subset { f n } n of K and an analytic subset A of [ N ] such that the fol lowing ar e satisﬁe d. (1) F or every L ∈ A t he ac cumulation p oints of the set { f n : n ∈ L } in R X is a su bset of C . (2) F or every g ∈ C which is an ac cumulation p oint of K ther e exists L ∈ A with g ∈ { f n } p n ∈ L . Observe that ev ery s eparable Ro s en thal co mpact K is an analytic subspace of itself with resp ect to any countable dense set. Let us p oint out that while the class of analytic subspaces is strictly wider than the clas s of separable ones , it sha res all the structural prop erties o f the separ able Rosenthal co mpacta. This will b ecome clear in the s equel. A CLASSIFICA TION OF S E P ARABLE ROSENTHAL COMP A CT A 5 A natur al question ra ised by the ab o ve deﬁnition is whether the concept o f an analytic s ubs pace dep ends o n the choice of the coun table dense s ubset of K . W e belie v e that it is indep enden t. This is supp orted b y the fact tha t it is indeed the case for ana lytic subspa c es of separ a ble Ros e n thal compacta in B 1 ( X ) with X compact metrizable. T o state o ur results concer ning ana lytic subspaces, we also need the following. Deﬁnition 5. L et K b e a sep ar able R osenthal c omp act, { f n } n a c ountable d ense subset o f K and C a close d subsp ac e of K . We say that one of the pr ototyp es K i (1 ≤ i ≤ 7) c anonic ally emb e ds into K with r esp e ct to { f n } n and C , if ther e exist s a subfamily { f t } t ∈ 2 < N of { f n } n which is e quivalent t o t he c anonic al dense family of K i and su ch that al l ac cumulation p oints of { f t } t ∈ 2 < N ar e in C . The following theorem de s cribes the str ucture of not ﬁrst c o un table ana lytic subspaces. Theorem 6. L et K b e a sep ar able R osenthal c omp act, C an analytic subsp ac e of K and { f n } n a c ountable dense s ubset of K witnessing the analyticity of C . L et also f ∈ C b e a non- G δ p oi nt of C . Then ˆ A (2 N ) c anonic al ly emb e ds int o K with re sp e ct to { f n } n and C and s u ch that f is the unique non- G δ p oi nt of its image. Theorem 6 is the last step of a se ries of results initiated by a fruitful pro blem concerning the character of p oint s in separable Ro sen thal compacta, p osed by R. Pol [Po1]. The ﬁrst decisive step tow ards the so lution of this problem was made by A. Kraw czyk [Kr]. He proved that a p oin t f ∈ K is non- G δ if and only if the set L f ,f = { L ∈ [ N ] : ( f n ) n ∈ L is p oin t wise conv ergent to f } is co-analy tic no n-Borel. His analysis revealed a fundamental co nstruction, which we call Kr awczyk tr e e ( K -tree) with resp ect to the given po in t f and any count able dense s ubs e t f = { f n } n of K . He actually show ed that there exists a subfamily { f t } t ∈ N < N of { f n } n such that the following are fulﬁlled. (P1) F or every σ ∈ N N , f / ∈ { f σ | n } p n . (P2) If A ⊆ N < N is such that f / ∈ { f t } p t ∈ A , then for n ∈ N there e x ist t 0 , ..., t k ∈ N n such tha t A is almos t included in the set of the successo r s of the t i ’s. Using K -trees, the seco nd na med author has shown that the set L f = { L ∈ [ N ] : ( f n ) n ∈ L is p oin t wise conv ergent } is complete co-a nalytic if there exists a non- G δ po in t f ∈ K ([Do ]). Let a lso p oin t out that the deep e ﬀ ective version of G. Debs’ theorem [De] yields that for any separable Rosenthal co mpact the set L f contains a Borel coﬁnal subset. There are str ong evidences, a s Debs’ theo r em mentioned a bov e, that separ able Rosenthal compacta ar e deﬁna ble o b jects, hence, they a re natura lly connected to descriptive s e t theo r y (see also [ADK1 ], [B], [Do]). One of the ﬁrst re sults illustrat- ing this co nnection was prov ed in the late 70’s by G. Go defroy [Go], asserting that 6 S. A. ARGYR OS, P . DODOS AND V. KANELLOPOULOS a sepa r able compact K is Rosenthal if a nd only if C ( K ) is an ana lytic subset of R D for every countable dens e subset D of K . Related to this, R. Pol has conjectured that a separa ble Ro s en thal compact K embeds int o B 1 (2 N ) if and only if C ( K ) is a Borel subset of R D (see [Ma] and [Po2]) . It is worth mentioning that for a separable K in B 1 (2 N ), for every countable dense subset { f n } n of K and every f ∈ K , there exists a Bo r el co ﬁnal s ubset of the co r responding set L f ,f , a prop ert y not shared by a ll separable Rosenthal compacta . The ﬁna l step to the solution of Pol’s problem was made by S. T o dorˇ cevi ´ c [T o1]. He prov ed that if f is a non- G δ po in t of K , then the space A (2 N ) is homeomorphic to a closed subset o f K with f as the unique limit p oin t. His remark able pro of inv olves metamathematical arguments like for cing metho d and abso lutenes s. Let us pro ceed to a discussion on the pr oof of Theo rem 6. The ﬁrst decisive step is the following theorem, concerning the existence of K -trees. Theorem 7. L et K , C , { f n } n and f ∈ C b e as in The or em 6. Then ther e exists a K -tre e { f t } t ∈ N < N with r esp e ct t o the p oint f and the dense se quenc e { f n } n such that for every σ ∈ N N al l ac cumulation p oints of t he set { f σ | n : n ∈ N } ar e in C . The proof of the ab ov e result is a rather direct extension of the results o f A. Kraw czyk fr om [Kr] and is based o n the k ey pr operty of bi-sequentialit y , established for separable Rosenthal co mpacta by R. Pol [Po3 ]. W e will brieﬂy co mment on some further pr operties of the K -tree { f t } t ∈ N < N obtained b y Theor em 7. T o this end, let us c a ll an an tichain { t n } n of N < N a fan if ther e exist s ∈ N < N and a strictly increasing sequence ( m n ) n in N such that s a m n ⊑ t n for every n ∈ N . Let us a lso say that an antic hain { t n } n conv erges to σ ∈ N N if for every k ∈ N the set { t n } n is almost co ntained in the set of the successor s of σ | k . Pr operty (P2) of K -tree s implies that for every fa n { t n } n of N < N the sequence ( f t n ) n m ust b e p oint wise conv ergent to f . This fact combined with the bi-sequentialit y of separa ble Ro sen thal compacta yields the following. (P3) F or every σ ∈ N N there exists an antic hain { t n } n of N < N which conv erges to σ and such that the sequence ( f t n ) n is p oint wise convergen t to f . In the second crucia l step, we us e the inﬁnite dimensio nal extensio n of Hindman’s theorem, due to K. Milliken [Mil1], to pa ss to a n inﬁnitely splitting subtree T of N < N such that for every σ ∈ [ T ] the co rrespo nding a n tic hain { t n } n , described in prop ert y (P3), is found in a canonica l wa y . W e should p oint out that, althoug h Milliken’s theore m is a result concerning partitions o f block sequences, it can b e also considered as a partition theorem for a certa in c la ss o f inﬁnitely s plitting s ubtrees of N < N . This fact w as ﬁrst realized b y W. Henson, in his alternative pro of of Stern’s theorem (see [O d ]), a nd it is used in the pro of of Theor em 6 in a similar s pirit. The pro of of Theore m 6 is co mpleted b y choos ing an appro priate dyadic subtree S of T and a pplying the canonica lization metho d (Theorem 2) to the family { f s } s ∈ S . A CLASSIFICA TION OF S E P ARABLE ROSENTHAL COMP A CT A 7 The following co nsequence of Theor e m 6 descr ibes the universal pr operty of ˆ A (2 N ) among a ll fundamental prototype s . Corollary 8. L et K b e a non-metrizable s ep ar able R osenthal c omp ac t and D = { f n } n a c ountable dense subset of K . Then the sp ac e ˆ A (2 N ) c anonic ally emb e ds into K − K with r esp e ct to D − D and with the c onstant function 0 as the unique non- G δ p oi nt. W e notice that the a bov e corollar y remains v alid within the cla ss of analytic subspaces. The embedding of ˆ A (2 N ) in an analytic subs pa ce C o f a separa ble Ro sen thal compact K yields unconditional families of elements of C as follows. Theorem 9. L et K b e a sep ar able R osenthal c omp act on a Polish sp ac e X c onsisting of b ounde d functions. L et also C b e an analytic subsp ac e of K having the c onstant function 0 as a non- G δ p oi nt. Then t her e exists a family { f σ : σ ∈ 2 N } in C which is 1-unc onditional in the su pr emum norm, p ointwise discr ete and having 0 as u n iq ue ac cumulation p oint. The pr oof o f Theorem 9 follows by Theorem 6 a nd the “p erfect unconditionality theorem” fo rm [ADK2]. A sec o nd application concerns re pr esen table Banach spaces, a cla ss intro duced in [GT ] and closely related to separa ble Rose n thal compa c ta . Theorem 10. L et X b e a non-s ep ar able r epr esentable Banach sp ac e. Then X ∗ c ontains an u nc onditional family of size | X ∗ | . W e also introduce the co ncept of spreading and level unconditional tree bas e s . This notion is implicitly contained in [ADK2] wher e their existence was established in ev ery separable Banach spac e not con taining ℓ 1 and with no n-separable dual. W e present some extensions o f this result in the framework of separable Ro s en thal compacta. W e pro ceed to discuss how this work is organiz e d. In § 2 , w e set up our no ta tions concerning trees and we pres e nt the Ra msey theor etic preliminaries needed in the rest of the pap er. In the next section we deﬁne a nd study the classes of increasing and decr easing antic hains. The main r esult in § 3 is Theo r em 10 which es ta blishes the Rams e y prop erties of these cla sses. Section 4 is exclusively dev oted to the pr o of of Theorem 2. It consists of four subsections. In the ﬁrst one, w e prov e a theo rem (Theorem 16 in the main text) which is the ﬁrs t s tep tow ards the pro of of Theorem 2. Theorem 16 is a consequence of the Ra msey and structural prop erties o f chains and o f increasing and decreas ing antic hains. In § 4.2, we intro duce the notion of equiv alence of families o f functions and w e provide a criterion for establishing it. As we have alr eady mentioned, in § 4.3 we describ e the seven minimal families. The pro of of Theorem 2 is completed in § 4.4. 8 S. A. ARGYR OS, P . DODOS AND V. KANELLOPOULOS In § 5.1 , we introduce the class of ana lytic subspaces of separa ble Rosenthal compacta and we present some of their prop erties, while in § 5.2 we study separ able Rosenthal compacta in B 1 (2 N ). In § 6 , we present parts (a), (b) and (c) of Theor em 3. Actually , Theorem 3 is prov ed for the wider class of analytic subspaces and within the context o f Deﬁnition 5. The precis e statement is as follows. Theorem 11. L et K b e a sep ar able R osenthal c omp act, C an analytic su bsp ac e of K and { f n } n a c ountable dense s ubset of K witnessing the analyticity of C . (a) If C is metrizabl e in the p ointwise top olo gy, c onsists of b ounde d functions and it is non-sep ar abl e in the su pr emum norm of B 1 ( X ) , then 2 6 N c anon- ic ally emb e ds into K with r esp e ct to { f n } n and C , such t h at its image is norm non-sep ar able. (b) If C is her e ditarily sep ar able and n on-metrizable , t h en either ˆ S + (2 N ) , or ˆ S − (2 N ) c anonic al ly emb e ds into K with r esp e ct to { f n } n and C . (c) If C is not her e ditarily sep ar able and ﬁrst c ountable, then either ˆ D (2 N ) , or ˆ D  S (2 N )  c anonic al ly emb e ds into K with r esp e ct to { f n } n and C . Section 7 is devoted to the study of not ﬁrst countable analy tic subspaces. In § 7.1 we prove Theo rem 7, while § 7 .2 is dev oted to the proo f of Theo r em 6. The ﬁnal section is devoted to applica tions a nd in particular to the pro ofs of Theor em 9 and Theo rem 10. W e thank Stevo T o dor ˇ cevi´ c for his v aluable remark s and comments. 2. Ramsey pr oper ties of perfect sets and of subtrees of the Cantor tree The aim of this section is to pre sen t the Ramsey theoretic preliminar ies needed in the re st of the pap er, as well a s, to set up our nota tio n concer ning tree s. Ramsey Theory for trees was initiated with the fun damental Halpern- L¨ auchli Partition Theorem [HL]. The orig inal proof was using metamathema tical argu- men ts. The pro of av oiding metamathematics was given in [AFK]. Partition theo- rems related to the ones pres en ted in this se ction can b e found in the work of K. Milliken [Mil2 ], A. Blass [Bl] and A. Louveau, S. Shelah and B. V eliˇ cko vi´ c [LSV]. 2.1. Notations. W e let N = { 0 , 1 , 2 , ... } . By [ N ] we denote the set of a ll inﬁnite subsets of N , while for every L ∈ [ N ] b y [ L ] we denote the set o f all inﬁnite s ubsets of L . If k ≥ 1 a nd L ∈ [ N ], then [ L ] k stands for the set o f all ﬁnite subsets of L of cardinality k . A. By 2 < N we deno te the set of all ﬁnite s equences o f 0 ’s and 1 ’s (the empt y sequence is included). W e view 2 < N as a tree equipp ed with the (stric t) partial order ⊏ o f extension. If t ∈ 2 < N , then the length | t | of t is deﬁned to b e the cardinality of the set { s : s ⊏ t } . If s, t ∈ 2 < N , then by s a t we denote their concatenation. Two no des s, t are s aid to b e c omp ar able if either s ⊑ t or t ⊑ s ; otherwise are said to b e inc omp ar able . A s ubset of 2 < N consisting of pairwise A CLASSIFICA TION OF S E P ARABLE ROSENTHAL COMP A CT A 9 comparable no des is said to be a chain while a s ubs et of 2 < N consisting of pa irwise incomparable no des is s aid to be a n antichain . F o r every x ∈ 2 N and every n ≥ 1 we set x | n =  x (0) , ..., x ( n − 1)  ∈ 2 < N while x | 0 = ( ∅ ). F or x, y ∈ (2 < N ∪ 2 N ) with x 6 = y we denote b y x ∧ y the ⊏ -maximal no de t of 2 < N with t ⊑ x and t ⊑ y . Moreov er, we write x ≺ y if w a 0 ⊑ x and w a 1 ⊑ y , where w = x ∧ y . The ordering ≺ restricted on 2 N is the usual lexicogr a phical order ing o f the Cantor set. B. W e view every subset of 2 < N as a su bt r e e with the induced partia l ordering. A subtree T of 2 < N is said to be prune d if for every t ∈ T there exists s ∈ T with t ⊏ s . It is said to be downwar ds close d if for every t ∈ T and e very s ⊏ t w e hav e that s ∈ T . F o r a s ubtr ee T of 2 < N (not necessar ily down w ards closed) we set ˆ T = { s : ∃ t ∈ T with s ⊑ t } . If T is down wards closed, then the b o dy [ T ] of T is the set { x ∈ 2 N : x | n ∈ T ∀ n } . C. Let T b e a (not necessa rily down wards clo sed) subtree of 2 < N . F or every t ∈ T by | t | T we denote the cardinality of the set { s ∈ T : s ⊏ t } and for every n ∈ N we set T ( n ) = { t ∈ T : | t | T = n } . Mor eo ver, for every t 1 , t 2 ∈ T by t 1 ∧ T t 2 we denote the ⊏ -maximal no de w of T such that w ⊑ t 1 and w ⊑ t 2 . Notice that t 1 ∧ T t 2 ⊑ t 1 ∧ t 2 . Given t wo subtree s S and T of 2 < N , w e say tha t S is a r e gular subtr e e of T if S ⊆ T a nd for every n ∈ N there exists m ∈ N such that S ( n ) ⊆ T ( m ). F or a regular subtree T of 2 < N , the level set L T of T is the set { l n : T ( n ) ⊆ 2 l n } ⊆ N . Notice that for every x ∈ [ ˆ T ] a nd every m ∈ N we have that x | m ∈ T if and only if m ∈ L T . Hence, the chains of T ar e natura lly iden tiﬁed with the pro duct [ ˆ T ] × [ L T ]. A pr uned subtree T of 2 < N is said to b e skew if for every n ∈ N there exis ts at most one splitting no de of T in T ( n ) with exactly t wo immediate successo rs in T ; it is said to b e dyadic if every t ∈ T has exa ctly t wo immediate successo r s in T . W e observe that a subtree T of the Ca n tor tree is regular dyadic if there exists a (necessar ily unique) bijection i T : 2 < N → T such that the following are satisﬁed. (1) F o r all t 1 , t 2 ∈ 2 < N we have | t 1 | = | t 2 | if and o nly if | i T ( t 1 ) | T = | i T ( t 2 ) | T . (2) F o r all t 1 , t 2 ∈ 2 < N we hav e t 1 ⊏ t 2 (resp ectiv ely t 1 ≺ t 2 ) if and only if i T ( t 1 ) ⊏ i T ( t 2 ) (resp ectiv ely i T ( t 1 ) ≺ i T ( t 2 )). When we write T = ( s t ) t ∈ 2 < N , where T is a r egular dyadic subtree of 2 < N , we mean that s t = i T ( t ) for all t ∈ 2 < N . Finally we notice the following. I f T is a regula r dyadic subtree o f 2 < N and R is a re gular dyadic subtree of T , then R is a regula r dyadic subtree of 2 < N to o. 2.2. P artitions of trees. W e b egin by r ecalling the following no tion from [Ka]. Deﬁnition 1. L et T b e a skew su bt r e e of 2 < N . We deﬁne f T : N → { 1 , 2 } < N as fol lo ws. F or every n ∈ N , let T ( n ) = { s 0 ≺ ... ≺ s m − 1 } b e the ≺ - incr e asing enumer ation of T ( n ) . We set f T ( n ) = ( e 0 , ..., e m − 1 ) ∈ { 1 , 2 } m , wher e for every i ∈ { 0 , ..., m − 1 } , e i is the c ar dinality of the set of the imme diate suc c essors of s i in T . The function f T wil l b e c al le d the c o de of the tre e T . If f : N → { 1 , 2 } < N is 10 S. A. ARGYR OS, P . DODOS AND V. KANELLOPOULOS a fun ct io n such t ha t ther e exists a s kew tr e e T with f = f T , t hen f wil l b e c al le d a skew t re e c o de. F o r instance, if f T ( n ) = (1) for all n ∈ N , then the tree T is a chain. Also, if f T (0) = (2) and f T ( n ) = (1 , 1) for all n ≥ 1, then T consis ts o f tw o chains. Moreov er, observe that if T and S are tw o skew s ubtrees o f 2 < N with f T = f S , then T and S are is omorphic with resp ect to b oth ≺ a nd ⊏ . If f is a skew tree co de and T is a regular dyadic subtree of 2 < N , then by [ T ] f we deno te the s et o f a ll reg ular skew subtree s of T of co de f . It is easy to see that the set [ T ] f is a Polish subspace of 2 T . Also o bserv e that if R is a reg ular dyadic tree of T , then [ R ] f = [ T ] f ∩ 2 R . W e will ne e d the following theorem, which is a co nsequence of Theor e m 4 6 in [Ka]. Theorem 2. L et T b e a r e gular dyadic subt r e e of 2 < N , f a skew tr e e c o de and A b e an analytic subset of [ T ] f . Then t he r e ex ist s a re gular dyadic su btr e e R of T s uch that either [ R ] f ⊆ A , or [ R ] f ∩ A = ∅ . F o r a re gular dyadic subtree T of 2 < N , denote by [ T ] chains the s et o f all inﬁnite chains o f T . Theor em 2 includes the following result due to J . Stern [Ste], A.W . Miller, S. T o dorˇ cevi´ c [Mi] and J. Pa wliko wski [Pa]. Theorem 3. L et T b e a r e gular dyadic su btr e e of 2 < N and A b e an analytic subset of [ T ] chains . Then ther e exists a r e gular dyadic s u btr e e R of T such that either [ R ] chains ⊆ A , or [ R ] chains ∩ A = ∅ . Theorem 2 will essentially b e a pplied to the following cla sses of skew subtrees. Deﬁnition 4. L et T b e a r e gular dyadic su btr e e of 2 < N . A su btr e e S of T wil l b e c al le d incr e asing (r esp e ctively de cr e asing) if the fol lowing ar e satisﬁe d. (a) S is uniquely r o ot e d, r e gular, skew and prune d. (b) F or every n ∈ N , ther e exists a splitting no de of S in S ( n ) , which is the ≺ - maximum (r esp e ctively ≺ -minimu m ) no de of S ( n ) and it has two imme diate suc c essors in S . The class of incr e asing (r esp e ctively de cr e asing) subtr e es of T wil l b e denote d by [ T ] Incr (r esp e ctively [ T ] Decr ). It is ea sy to se e tha t every incr e asing (respec tiv ely decreasing) subtree is o f ﬁxed co de. Th us Theorem 2 can b e applied to give the following. Corollary 5. L et T b e a r e gular dyadi c subtr e e of 2 < N and A b e an analytic subset of [ T ] Incr . Then ther e exists a r e gular dyadic subtr e e R of T such that either [ R ] Incr ⊆ A , or [ R ] Incr ∩ A = ∅ . Similarly for the c ase of [ T ] Decr . The a bov e c o rollary may b e co nsidered as a parameter ized version o f the Louveau- Shelah-V eliˇ ck ovi ´ c theorem [LSV]. A CLASSIFICA TION OF S E P ARABLE ROSENTHAL COMP A CT A 11 2.3. P artitions of p erfect se ts . F o r every subset X o f 2 N , by [ X ] 2 we denote the set o f all doubletons of X . W e ident ify [ X ] 2 with the set of all ( σ , τ ) ∈ X 2 with σ ≺ τ . W e will need the following partition theorem due to F. Galvin (see [Ke], Theorem 1 9.7). Theorem 6. L et P b e a p erfe ct su bset of 2 N . If A is a subset of [ P ] 2 with the Bair e pr op erty, then ther e exist s a p erfe ct subset Q of P such that either [ Q ] 2 ⊆ A , or [ Q ] 2 ∩ A = ∅ . 3. Increasing and decreasing antichains of a regular dy adic tree In this section we deﬁne the incr easing and decreas ing antic hains and we es tablish their fundamental Ramsey prop erties. As we have a lready seen in § 2 the class o f inﬁnite chains o f the Cantor tree is Ramsey . On the other hand an analog ue of Theor e m 3 for inﬁnite antic hains is not v a lid. F or instance, co lor a n antic hain ( t n ) n of 2 < N red if t 0 ≺ t 1 ; otherwise color it blue. It is easy to s ee that this is an o pen partition, yet there is no dyadic subtree of 2 < N all of who s e an tichains are mono chromatic. So, it is necess ary , in or der to hav e a Ramsey result for an tichains, to restrict our a ttention to those which are monotone with respect to ≺ . Still, how ev er, this is not enough. T o see this, consider the s e t o f a ll ≺ -increasing ant ichains and color such a n antic hain ( t n ) n red if | t 0 | ≤ | t 1 ∧ t 2 | ; otherwise co lor it blue. Again w e s ee that this is a n open partition which is not Ra ms ey . The following deﬁnition inco rpora tes all the r estrictions indicated by the ab o ve discussion and whic h are, as we shall see, ess e ntially the only obsta cles to a Ra msey result for ant ichains. Deﬁnition 7. L et T b e a r e gu la r dyadic subtr e e of the Cantor tr e e 2 < N . An inﬁnite antichain ( t n ) n of T wil l b e c al le d incr e asi ng if the fol lowing c onditio ns ar e satisﬁe d. (1) F or al l n, m ∈ N with n < m , | t n | T < | t m | T . (2) F or al l n, m, l ∈ N with n < m < l , | t n | T ≤ | t m ∧ T t l | T . (3I) F or al l n, m ∈ N with n < m , t n ≺ t m . The set of al l incr e asing antichains of T wil l b e denote d by Incr( T ) . Similarly, an inﬁnite antichain ( t n ) n of T wil l b e c al le d de cr e asing if (1) and (2) ab ove ar e satisﬁe d and (3I) is r eplac e d by the fol lowing. (3D) F or al l n, m ∈ N with n < m , t m ≺ t n . The set of al l de cr e asing antichains of T wil l b e denote d by Decr( T ) . The classes of increasing and decreasing ant ichains of T hav e the following cr ucial stability prop erties. Lemma 8. L et T b e a r e gular dyadic subtr e e of 2 < N . Then the fol lowing hold. 12 S. A. ARGYR OS, P . DODOS AND V. KANELLOPOULOS (1) (Her e di tariness) L et ( t n ) n ∈ Incr( T ) and L = { l 0 < l 1 < ... } b e an inﬁnite subset of N . Then ( t l n ) n ∈ Incr( T ) . Similarly, if ( t n ) n ∈ D ecr( T ) , then ( t l n ) n ∈ Decr( T ) . (2) (Coﬁnality) L et ( t n ) n b e an inﬁnite antichain of T . Then ther e exists L = { l 0 < l 1 < ... } ∈ [ N ] su ch that either ( t l n ) n ∈ Incr( T ) or ( t l n ) n ∈ Decr( T ) . (3) (Coher enc e) We have Incr( T ) = Incr(2 < N ) ∩ 2 T and similarly for the de- cr e asing ant ichains. Pr o of. (1) It is s traigh tforward. (2) The point is that all three prop erties in the deﬁnition of increasing and de- creasing antic hains are co ﬁna l in the s e t o f all a n tic hains of T . Indeed, let ( t n ) n be a n inﬁnite antic hain of T . Clear ly there exis ts N ∈ [ N ] s uch that the sequence  | t n | T  n ∈ N is str ic tly incr easing. Moreov er, by Ramsey’s theor e m, there e xists M ∈ [ N ] such that the s e quence ( t n ) n ∈ M is either ≺ - increasing or ≺ -decreasing. Finally , to see that condition (2) in Deﬁnition 7 is coﬁnal, let A =  ( n, m, l ) ∈ [ M ] 3 : | t n | T ≤ | t m ∧ T t l | T  By Ramsey’s Theorem aga in, there ex ists L ∈ [ M ] such that either [ L ] 3 ⊆ A o r [ L ] 3 ∩ A = ∅ . W e claim that [ L ] 3 ⊆ A , which clea r ly co mpletes the pr o of. Assume not, i.e. [ L ] 3 ∩ A = ∅ . Let n = min L and L ′ = L \ { n } ∈ [ L ]. Let a lso k = | t n | T . Then for every ( m, l ) ∈ [ L ′ ] 2 we hav e that | t m ∧ T t l | T < k . The set { t ∈ T : | t | T < k } is ﬁnite. Hence, by a nother a pplication of Rams ey’s theorem, there exist s ∈ T with | s | T < k and L ′′ ∈ [ L ′ ] such that for every ( m, l ) ∈ [ L ′′ ] 2 we hav e that s = t m ∧ T t l . But this is c learly imp o ssible a s the tree T is dyadic. (3) First we observe the following. As the tree T is regular , for every t, s ∈ T we hav e | t | T < | s | T (resp ectiv ely | t | T = | s | T ) if a nd only if | t | < | s | (resp ectiv ely | t | = | s | ). Now, let ( t n ) n ∈ Incr( T ). In or der to show that ( t n ) n ∈ Incr(2 < N ) ∩ 2 T it is enough to prove that for every n < m < l we hav e | t n | ≤ | t m ∧ t l | . By the ab o ve remarks , we have that | t n | ≤ | t m ∧ T t l | . As t m ∧ T t l ⊑ t m ∧ t l , we a re done. Conv ersely assume that ( t n ) n ∈ Incr(2 < N ) ∩ 2 T . Again it is eno ug h to chec k that condition (2) in Deﬁnition 7 is satisﬁed. So let n < m < l . There exist s m , s l ∈ T with | s m | T = | s l | T = | t n | T , s m ⊑ t m and s l ⊑ t l . W e cla im that s m = s l . Indeed, if not, then | t m ∧ t l | = | s m ∧ s l | < | t n | co n tradicting the fact that the antic hain ( t n ) n is increasing in 2 < N . It follows that t m ∧ T t l ⊒ s m , and s o , | s m | T = | t n | T ≤ | t m ∧ T t l | T , as desired. The pro of for the decrea sing antic hains is identical.  A corollar y of prop erty (3) of Lemma 8 is the following. Corollary 9. L et T b e a r e gular dyadic subt r e e of 2 < N and R a r e gular dyadic subtr e e of T . Then Incr ( R ) = Incr( T ) ∩ 2 R and Decr( R ) = Decr( T ) ∩ 2 R . A CLASSIFICA TION OF S E P ARABLE ROSENTHAL COMP A CT A 13 W e notice that for every reg ular dyadic subtree T of the Cantor tree 2 < N the sets Incr( T ) and Decr( T ) are Polish subspaces of 2 T . The main result of this section is the following. Theorem 10. L et T b e a r e gular dyadic su btr e e of 2 < N and A b e an analytic subset of Incr( T ) (r esp e ctively of Decr( T ) ). Then ther e exists a r e gu la r dyadic subtr e e R of T such that either Incr( R ) ⊆ A , or Incr( R ) ∩ A = ∅ (r esp e ctively, either Decr( R ) ⊆ A , or Decr ( R ) ∩ A = ∅ ). W e notice that, after a ﬁrst dra ft of the present pap er, S. T o dorˇ cevi´ c informed us that he is als o aw are of the abov e res ult with a pro of ba sed on K. Millik en’s theorem for strong s ubtrees ([T o2]). The pro of o f Theor e m 10 is based on Corolla ry 5. The metho d is to r educe the coloring o f Incr( T ) (resp ectiv ely of Decr( T )) in Theorem 10, to a colo ring of the class [ T ] Incr (resp ectiv ely [ T ] Decr ) of increasing (resp ectiv ely decreas ing) regular subtrees of T (see Deﬁnition 4). T o this end, w e need the following ea sy fact concerning the classes [ T ] Incr and [ T ] Decr . F act 11. L et T b e a r e gular dyadic subt r e e of 2 < N . If S ∈ [ T ] Incr or S ∈ [ T ] Decr , then for every n ∈ N we have | S ( n ) | = n + 1 . As we hav e indicated, the cr uc ia l fact in the present setting is that there is a canonical cor respondence b et w een [ T ] Incr and Incr( T ) (and s imilarly for the de- creasing an tich ains) whic h w e are ab out to describ e. F or every S ∈ [2 < N ] Incr or S ∈ [2 < N ] Decr and every n ∈ N , let { s n 0 ≺ ... ≺ s n n } be the ≺ -incre asing e n umeration of S ( n ). Deﬁne Φ : [2 < N ] Incr → Incr(2 < N ) b y Φ( S ) = ( s n +1 n ) n . It is ea sy to s e e that Φ is a well-deﬁned contin uous map. Resp ectiv ely , deﬁne Ψ : [2 < N ] Decr → Decr (2 < N ) by Ψ( S ) = ( s n +1 1 ) n . Again it is ea s y to see that Ψ is well-deﬁned and contin uous. Lemma 12. Le t T b e a r e gular dyadic subtr e e of 2 < N . Then Φ  [ T ] Incr  = Incr( T ) and Ψ  [ T ] Decr  = Decr( T ) . Pr o of. W e sha ll g iv e the pr oof o nly for the case of increasing s ubtrees. The pro of of the o ther cas e is similar. First, we notice that for every S ∈ [ T ] Incr we hav e Φ( S ) ∈ Incr(2 < N ) ∩ 2 T , and so, by Lemma 8(3) we get that Φ  [ T ] Incr  ⊆ Incr( T ). Conv ersely , let ( t n ) n ∈ Incr( T ). Claim 1. F or every n < m < l we have t n ∧ T t m = t n ∧ T t l . Pr o of of the claim. Let n < m < l . B y condition (2) in Deﬁnition 7, there exists s ∈ T with | s | T = | t n | T and such tha t s ⊑ t m ∧ T t l . Moreov er, o bserv e that t n ≺ s , as t n ≺ t m . It follows that t n ∧ T t m = t n ∧ T s = t n ∧ T t l , as cla imed. ♦ 14 S. A. ARGYR OS, P . DODOS AND V. KANELLOPOULOS F o r every n ∈ N , we s et c n = t n ∧ T t n +1 . Claim 2. F or every n < m we ha ve c n ⊏ c m . That is, the se quenc e ( c n ) n is an inﬁnite chain of T . Pr o of of the claim. Let n < m . By Claim 1, we get that c n and c m are compatible, since c n = t n ∧ T t m and, by deﬁnition, c m = t m ∧ T t m +1 . No w notice that | c n | T < | t n | T ≤ | t m ∧ T t m +1 | T = | c m | T . ♦ F o r every n ≥ 1, let c ′ n be the unique no de of T such that c ′ n ⊑ c n and | c ′ n | T = | t n − 1 | T . W e deﬁne recursively S ∈ [ T ] Incr as follows. W e set S (0) = { c 0 } and S (1) = { t 0 , c ′ 1 } . Assume that S ( n ) = { s n 0 ≺ ... ≺ s n n } has b een deﬁned so a s s n n − 1 = t n − 1 and s n n = c ′ n . F or e very 0 ≤ i ≤ n − 1, we chose no des s n +1 i such that s n i ⊏ s n +1 i and | s n +1 i | T = | t n | T . W e set S ( n + 1 ) = { s n +1 0 ≺ ... ≺ s n +1 n − 1 ≺ t n ≺ c ′ n +1 } . It is easy to chec k that S ∈ [ T ] Incr and tha t Φ( S ) = ( t n ) n . The pro of is completed.  W e a re ready to give the pro of of Theor em 10. Pr o of of The or em 10. Let A b e an analy tic s ubs et of Incr( T ). By Lemma 1 2 , the set B = Φ − 1 ( A ) ∩ [ T ] Incr is an analytic subset o f [ T ] Incr . By Corollar y 5, there exists a regular dyadic subtre e R of T such that either [ R ] Incr ⊆ B or [ R ] Incr ∩ B = ∅ . By Lemma 1 2, the ﬁr st case implies that Incr( R ) = Φ  [ R ] Incr  ⊆ Φ( B ) ⊆ A , while the second that Incr( R ) ∩ A = Φ  [ R ] Incr  ∩ A = ∅ . The pro of for the case of decreasing antic hains is similar .  4. Canonicalizing sequential comp actness of trees of functions The present section consists of four subsections. In the ﬁr s t o ne, using the Ramsey pro perties of c hains and of inc r easing and decreasing antic hains, we prove a str engthening of a res ult of J. Stern [Ste]. In the second one, we introduce the notion of equiv alence of families of functions and we pro vide a cr iterion for establishing it. In the third subsectio n, we deﬁne the seven minima l families. The last subsection is devoted to the pro of of the main r e s ult of the sectio n, concerning the canonical embedding in any separable Rosenthal co mpa ct of one of the minimal families. 4.1. Sequen tial com pa ctness of trees of functions. W e star t with the follow- ing de ﬁnitio n. Deﬁnition 13 . L et L b e an inﬁnite subset of 2 < N and σ ∈ 2 N . We say that L c onver ges to σ if for every k ∈ N the set L is almost include d in the set { t ∈ 2 < N : σ | k ⊑ t } . The element σ wi l l b e c al le d the limit of the set L . We write L → σ to denote that L c onver ges to σ . It is clear that the limit o f a subset L of 2 < N is unique , if it e xists. A CLASSIFICA TION OF S E P ARABLE ROSENTHAL COMP A CT A 15 F act 14. L et ( t n ) n b e an incr e asing (r esp e ctively de cr e asing) antichain of 2 < N . Then ( t n ) n c onver ges to σ , wher e σ is the unique element of 2 N determine d by the chain ( c n ) n with c n = t n ∧ t n +1 (se e the pr o of of L emma 12). W e will a lso need the following notations . Notation 1. F o r every L ⊆ 2 < N inﬁnite and every σ ∈ 2 N we wr ite L ≺ ∗ σ if the set L is a lmost included in the set { t : t ≺ σ } . Resp ectiv ely , we write L  ∗ σ if L is almost included in the set { t : t ≺ σ } ∪ { σ | n : n ∈ N } . The notations σ ≺ ∗ L and σ  ∗ L hav e the obvious mea ning . W e also write L ⊆ ∗ σ if for all but ﬁnitely many t ∈ L we hav e t ⊏ σ , while by L ⊥ σ we mea n tha t the set L ∩ { σ | n : n ∈ N } is ﬁnite. The following fact is essentially a consequence of Lemma 8(2). F act 15. If L is an inﬁnite subset of 2 < N and σ ∈ 2 N ar e such that L → σ and L ≺ ∗ σ (r esp e ctively σ ≺ ∗ L ), t he n every inﬁ n ite subset of L c ontains an incr e asing (r esp e ctively de cr e asing) antichain c onver ging t o σ . The aim of this subsection is to give a pro of of the following result. Theorem 16. L et X b e a Polish sp ac e and { f t } t ∈ 2 < N b e a family r elatively c om- p act in B 1 ( X ) . Then ther e exist a r e gular dyadic subt re e T of 2 < N and a famil y { g 0 σ , g + σ , g − σ : σ ∈ P } , wher e P = [ ˆ T ] , such that for every σ ∈ P the fol lowing ar e satisﬁe d. (1) The se quenc e ( f σ | n ) n ∈ L T c onver ges p ointwise to g 0 σ (r e c al l that L T stands for the level set of T ). (2) F or every se quenc e ( σ n ) n in P c onver ging to σ such that σ n ≺ σ for al l n ∈ N , the se quenc e ( g ε n σ n ) n c onver ges p ointwise t o g + σ for any choic e of ε n ∈ { 0 , + , − } . If su ch a se quenc e ( σ n ) n do es not exist , then g + σ = g 0 σ . (3) F or every se quenc e ( σ n ) n in P c onver ging to σ such that σ ≺ σ n for al l n ∈ N , the se quenc e ( g ε n σ n ) n c onver ges p ointwise to g − σ for any choic e of ε n ∈ { 0 , + , − } . If su ch a se quenc e ( σ n ) n do es not exist , then g − σ = g 0 σ . (4) F or every inﬁn i te subset L of T c onver ging to σ with L ≺ ∗ σ , the se quenc e ( f t ) t ∈ L c onver ges p ointwise t o g + σ . (5) F or every inﬁn i te subset L of T c onver ging to σ with σ ≺ ∗ L , the se quenc e ( f t ) t ∈ L c onver ges p ointwise t o g − σ . Mor e over, the functions 0 , + , − : P × X → R deﬁne d by 0( σ , x ) = g 0 σ ( x ) , +( σ, x ) = g + σ ( x ) , − ( σ, x ) = g − σ ( x ) ar e al l Bor el. Before we pro ceed to the pro of of Theorem 16 we notice the following fact (the pro of of which is left to the reader). 16 S. A. ARGYR OS, P . DODOS AND V. KANELLOPOULOS F act 17. (1) L et A 1 = ( t 1 n ) n and A 2 = ( t 2 n ) n b e two incr e asing (r esp e ctively de- cr e asing) anticha ins o f 2 < N c onver ging to the same σ ∈ 2 N . Then ther e exists an incr e asing (r esp e ctively de cr e asing) antichain ( t n ) n of 2 < N c onver ging to σ such that t 2 n ∈ A 1 and t 2 n +1 ∈ A 2 for every n ∈ N . (2) L et ( σ n ) n b e a se quenc e in 2 N c onver ging to σ ∈ 2 N . F or every n ∈ N , let N n = ( t n k ) k b e a se quenc e in 2 < N c onver ging t o σ n . If σ n ≺ σ (r esp e ctively σ n ≻ σ ) for al l n , then ther e exist an incr e asing (r esp e ctively de cr e asing) ant ichain ( t m ) m and L = { n m : m ∈ N } such that ( t m ) m c onver ges to σ and t m ∈ N n m for every m ∈ N . Pr o of of The or em 16. Our hypotheses imply that for every sequence ( g n ) n belo ng- ing to the closur e of { f t } t ∈ 2 < N in R X , there exists a subsequenc e of ( g n ) n which is po in t wise conv ergent. Consider the following subse t Π 1 of [2 < N ] chains deﬁned by Π 1 =  c ∈ [2 < N ] chains : the sequence ( f t ) t ∈ c is p oint wise convergen t  . Then Π 1 is a co-ana ly tic subset of [2 < N ] chains (see [Ste ]). Applying Theorem 3 and inv oking our hyp o theses, we get a r egular dyadic subtree T 1 of 2 < N such tha t [ T 1 ] chains ⊆ Π 1 . Now consider the subset Π 2 of Incr( T 1 ), deﬁned b y Π 2 =  ( t n ) n ∈ Incr( T 1 ) : the sequence ( f t n ) n is p oint wise convergen t  . Again Π 2 is co-a na lytic (this can b e chec ked with similar arguments as in [Ste]). Applying Theor e m 1 0, we get a regular dyadic subtree T 2 of T 1 such that Incr( T 2 ) ⊆ Π 2 . Finally , applying Theo rem 10 for the decreasing antic hains of T 2 and the co lo r Π 3 =  ( t n ) n ∈ Decr( T 2 ) : the sequence ( f t n ) n is p oint wise convergen t  , we obtain a r e gular dyadic subtree T of T 2 such tha t, setting P = [ ˆ T ], the fo llowing are s a tisﬁed. (i) F or every increasing antic hain ( t n ) n of T , the sequence ( f t n ) n is p oin t wise conv ergent. (ii) F or every decreasing ant ichain ( t n ) n of T , the sequence ( f t n ) n is p oint wise conv ergent. (iii) F or every σ ∈ P , the s equence ( f σ | n ) n ∈ L T is p oint wise convergen t to a function g 0 σ . W e no tice the following. By F act 1 7(1), if ( t 1 n ) n and ( t 2 n ) n are tw o increa sing (resp ectiv ely decreasing ) antic hains of T conv erging to the s ame σ , then ( f t 1 n ) n and ( f t 2 n ) n are b oth p oin twise conv ergent to the same function. F or every σ ∈ P , w e deﬁne g + σ as follows. If there ex is ts an incr easing antic hain ( t n ) n of T conv erging to σ , then we s e t g + σ to b e the point wise limit of ( f t n ) n (b y the ab o ve remarks g + σ is indep enden t of the choice o f ( t n ) n ). Otherwis e w e set g + σ = g 0 σ . Similarly w e deﬁne g − σ to b e the po in t wise limit of ( f t n ) n , with ( t n ) n a decreasing antic hain o f T c o n verging to σ , if such an ant ichain exists. Other wise we set g − σ = g 0 σ . By F a ct 15 a nd the ab o ve discussion, prop erties (i) and (ii) can be strengthened a s follows. A CLASSIFICA TION OF S E P ARABLE ROSENTHAL COMP A CT A 17 (iv) F or every σ ∈ P and every inﬁnite L ⊆ T conv erging to σ with L ≺ ∗ σ , the sequence ( f t ) t ∈ L is p oin t wise conv ergent to g + σ . (v) F or every σ ∈ P and every inﬁnite L ⊆ T conv erging to σ with σ ≺ ∗ L , the sequence ( f t ) t ∈ L is p oin t wise conv ergent to g − σ . W e cla im tha t the tre e T and the family { g 0 σ , g + σ , g − σ : σ ∈ P } ar e as desir e d. First we chec k that pro perties (1)-(5) a re satisﬁed. C le a rly we only have to chec k (2) and (3). W e will prov e only pro perty (2) (the argument is symmetric). W e argue by contradiction. So , assume that ther e exist a se q uence ( σ n ) n in P , σ ∈ P and ε n ∈ { 0 , + , −} such that σ n ≺ σ , ( σ n ) n conv erges to σ while the s equence ( g ε n σ n ) n do es not converge p oin twise to g + σ . Hence there exis t L ∈ [ N ] a nd an op en neighborho o d V of g + σ in R X such that g ε n σ n / ∈ V for all n ∈ L . By deﬁnition, for every n ∈ L we may select a sequence ( t n k ) k in T such that for every n ∈ L the following ho ld. (a) The sequence N n = ( t n k ) k conv erges to σ n . (b) The sequence ( f t n k ) k conv erges p oin twise to g ε n σ n . (c) F or all k ∈ N , we hav e f t n k / ∈ V . (d) The sequence ( σ n ) n ∈ L conv erges to σ a nd σ n ≺ σ . By F a ct 17(2), there exist a diagona l inc r easing a n tic hain ( t m ) m conv erging to σ . By (c) ab o ve, we see tha t ( f t m ) m is not p oint wise convergen t to g + σ . This leads to a contradiction by the deﬁnition o f g + σ . Now we will chec k the Borelnes s of the maps 0 , + and − . Let L T = { l 0 < l 1 < ... } be the increasing en umeration of the le vel set L T of T . F or every n ∈ N deﬁne h n : P × X → R by h n ( σ , x ) = f σ | l n ( x ). Clea rly h n is Bo rel. As for a ll ( σ, x ) ∈ P × X we have 0( σ , x ) = g 0 σ ( x ) = lim n ∈ N h n ( σ , x ) the Borelnes s of 0 is clear. W e will only chec k the Borelness of the function + (the argument for the map − is symmetric). F o r every n ∈ N and every σ ∈ P , let l n ( σ ) b e the lexico graphically minim um of the clos e d s e t { τ ∈ P : σ | l n ⊏ τ } . The function P ∋ σ 7→ l n ( σ ) ∈ P is clear ly co n tin uous. Inv oking the deﬁnitio n of g + σ and prop erty (2) in the statement of the theorem we see that for a ll ( σ, x ) ∈ P × X we have +( σ , x ) = g + σ ( x ) = lim n ∈ N g 0 l n ( σ ) ( x ) = lim n ∈ N 0  l n ( σ ) , x  . Thu s + is Borel to o a nd the pr oof is completed.  Remark 1. W e would like to po in t out that in o rder to apply the Ra msey theor y for tr e es in the present setting one has to know that all the colo r s are suﬃciently deﬁnable. This is also the rea son why the Bo relness of the functions 0 , + and − is emphasized in Theor em 16. As a ma tter of fac t, we will need the full s trength o f the Rams e y theory for tree s and p erfect sets, in the sense that in certain s itua tions the color will b elong to the σ -algebra gener a ted by the ana lytic sets. It should be 18 S. A. ARGYR OS, P . DODOS AND V. KANELLOPOULOS noted that this is in contrast with the cla ssical Silv er’s theorem [Si] for which, most applications, in volv e Bo rel partitions. 4.2. Equiv alence o f families of functions. Le t us give the following deﬁnition. Deﬁnition 18. L et I b e a c ountable set and X , Y b e Polish s p ac es. L et also { f i } i ∈ I and { g i } i ∈ I b e two p ointwise b ounde d families of r e al-value d functions on X and Y r esp e ctively, indexe d by the set I . We say that { f i } i ∈ I is e quivalent to { g i } i ∈ I if the m ap f i 7→ g i is ex tende d t o a top olo gic al home omorph ism b etwe en { f i } p i ∈ I and { g i } p i ∈ I . The equiv alence of the families { f i } i ∈ I and { g i } i ∈ I is stronger than saying that { f i } p i ∈ I is homeomor phic to { g i } p i ∈ I (such an example will b e given in the nex t subsection). The crucial p oin t in Deﬁnition 1 8 is that the equiv alence of { f i } i ∈ I and { g i } i ∈ I gives a natural homeomor phism b et ween their closur e s . The following lemma provides a n e ﬃcien t cr iterion for chec king the equiv alence of families of Borel functions. W e mention that in its pro of we will often mak e use o f the Bour g ain-F remlin-T a lagrand theorem [BFT] without ma k ing an explicit reference. F rom the context it will b e clea r that this is what we use. Lemma 19. L et I b e a c ountable set and X , Y b e Polish sp ac es. Le t K 1 and K 2 b e two sep ar able R osenthal c omp acta on X and Y r esp e ctively. L et { f i } i ∈ I and { g i } i ∈ I b e two dense families of K 1 and K 2 r esp e ctively. Assu me that for every i ∈ I the functions f i and g i ar e isolate d in K 1 and K 2 r esp e ctively. Then the fol lowing ar e e quivale nt. (1) The families { f i } i ∈ I and { g i } i ∈ I ar e e quivalent. (2) F or every L ⊆ I inﬁnite, the se qu enc e ( f i ) i ∈ L c onver ges p ointwise if and only if the se quenc e ( g i ) i ∈ L do es. Pr o of. The direction (1) ⇒ (2 ) is obvious. What remains is to prove the c on verse. So as sume that (2) holds. Le t M ⊆ I inﬁnite. W e set K M 1 = { f i } p i ∈ M and K M 2 = { g i } p i ∈ M . Notice that b oth K M 1 and K M 2 are s eparable Rosenthal compacta. Our assumptions imply that the iso lated p oin ts of K M 1 is prec is ely the set { f i : i ∈ M } and s imilarly for K M 2 . Deﬁne Φ M : K M 1 → K M 2 as follows. First, for every i ∈ M we set Φ M ( f i ) = g i . If h ∈ K M 1 with h / ∈ { f i : i ∈ M } , then there exists L ⊆ M inﬁnite such tha t h is the p oin t wise limit of the sequence ( f i ) i ∈ L . Deﬁne Φ M ( h ) to b e the po in t wise limit of the seq ue nce ( g i ) i ∈ L (b y o ur ass umptions this limit exists). T o simplify notation, let Φ = Φ I . Claim. L et M ⊆ I inﬁnite. Then t he fol lowing hold. (1) The map Φ M is wel l-deﬁne d, 1-1 and onto. (2) We have Φ | K M 1 = Φ M . A CLASSIFICA TION OF S E P ARABLE ROSENTHAL COMP A CT A 19 Pr o of of the claim. (1) Fix M ⊆ I inﬁnite. T o see that Φ M is well-deﬁned, notice that fo r every h ∈ K M 1 with h / ∈ { f i : i ∈ M } and every L 1 , L 2 ⊆ M inﬁnite with h = lim i ∈ L 1 f i = lim i ∈ L 2 f i it holds that lim i ∈ L 1 g i = lim i ∈ L 2 g i . F or if not, we would have that the seq uenc e ( f i ) i ∈ L 1 ∪ L 2 conv erges p oin t wise while the sequence ( g i ) i ∈ L 1 ∪ L 2 do es not, contradicting our ass umpt ions. W e observe the following consequence o f our assumptions a nd the deﬁnition of the map Φ M . F or e very h ∈ K M 1 , the p oin t h is isolated in K M 1 if and only if Φ M ( h ) is is olated in K M 2 . Using this we will show that Φ M is 1 -1. Indeed, let h 1 , h 2 ∈ K M 1 with Φ M ( h 1 ) = Φ M ( h 2 ). Then, either Φ M ( h 1 ) is iso lated in K M 2 or not. In the ﬁrst case, there exists an i 0 ∈ M w ith Φ M ( h 1 ) = g i 0 = Φ M ( h 2 ). Thus, h 1 = f i 0 = h 2 . So, assume that Φ M ( h 1 ) is not iso lated in K m 2 . Hence, neither Φ M ( h 2 ) is. It follows that bo th h 1 and h 2 are not isolated p oin ts of K M 1 . Pick L 1 , L 2 ⊆ M inﬁnite with h 1 = lim i ∈ L 1 f i and h 2 = lim i ∈ L 2 f i . As the seq uence ( g i ) i ∈ L 1 ∪ L 2 is p oin t wise conv ergent to Φ M ( h 1 ) = Φ M ( h 2 ), our assumptions yield that h 1 = lim i ∈ L 1 f i = lim i ∈ L 1 ∪ L 2 f i = lim i ∈ L 2 f i = h 2 which proves that Φ M is 1-1. Finally , to see that Φ M is onto, let w ∈ K M 2 with w / ∈ { g i : i ∈ M } . Let L ⊆ M inﬁnit e with w = lim i ∈ L g i . By our ass umpt ions, the sequence ( f i ) i ∈ L conv erges p oin twise to an h ∈ K M 1 and clearly Φ M ( h ) = w . (2) By similar a rgumen ts as in (1 ). ♦ By the above claim, it is enoug h to show that the map Φ is co n tin uous. Notice that it is enough to show that if ( h n ) n is a sequence in K 1 that co n v erg e s p oin twise to a n h ∈ K 1 , then the seq uence  Φ( h n )  n conv erges to Φ( h ). Assume on the contrary . Hence, there exist a sequence ( h n ) n in K 1 , h ∈ K 1 and w ∈ K 2 such that h = lim n h n , w = lim n Φ( h n ) and w 6 = Φ ( h ). As the map Φ is onto, ther e exists z ∈ K 1 such that z 6 = h and Φ ( z ) = w . Pick x ∈ X a nd ε > 0 such that | h ( x ) − z ( x ) | > ε . As the sequence ( h n ) n conv erges p oin twise to h we may assume that for all n ∈ N we have | h n ( x ) − z ( x ) | > ε . Let M =  i ∈ I : | f i ( x ) − z ( x ) | ≥ ε 2  . Observe the following. (O1) F or all n ∈ N , h n ∈ K M 1 . (O2) z / ∈ K M 1 . By part (2) of the ab o ve cla im a nd (O1 ), we get that Φ( h n ) = Φ M ( h n ) ∈ K M 2 for all n ∈ N and so w ∈ K M 2 . As Φ M is o n to, there exists h ′ ∈ K M 1 such tha t Φ M ( h ′ ) = w . Hence by (O2) and inv oking the claim o nc e mor e, we hav e that z 6 = h ′ while Φ M ( h ′ ) = Φ( h ′ ) = Φ( z ), co n tradicting that Φ is 1-1. The pr oof o f the lemma is completed.  20 S. A. ARGYR OS, P . DODOS AND V. KANELLOPOULOS 4.3. Sev en families of functions. The a im of this subse ction is to describ e seven families { d i t : t ∈ 2 < N } (1 ≤ i ≤ 7) of functions indexed by the Cantor tr ee. F or every i ∈ { 1 , ..., 7 } , the closure of the family { d i t : t ∈ 2 < N } in the p oint wise top ology is a sepa rable Rosenthal co mpa ct K i . Eac h one of them is minimal , namely , for every dyadic (not necessarily regular ) subtree S = ( s t ) t ∈ 2 < N of 2 < N and ev ery i ∈ { 1 , ..., 7 } the families { d i t } t ∈ 2 < N and { d i s t } t ∈ 2 < N are equiv alen t in the sense of Deﬁnition 18. Although the families are m utually no n-equiv alent, the co rrespo nding compacta might be homeomo rphic. In all cases, the family { d i t : t ∈ 2 < N } will b e discrete in its closure. F or any o f the corres p onding compacta K i (1 ≤ i ≤ 7), b y L ( K i ) w e shall denote the set of all inﬁnite subsets L of 2 < N for whic h the s equence ( d i t ) t ∈ L is p oint wise conv erge n t. W e will name the corres p onding compacta (all of them are homeomorphic to closed subspaces of well-known co mpacta – see [AU], [E]) and we will refer to the families of functions as the ca no nical dense sequences of them. W e will use the following notations. If σ ∈ 2 N , then δ σ is the Dirac function a t σ . By x + σ we deno te the ch ara cteristic function of the se t { τ ∈ 2 N : σ  τ } , while b y x − σ the characteristic function of the set { τ ∈ 2 N : σ ≺ τ } . No tice that if t ∈ 2 < N , then t a 0 ∞ ∈ 2 N , and so, the function x + t a 0 ∞ is well-deﬁned. It is useful at this po in t to is olate the following prop ert y of the functions x + σ and x − σ which will justify the notation g + σ and g − σ in Theorem 16. If ( σ n ) n is a sequence in 2 N conv erging to σ with σ n ≺ σ (re s pectively σ ≺ σ n ) for all n ∈ N , then sequence ( x ε n σ n ) n conv erges p o in t wise to x + σ (resp ectiv ely to x − σ ) for any choice o f ε n ∈ { + , − } . By iden tifying the Cantor set with a subset of the unit interv al, we will identif y every σ ∈ 2 N with the real-v alued function o n 2 N which is equal everywhere with σ . Notice that for every t ∈ 2 < N , we hav e t a 0 ∞ ∈ 2 N , and s o, the function t a 0 ∞ is well-deﬁned. F o r every t ∈ 2 < N , v t stands for the characteristic function of the clop en set V t = { σ ∈ 2 N : t ⊏ σ } . By 0 we denote the co nstan t function o n 2 N which is e qual everywhere with zero. W e will als o need to deal with functions on 2 N ⊕ 2 N . In this cas e when we wr ite, for instance, ( δ σ , x + σ ) we mean that this function is the function δ σ on the ﬁr st copy of 2 N while it is the function x + σ on the s econd copy . W e also ﬁx a regular dyadic subtree R = ( s t ) t ∈ 2 < N of 2 < N with the follo wing prop ert y . (Q) F o r every s, s ′ ∈ R , we hav e that s a 0 ∞ 6 = s ′ a 0 ∞ and s a 1 ∞ 6 = s ′ a 1 ∞ . Hence, the set [ ˆ R ] do es not contain the even tually constant s equences. In what follows by P we shall denote the p erfect set [ ˆ R ]. By P + we shall deno te the s ubset of P cons is ting of a ll σ ’s for which there exists an increa sing a n tic hain ( s n ) n of R conv erging to σ in the sens e of Deﬁnition 13. Resp ectiv ely , by P − we shall denote the subse t of P consisting of a ll σ ’s for which there exists a decre asing antic hain ( s n ) n of R conv erging to σ . A CLASSIFICA TION OF S E P ARABLE ROSENTHAL COMP A CT A 21 4.3.1. The Alexandr oﬀ c omp actiﬁc ation of the Cantor tr e e A (2 < N ) . It is the p oin t- wise closure o f the family n 1 | t | + 1 v t : t ∈ 2 < N o . Clearly the spa ce A (2 < N ) is countable co mpact, as the whole family accumulates to 0 . Setting d 1 t = 1 | t | +1 v t for all t ∈ 2 < N , w e see that the family { d 1 t : t ∈ 2 < N } is a dense discrete subset of A (2 < N ). In this case the description of L  A (2 < N )  is trivial as L ∈ L  A (2 < N )  ⇔ L ⊆ 2 < N . 4.3.2. The sp ac e 2 6 N . It is the p oin twise closure of the family { s a 0 ∞ : s ∈ R } . The a ccum ulation p oints of 2 6 N is the set { σ : σ ∈ P } which is cle arly homeomorphic to 2 N . Th us, the space 2 6 N is unco un table co mpact metrizable. Setting d 2 t = s a t 0 ∞ for all t ∈ 2 < N and inv oking prop ert y (Q ) ab o ve, we see that the family { d 2 t : t ∈ 2 < N } is a dense discrete subset o f 2 6 N . The description of L  2 6 N  is g iv en by L ∈ L  2 6 N  ⇔ ∃ σ ∈ 2 N with L → σ . 4.3.3. The extende d split Cantor set ˆ S + (2 N ) . It is the p oin twise c losure of the family { x + s a 0 ∞ : s ∈ R } . Notice that ˆ S + (2 N ) can b e r ealized as a c lo sed subspa ce of the split interv al S ( I ). Thu s, it is hereditarily separable. F o r every σ ∈ P , the function x + σ belo ngs to ˆ S + (2 N ). How ever, for a n element σ ∈ P , the function x − σ belo ngs to ˆ S + (2 N ) if and only if there exis ts a decr easing antic hain ( s n ) n of R converging to σ . Finally observe that the fa mily { x + s a 0 ∞ : s ∈ R } is a discrete subset of ˆ S + (2 N ) (this is essentially a consequence of pr operty (Q) ab o ve). Hence, the accumulation p oin ts of ˆ S + (2 N ) is the set { x + σ : σ ∈ P } ∪ { x − σ : σ ∈ P − } . Setting d 3 t = x + s a t 0 ∞ for all t ∈ 2 < N , we see that the family { d 3 t : t ∈ 2 < N } is a dense discrete subset of ˆ S + (2 N ). Moreover, we have the follo wing description of L  ˆ S + (2 N )  L ∈ L  ˆ S + (2 N )  ⇔ ∃ σ ∈ 2 N with L → σ and (either L  ∗ σ or σ ≺ ∗ L ) . 22 S. A. ARGYR OS, P . DODOS AND V. KANELLOPOULOS 4.3.4. The mirr or image ˆ S − (2 N ) of the extende d split Cantor set. The space ˆ S + (2 N ) has a na tural mirro r imag e ˆ S − (2 N ) which is the po in t wise closure of the set { x − s a 1 ∞ : s ∈ R } . The spaces ˆ S + (2 N ) a nd ˆ S − (2 N ) are homeomor phic. T o se e this, for every t ∈ 2 < N let ¯ t ∈ 2 < N be the ﬁnite sequence obtained by reversing 0 with 1 and 1 with 0 in the ﬁnite sequence t . Deﬁne φ : R → R by φ ( s t ) = s ¯ t for all t ∈ 2 < N . Then it is easy to see that the map ˆ S + (2 N ) ∋ x + s a t 0 ∞ 7→ x − φ ( s t ) a 1 ∞ ∈ ˆ S − (2 N ) is extended to a top ological homeomor phism b et ween ˆ S + (2 N ) and ˆ S − (2 N ). How ever, the cano nical dense sequences in them a re n o t equiv alen t. Notice that for e v ery σ ∈ P the function x − σ belo ngs to ˆ S − (2 N ), while the function x + σ belo ngs to ˆ S − (2 N ) if and only if ther e exists an incr easing a ntich ain ( s n ) n of R c o n v erg ing to σ . It follows that the a ccum ulation p oints of ˆ S − (2 N ) is the se t { x − σ : σ ∈ P } ∪ { x + σ : σ ∈ P + } . As b efore, setting d 4 t = x − s a t 1 ∞ for all t ∈ 2 < N , we see that the fa mily { d 4 t : t ∈ 2 < N } is a dense dis crete subset of L  ˆ S − (2 N )  and moreover L ∈ L  ˆ S − (2 N )  ⇔ ∃ σ ∈ 2 N with L → σ and (either L ≺ ∗ σ or σ  ∗ L ) . 4.3.5. The extende d Alexandr oﬀ c omp actiﬁc ation of the Cantor set ˆ A (2 N ) . The space ˆ A (2 N ) is the p oint wise clos ure of the family { v t : t ∈ 2 < N } . F o r every σ ∈ 2 N the function δ σ belo ngs in ˆ A (2 N ), the family { δ σ : σ ∈ 2 N } is discrete and accumulates to 0. The function 0 is the only no n- G δ po in t of ˆ A (2 N ) and this is witnessed in the most extr e me way . T he accumulation po in ts of ˆ A (2 N ) is the s et { δ σ : σ ∈ 2 N } ∪ { 0 } Setting d 5 t = v t for all t ∈ 2 < N , the family { d 5 t : t ∈ 2 < N } is a dense discre te subset of ˆ A (2 N ) and L ∈ L  ˆ A (2 N )  ⇔ ( ∃ σ ∈ 2 N with L ⊆ ∗ σ ) or ( ∀ σ ∈ 2 N L ⊥ σ ) . 4.3.6. The ex tende d duplic ate of the Cantor set ˆ D (2 N ) . The space ˆ D (2 N ) is the po in t wise closure of the family { ( v t , t a 0 ∞ ) : t ∈ 2 < N } . This is the separ a ble extensio n of the space D (2 N ), as it w as described in [T o 1]. The a ccum ulation p oints of ˆ D (2 N ) is the se t { ( δ σ , σ ) : σ ∈ 2 N } ∪ { (0 , σ ) : σ ∈ 2 N } , A CLASSIFICA TION OF S E P ARABLE ROSENTHAL COMP A CT A 23 which is homeomor phic to the Alexa ndroﬀ duplicate o f the Cantor set. T o dorˇ cevi´ c was the ﬁrs t to rea lize that this cla ssical construction can be represented as a compact subse t of the ﬁrst Baire class. The spa ce ˆ D (2 N ) is not only ﬁrst countable but it is also pr e-metric of degree at most t w o, in the sens e of [T o1]. As in the previous cases, se tting d 6 t = ( v t , t a 0 ∞ ) for every t ∈ 2 < N , w e see that the family { d 6 t : t ∈ 2 < N } is a dense dis c r ete subset of ˆ D (2 N ) and L ∈ L  ˆ D (2 N )  ⇔ ∃ σ ∈ 2 N with L → σ and (either L ⊆ ∗ σ or L ⊥ σ ) . 4.3.7. The ex t ende d duplic ate of the split Cantor set ˆ D  S (2 N )  . It is the p oin t wise closure of the family { ( v s , x + s a 0 ∞ ) : s ∈ R } . The space ˆ D  S (2 N )  is homeo morphic to a subspace o f the Helly spa ce H . T o see this, let { ( a t , b t ) : t ∈ 2 < N } be a family in [0 , 1] 2 such that (i) a t = a t a 0 < b t a 0 < a t a 1 < b t a 1 = b t , a nd (ii) b t − a t ≤ 1 3 | t | for every t ∈ 2 < N . Deﬁne h t : [0 , 1] → [0 , 1] by h t ( x ) =      1 : b t < x, 1 2 : a t ≤ x ≤ b t , 0 : x < a t . It is ea sy to see that the map ˆ D  S (2 N )  ∋ ( v s t , x + s a t 0 ∞ ) 7→ h t ∈ H is extended to a ho meomorphic embedding. It is follows that the space ˆ D  S (2 N )  is ﬁrst countable. W e notice , how ever, that it is no t pre-metric of degr ee at most t wo. As in all prev ious ca ses, we will describ e the accumulation p oin ts of ˆ D  S (2 N )  . First we observe tha t if ( s n ) n is a c hain of R conv erging to σ ∈ P , then the se- quence  ( v s n , x + s n a 0 ∞ )  n is p oin twise co n v ergent to ( δ σ , x + σ ). If ( s n ) n is an incr eas- ing antic hain of R conv erging to σ , then the sequence  ( v s n , x + s n a 0 ∞ )  n is p oin twise conv ergent to (0 , x + σ ), while if it is decreasing, then it is po in t wise convergen t to (0 , x − σ ). Thus, the accumulation p oin ts of ˆ D  S (2 N )  is the s et { ( δ σ , x + σ ) : σ ∈ P } ∪ { (0 , x + σ ) : σ ∈ P + } ∪ { (0 , x − σ ) : σ ∈ P − } . Finally , setting d 7 t = ( v s t , x + s t a 0 ∞ ) for all t ∈ 2 < N , w e see that the family { d 7 t : t ∈ 2 < N } is a dense discrete subset of ˆ D  S (2 N )  . The descriptio n of L  ˆ D  S (2 N )  is given by L ∈ L  ˆ D  S (2 N )  ⇔ ∃ σ ∈ 2 N with L → σ and ( L ≺ ∗ σ or L ⊆ ∗ σ or σ ≺ ∗ L ) . W e close this subsection by noticing the following minimalit y pr operty of the ab o ve describ ed families. 24 S. A. ARGYR OS, P . DODOS AND V. KANELLOPOULOS Prop osition 2 0 . L et { d i t : t ∈ 2 < N } with i ∈ { 1 , ..., 7 } b e one of the seven families of functions and let S = ( s t ) t ∈ 2 < N b e a dya dic (not ne c essarily r e gu la r) subt r e e of 2 < N . Then the family { d i t : t ∈ 2 < N } and the c orr esp onding family { d i s t : t ∈ 2 < N } determine d by t he tr e e S ar e e quivalent. W e also obs e rv e tha t any t wo of the seven families are not equiv alen t. Moreov er, bes ide the case of ˆ S + (2 N ) and ˆ S − (2 N ), the cor responding co mpa cta are not mut ually homeomorphic either. 4.4. Canonicalization. The main result o f this s e ction is the following. Theorem 21 . L et { f t } t ∈ 2 < N b e a family of r e al-value d functions on a Polish sp ac e X which is r elatively c omp act in B 1 ( X ) . L et also { d i t } t ∈ 2 < N (1 ≤ i ≤ 7 ) b e the families describ e d in t h e pr evious subse ction. Then ther e exist a r e gular dyadic subtr e e S = ( s t ) t ∈ 2 < N of 2 < N and i 0 ∈ { 1 , ..., 7 } such that { f s t } t ∈ 2 < N is e quivalent to { d i 0 t } t ∈ 2 < N . Pr o of. The family { f t } t ∈ 2 < N satisﬁe s all hypo theses of Theorem 16. Th us, there exist a r egular dyadic subtree T of 2 < N and a family o f functions { g 0 σ , g + σ , g − σ : σ ∈ P } , with P = [ ˆ T ], as described in Theorem 16. Let also 0 , + and − b e the corres p onding Bo rel functions. W e reca ll that for every subse t X of 2 N we identify the set [ X ] 2 of doubletons o f X with the set of all ( σ, τ ) ∈ X 2 with σ ≺ τ . F or every ε ∈ { 0 , + , −} let A ε,ε = { ( σ 1 , σ 2 ) ∈ [ P ] 2 : g ε σ 1 6 = g ε σ 2 } . Then A ε,ε is a n analy tic s ubset of [ P ] 2 . T o see this, notice that ( σ 1 , σ 2 ) ∈ A ε,ε ⇔ ∃ x ∈ X with g ε σ 1 ( x ) 6 = g ε σ 2 ( x ) ⇔ ∃ x ∈ X with ε ( σ 1 , x ) 6 = ε ( σ 2 , x ) . Inv oking the Bor elness of the functions 0 , + , − we see that A ε,ε is ana lytic, a s desired. Notice that for every Q ⊆ P p e r fect and every ε ∈ { 0 , + , −} , the set A ε,ε ∩ [ Q ] 2 is a nalytic in [ Q ] 2 . Th us, applying Theorem 6 succes siv ely three times, we g et a per fect subset Q 0 of P such that for all ε ∈ { 0 , + , −} we hav e that either [ Q 0 ] 2 ⊆ A ε,ε or A ε,ε ∩ [ Q 0 ] 2 = ∅ . Case 1. A 0 , 0 ∩ [ Q 0 ] 2 = ∅ . Is this case, we hav e that g 0 σ 1 = g 0 σ 2 for all ( σ 1 , σ 2 ) ∈ [ Q 0 ] 2 . Thus, there ex ists a function g such that g 0 σ = g fo r a ll σ ∈ Q 0 . By prop erties (2) a nd (3) in Theo r em 16 and the ho mogeneit y of Q 0 , we see that g + σ = g − σ = g 0 σ = g for all σ ∈ Q 0 . Pick a regula r dyadic subtree S = ( s t ) t ∈ 2 < N of T such that [ ˆ S ] ⊆ Q 0 and f s 6 = g for all s ∈ S . Inv oking pr operties (1), (4) and (5) of Theore m 16 a s well as Lemma 8(2), we s ee that for every inﬁnite subset A of S , the sequence ( f t ) t ∈ A accumulates to g . It follows tha t { f s } p s ∈ S = { f s } s ∈ S ∪ { g } , and s o, { f s t } t ∈ 2 < N is equiv alent to the canonica l dense family of A (2 < N ). A CLASSIFICA TION OF S E P ARABLE ROSENTHAL COMP A CT A 25 Case 2. [ Q 0 ] 2 ⊆ A 0 , 0 . Then for every ( σ 1 , σ 2 ) ∈ [ Q 0 ] 2 we have that g 0 σ 1 6 = g 0 σ 2 . By passing to a further p erfect subset of Q 0 if necessary , w e may also assume that (P1) g 0 σ 6 = f t for every σ ∈ Q 0 and e very t ∈ T . Case 2.1. E ither A + , + ∩ [ Q 0 ] 2 = ∅ , or A − , − ∩ [ Q 0 ] 2 = ∅ . Assume ﬁrst that A + , + ∩ [ Q 0 ] 2 = ∅ . In this c ase we have that there ex ists a function g suc h tha t g + σ = g for all σ ∈ Q 0 . By pr o perty (3) in Theorem 16 and the homogeneity of Q 0 , we must als o have that g − σ = g for a ll σ ∈ Q 0 . This means that A − , − ∩ [ Q 0 ] 2 = ∅ . Thu s, by symmetry , this ca se is e quiv alent to say that A + , + ∩ [ Q 0 ] 2 = ∅ and A − , − ∩ [ Q 0 ] 2 = ∅ . It follows that there exists a function g such that g + σ = g − σ = g for all σ ∈ Q 0 . By pas sing to a further perfect subset of Q 0 if necess ary , we may also assume that g 0 σ 6 = g for a ll σ ∈ Q 0 . W e select a re g ular dyadic subtree S = ( s t ) t ∈ 2 < N of T such that [ ˆ S ] ⊆ Q 0 and f s 6 = g for all s ∈ S . This prop ert y c o m bined with (P1) implies that for every s ∈ S the function f s is is olated in { f s } p s ∈ S . W e claim that { f s t } t ∈ 2 < N is equiv alent to the canonica l dens e family of ˆ A (2 N ). W e will give a detailed ar g umen t whic h will s erv e as a prototype for the other c a ses as well. First, we notice that, by Lemma 1 9 and the description of L  ˆ A (2 N )  , it is enough to show tha t for a subset A of S , the se q uence ( f s ) s ∈ A conv erges p oin twise if and only if either A is a lmost included in a chain, or A do es not contain an inﬁnite chain. F or the if pa rt, we obse rv e that if A is almost contained in a ch ain, then by prop ert y (1) of Theorem 1 6, the sequence ( f s ) s ∈ A is p oint wise co n v erge n t. Ass ume that A do es not contain an inﬁnite chain. Since g + σ = g − σ = g for all σ ∈ Q 0 , we see that for every increasing a nd every decr e a sing antic hain ( s n ) n of S , the seq uence ( f s n ) n conv erges p oin t wise to g . Th us, ( f s ) s ∈ A is p oin twise conv ergent to g . F or the only if part we ar gue by con tradictio n. If ther e e x ist σ 1 6 = σ 2 contained in [ ˆ S ] with A ∩ { σ 1 | n : n ∈ N } and A ∩ { σ 2 | n : n ∈ N } inﬁnite, then the fact that g 0 σ 1 6 = g 0 σ 2 implies that the s equence ( f s ) s ∈ A is not p oint wise convergen t. Finally , if A contains an inﬁnite chain and an inﬁnite antic hain, then the fact that g 0 σ 6 = g for all σ ∈ [ ˆ S ] implies that ( f s ) s ∈ A is no t p oin twise conv ergent to o. Case 2.2 . [ Q 0 ] 2 ⊆ A + , + and [ Q 0 ] 2 ⊆ A − , − . In this case w e hav e that (P2) g ε σ 1 6 = g ε σ 2 for all ( σ 1 , σ 2 ) ∈ [ Q 0 ] 2 and ε ∈ { 0 , + , − } . Moreov er, by passing to a further perfect subset of Q 0 if necess a ry , we may s trengthen (P1) to (P3) g ε σ 6 = f t for all σ ∈ Q 0 , ε ∈ { 0 , + , −} and t ∈ T . Observe that (P3 ) implies the following. F o r every regular dyadic s ubtree S of T with [ ˆ S ] ⊆ Q 0 and every s ∈ S , the function f s is isola ted in the closur e of { f s } s ∈ S in R X . Th us, as in Case 2.1, in what follows Lemma 19 will be a pplicable. F o r every ε 1 , ε 2 ∈ { 0 , + , − } with ε 1 6 = ε 2 let A ε 1 ,ε 2 = { ( σ 1 , σ 2 ) ∈ [ Q 0 ] 2 : g ε 1 σ 1 6 = g ε 2 σ 2 } . 26 S. A. ARGYR OS, P . DODOS AND V. KANELLOPOULOS Then A ε 1 ,ε 2 is a n analy tic subset of [ Q 0 ] 2 . Applying Theorem 6 successively six times, we ﬁnd Q 1 ⊆ Q 0 per fect such tha t for all ε 1 , ε 2 ∈ { 0 , + , − } with ε 1 6 = ε 2 we hav e that either [ Q 1 ] 2 ⊆ A ε 1 ,ε 2 or A ε 1 ,ε 2 ∩ [ Q 1 ] 2 = ∅ . W e cla im tha t for ea c h pair ε 1 , ε 2 the ﬁrst alter nativ e must o ccur. Assume on the contrary that there ex ist ε 1 , ε 2 with ε 1 6 = ε 2 and s uc h that A ε 1 ,ε 2 ∩ [ Q 1 ] 2 = ∅ . Let τ be the lexicogra phical minimum o f Q 1 . Then for every σ , σ ′ ∈ Q 1 with τ ≺ σ ≺ σ ′ we hav e g ε 2 σ = g ε 1 τ = g ε 2 σ ′ which contradicts (P2 ). Summing up, by pa s sing to Q 1 , we have str engthen (P2) to (P4) g ε 1 σ 1 6 = g ε 2 σ 2 for all ( σ 1 , σ 2 ) ∈ [ Q 1 ] 2 and ε 1 , ε 2 ∈ { 0 , + , −} . F o r every ε ∈ { + , −} , deﬁne B 0 ,ε ⊆ Q 1 by B 0 ,ε = { σ ∈ Q 1 : g 0 σ 6 = g ε σ } . It is easy to see that B 0 ,ε is a n analytic subset of Q 1 . Th us, by the class ical p erfect set theo rem, we ﬁnd Q 2 ⊆ Q 1 per fect such that for every ε ∈ { + , −} we have either Q 2 ⊆ B 0 ,ε or B 0 ,ε ∩ Q 2 = ∅ . Case 2.2.a. B 0 , + ∩ Q 2 = ∅ and B 0 , − ∩ Q 2 = ∅ . In this case, for every σ ∈ Q 2 there exis ts a function g σ such tha t g σ = g 0 σ = g + σ = g − σ . More over, g σ 1 6 = g σ 2 for all σ 1 6 = σ 2 in Q 2 , as Q 2 ⊆ Q 1 . In voking pr operties (2) and (3) in T heo rem 16, we s ee that the set { g σ : σ ∈ Q 2 } is homeomor phic to Q 2 . W e select a reg ular dyadic subtree S = ( s t ) t ∈ 2 < N of T suc h that [ ˆ S ] ⊆ Q 2 ⊆ Q 0 . It follows that { f s } p s ∈ S = { f s } s ∈ S ∪ { g σ : σ ∈ [ ˆ S ] } , and so, the family { f s t } t ∈ 2 < N is equiv alent to the canonical dense family o f 2 6 N . Case 2.2.b. B 0 , + ∩ Q 2 = ∅ a nd Q 2 ⊆ B 0 , − . This means that g 0 σ = g + σ and g 0 σ 6 = g − σ for all σ ∈ Q 2 . Let S = ( s t ) t ∈ 2 < N b e a r e g ular dyadic subtree of T s uc h that [ ˆ S ] ⊆ Q 2 ⊆ Q 0 . Inv oking (P3) and the re marks following it, the description of L  ˆ S + (2 N )  and Lemma 19, a rguing precisely as in Cas e 2.1, we see that { f s t } t ∈ 2 < N is e q uiv alent to the canonica l dense family of ˆ S + (2 N ). Case 2.2.c. Q 2 ⊆ B 0 , + and B 0 , − ∩ Q 2 = ∅ . This means that g 0 σ = g − σ and g 0 σ 6 = g + σ for all σ ∈ Q 2 . As in the pr evious cas e, let S = ( s t ) t ∈ 2 < N b e a r egular dyadic subtree of T such that [ ˆ S ] ⊆ Q 2 ⊆ Q 0 . In this case { f s t } t ∈ 2 < N is equiv alen t to ca nonical dense family of the mir ror ima ge ˆ S − (2 N ) of the extended split Cantor set (the a r gumen t is as in Case 2.1). Case 2.2 . d. Q 2 ⊆ B 0 , + and Q 2 ⊆ B 0 , − . In this case we hav e (P5) g 0 σ 6 = g + σ and g 0 σ 6 = g − σ for all σ ∈ Q 2 . A CLASSIFICA TION OF S E P ARABLE ROSENTHAL COMP A CT A 27 Let B + , − = { σ ∈ Q 2 : g + σ 6 = g − σ } Again, B + , − is an analytic subset of Q 2 . Thus there exists Q 3 ⊆ Q 2 per fect such that either Q 3 ⊆ B + , − or Q 3 ∩ B + , − = ∅ . Case 2. 2.d.I. Q 3 ∩ B + , − = ∅ . This means that for every σ ∈ Q 3 there exists a function g σ such that g σ = g + σ = g − σ and g σ 6 = g 0 σ . Moreover, by prop ert y (P4) ab ov e, we hav e that g σ 1 6 = g σ 2 and g 0 σ 1 6 = g 0 σ 2 for all ( σ 1 , σ 2 ) ∈ [ Q 3 ] 2 , as Q 3 ⊆ Q 2 ⊆ Q 1 . Let S = ( s t ) t ∈ 2 < N b e a regular dyadic subtree o f T such that [ ˆ S ] ⊆ Q 3 ⊆ Q 0 . In this case { f s t } t ∈ 2 < N is equiv alent to the canonica l dense family of ˆ D (2 N ). The veriﬁcation is similar to the previo us cases. Case 2.2. d.II. Q 3 ⊆ B + , − . This means that g + σ 6 = g − σ for all σ ∈ Q 3 . Combining this with (P4) and (P5), w e see that g ε 1 σ 1 6 = g ε 2 σ 2 if either ε 1 6 = ε 2 or σ 1 6 = σ 2 . As befo re, let S = ( s t ) t ∈ 2 < N b e a r egular dy adic subtree o f T suc h tha t [ ˆ S ] ⊆ Q 3 ⊆ Q 0 . Then { f s t } t ∈ 2 < N is equiv alen t to the canonical dense family of ˆ D  S (2 N )  . All the ab o ve case s are exhaustive and the pro of is completed.  By Theorem 21 and P ropos ition 20 we get the following cor ollary . Corollary 22. L et X b e a Polish sp ac e and { f t } t ∈ 2 < N b e family of functions r el- atively c omp act in B 1 ( X ) . Then for every r e gular dyad ic su btr e e T of 2 < N ther e exist a r e gular dyadic subt r e e S of T and i 0 ∈ { 1 , ..., 7 } such that for every r e gular dyadic su btr e e R = ( r t ) t ∈ 2 < N of S , the family { f r t } t ∈ 2 < N is e quivalent to { d i 0 t } t ∈ 2 < N . 5. Anal ytic subsp a ces of sep arable Rosenthal comp act a In this section we in tro duce a cla ss o f subspaces of se pa rable Rosenth al compa c ta and we pr esen t so me of their basic prop erties. 5.1. Deﬁnitions and basi c prop erties. Let K be a separable Rosenthal compact on a Polish space X . F or every subset F o f K by Acc ( F ) we denote the set of accumulation p oints of F in R X . W e start with the following deﬁnition. Deﬁnition 23. Le t K b e a sep ar able R osenthal c omp act on a Polish sp ac e X and C a close d subsp ac e of K . We say t ha t C is an analytic s u bsp ac e of K if ther e exist a c ountable dense subset { f n } n of K and an analytic subset A of [ N ] such that the fol lowing ar e satisﬁe d. (1) F or every L ∈ A we have that Acc  { f n : n ∈ L }  ⊆ C . (2) F or every g ∈ C ∩ Acc ( K ) ther e exists L ∈ A with g ∈ { f n } p n ∈ L . 28 S. A. ARGYR OS, P . DODOS AND V. KANELLOPOULOS Let us mak e so me rema r ks concer ning the above notion. First w e no tice that the analytic set A witnes s ing the analyticity of C can always b e assumed to b e hereditary . W e also o bserv e that an a na lytic subspa ce of K is not neces s arily sepa- rable. F or ins tance, if K = ˆ A (2 N ) and C = A (2 N ), then it is easy to see that C is a n analytic subs pa ce of K . The following prop osition g iv es some examples of analytic subspaces. Prop osition 24. L et K b e a sep ar able Rosentha l c omp act. Then the fol lowi ng hold. (1) K is analytic with r esp e ct t o any c ountable dense subset { f n } n of K . (2) Every close d G δ subsp ac e C of K is analytic. (3) Every close d sep ar able subsp ac e C of K is analytic. Pr o of. (1) T ake A = [ N ]. (2) Let ( U k ) k be a sequence of op en subsets o f K with U k +1 ⊆ U k for all k ∈ N a nd such that C = T k U k . Let also { f n } n be a countable dense s ubs et o f K . F or every k ∈ N , let M k = { n ∈ N : f n ∈ U k } . Notice that the seque nc e ( M k ) k is decreasing . Let A ⊆ [ N ] b e deﬁned by L ∈ A ⇔ ∀ k ∈ N ( L ⊆ ∗ M k ) . Clearly A is Bor el. It is easy to see tha t A satisﬁes condition (1) of Deﬁnition 23 for C . T o see that condition (2) is also satisﬁed, le t g ∈ C ∩ Acc ( K ). By the Bourga in- F remlin-T alag rand theorem [BFT], there exists an inﬁnite subset L on N such that g is the p oin t wise limit of the s e q uence ( f n ) n ∈ L . As g ∈ U k for all k ∈ N , we s ee that L ⊆ ∗ M k for all k . Hence the set A witness the analy ticit y of C . (3) Le t D 1 be a co un table dense subs et of K and D 2 a countable dense s ubset of C . Let { f n } n be an e numeration of the s e t D 1 ∪ D 2 and set L = { n ∈ N : f n ∈ D 2 } . Let a lso M = { k ∈ L : f k ∈ Acc ( K ) } and for every k ∈ M select L k ∈ [ N ] such that f k is the po in t wise limit o f the sequence ( f n ) n ∈ L k . Deﬁne A = [ L ] ∪  S k ∈ M [ L k ]  . The countable dense subset { f n } n of K and the set A verify the a nalyticit y of C .  T o pro ceed with o ur disc us sion on the pro p erties of ana lytic subspaces we need some pieces o f notation. Let K b e a se pa rable Rosenthal compac t and f = { f n } n a countable dense subset of K . W e se t L f = { L ∈ [ N ] : ( f n ) n ∈ L is p oint wise conv ergent } . Moreov er, for every accumulation p oint f of K we let L f ,f = { L ∈ [ N ] : ( f n ) n ∈ L is p oint wise convergen t to f } . W e notice that b oth L f and L f ,f are co- analytic. The ﬁrst result r elating the top ological be havior of a p oin t f in K with the descr iptiv e set-theoretic prop erties of the set L f ,f is the result of A. Kr a wczyk fro m [Kr] asserting that a p oint f ∈ K is G δ if and only if the set L f ,f is Borel. Another impo rtan t structural prop ert y is the following consequence of th e eﬀectiv e version of the Bourgain-F remlin-T ala grand theorem, prov ed by G. Debs in [De]. A CLASSIFICA TION OF S E P ARABLE ROSENTHAL COMP A CT A 29 Theorem 25. L et K b e a sep ar able R osenthal c omp act. Then for every c ountable dense subset f = { f n } n of K , t h er e exists a Bor el, her e ditary and c oﬁnal subset C of L f . W e refer the reader to [Do] for an expla nation o f how Debs’ theor em yields the ab o ve res ult. Let K and f = { f n } n as ab o ve. F or every A ⊆ L f we s et K A, f = { g ∈ K : ∃ L ∈ A with g = lim n ∈ L f n } . W e have the following characterization o f a nalytic subspaces which is essentially a consequence of Theorem 25. Prop osition 26. L et K b e a sep ar able R osenthal c omp act and C a close d su bsp ac e of K . Then C is analytic if and only if t he r e exist a c ountable dense subset f = { f n } n of K and a her e ditary and analytic subset A ′ of L f such that K A ′ , f = C ∩ Acc ( K ) . Pr o of. The direction ( ⇐ ) is immediate. Co n v erse ly , a ssume that C is a na lytic and let f = { f n } n and A ⊆ [ N ] v erifying its ana ly ticit y . As we hav e alrea dy remarked, we may assume that A is hereditary . By Theorem 2 5, there exists a Borel, hereditar y and coﬁna l s ubs e t C of L f . W e set A ′ = A ∩ C . W e cla im that A ′ is the desired set. Clea rly A ′ is a he r editary and analytic subset of L f . Also o bserv e that, by condition (1) of Deﬁnition 23, fo r ev ery L ∈ A ′ the sequence ( f n ) n ∈ L m ust b e po in t wise c on vergen t to a function g ∈ C . Hence K A ′ , f ⊆ C ∩ Acc ( K ). Conv ersely let g ∈ C ∩ Acc ( K ). Ther e exists M ∈ A with g ∈ { f n } p n ∈ M . By the Bo urgain- F r emlin-T alagr and theorem, there exists N ∈ [ M ] such that g is the p oin t wise limit of the sequence ( f n ) n ∈ N . Clearly N ∈ L f . As C is coﬁnal in L f , there exists L ∈ [ N ] with L ∈ C . As A is hereditary , we see tha t L ∈ A ∩ C = A ′ . The pro of is completed.  5.2. Separable Rosenthal compacta in B 1 (2 N ) . Let K b e separa ble Rosenthal compact on a Polish space X a nd f = { f n } n a countable dense subset of K . By Theorem 25, there exists a Bor el coﬁnal subse t of L f . The following prop osition shows that if X is compact metrizable, then the glo bal pr o perty of L f (namely that it contains a Bo r el coﬁnal set) is also v alid lo cally . W e no tice that in the argument below w e make use of the Arsenin-Kunugui theorem in a spirit similar as in [Po2]. Prop osition 27. L et X b e a c omp act met ri zable sp ac e, K a sep ar able R osenthal c omp act on X and f = { f n } n a c ountable dense subset of K . Then for every f ∈ K ther e exists an analytic her e ditary subset B of L f ,f which is c oﬁnal in L f ,f . Pr o of. W e apply Theore m 2 5 and we get a her editary , Borel and co ﬁnal subset C of L f . Consider the function Φ : C × X → R deﬁned by Φ( L, x ) = f L ( x ), wher e by f L we denote the p oint wise limit of the seq uence ( f n ) n ∈ L . Then Φ is B o rel. T o see this, for every n ∈ N let Φ n : C × X → R b e deﬁned by Φ n ( L, x ) = f l n ( x ), where l n is the n th element of the incr easing enumeration of L . Clea rly Φ n is Bo r el. As 30 S. A. ARGYR OS, P . DODOS AND V. KANELLOPOULOS Φ( L, x ) = lim n Φ n ( L, x ) for all ( L, x ) ∈ C × X , the B orelness of Φ is shown. F or every m ∈ N de ﬁne P m ⊆ C × X by ( L, x ) ∈ P m ⇔ | f L ( x ) − f ( x ) | > 1 m + 1 ⇔ ( c, x ) ∈ Φ − 1  ( −∞ , − 1 m + 1 ) ∪ ( 1 m + 1 , + ∞ )  . Clearly P m is Borel. F or every L ∈ C the function x 7→ | f L ( x ) − f ( x ) | is Bair e-1. Hence, for every L ∈ C the section ( P m ) L = { x ∈ X : ( c, x ) ∈ P m } of P m at L is F σ , and as X is compact metrizable, it is K σ . By the Arsenin-Kunugui theorem (see [K e], Theorem 35.46 ), the set G m = pro j C P m is Borel. It follows that the se t G = S m G m is a Borel subset of C . Put D = C \ G . Now obser v e that for ev ery L ∈ C we have that L ∈ L f ,f if and only if L / ∈ G . Hence, the se t D is a Bo rel subset o f L f ,f , and as C is co ﬁna l, we get tha t D is coﬁnal in L f ,f . Hence, setting B to b e the hereditary clo sure o f D , we see that B is a s desired.  Remark 2 . (1) W e notice that Prop osition 27 is not v a lid fo r an ar bitrary s e parable Rosenthal compa ct. A counterexample, taken from [P o2] (see also [Ma]), is the following. Let A b e a n analytic non- Borel subset o f 2 N and deno te b y K A the separable Rosen thal co mpact obtained b y restrict every function of ˆ A (2 N ) on A . Clearly the function 0 | A belo ngs to K A and is a non- G δ po in t of K A . It is easy to chec k that, in this case, there do es not exist a Bo r el co ﬁnal subset of L 0 | A . (2) W e should point out that the hereditar y and coﬁna l subset B o f L f ,f , o bta ined by Prop osition 27, can b e chosen to be Bo r el. T o see this, star t with a n ana lytic and coﬁnal s ubs et A 0 of L f ,f . Using Souslin’s separatio n theorem we construct tw o sequences ( B n ) n and ( C n ) n such tha t B n is Borel, C n is the hereditar y closure of B n and A 0 ⊆ B n ⊆ C n ⊆ B n +1 ⊆ L f ,f for all n ∈ N . Setting B = S n B n , w e see that B is as desir ed. The ar gumen ts in the pr o of of Pr oposition 27 can b e used to der ive certain prop erties o f ana lytic subspa ces of separ able Rosenthal c o mpacta. T o state them we need one more piece o f notation. F or a separable Rose nthal co mpact K on a Polish space X , f = { f n } n a co un table dense subset o f K and C a clo sed subspa ce of K we set L f , C = { L ∈ [ N ] : ∃ g ∈ C with g = lim n ∈ L f n } . Clearly L f , C is a subset of L f . Also notice that if C = { f } for so me f ∈ K , then L f , C = L f ,f . Part (1) of the follo wing propo sition extends Prop osition 27 for analytic sub- spaces. T he second part s ho ws that the notion o f an analytic subspace of K is independent of the choice of the dense se q uence, for every separa ble Rosenthal compact K in B 1 (2 N ). A CLASSIFICA TION OF S E P ARABLE ROSENTHAL COMP A CT A 31 Prop osition 28. L et X b e a c omp act metrizable sp ac e, K b e a sep ar able R osenthal c omp act on X and C and analytic subsp ac e of K . L et f = { f n } n b e a c ountable dense subset of K and A ⊆ [ N ] witnessing t he analyticity of C . Then the fol lowing hold. (1) Ther e exists an analytic c oﬁnal subset A 1 of L f , C . (2) F or every c ountable de nse subset g = { g n } n of K ther e exists an analytic subset A 2 of L g such that K A 2 , g = C ∩ Acc ( K ) . Pr o of. (1) By Prop osition 26, there exists a hereditar y and analytic subset A ′ of L f such that K A ′ , f = C ∩ Acc ( K ). Applying Theorem 25, we get a Borel, here dita ry a nd coﬁnal subse t C of L f . As in Prop osition 27, for every L ∈ C by f L we deno te the po in t wise limit of the sequence ( f n ) n ∈ L . Let A ′′ = A ′ ∩ C . Clearly A ′′ is ana lytic and hereditar y . Moreov er, it is ea sy to see that K A ′′ , f = C ∩ Acc ( K ) (i.e. the set A ′′ co des all function in Acc ( K ) ∩ C ). Conside r the following eq uiv alence relatio n ∼ on C , deﬁned by L ∼ M ⇔ f L = f M ⇔ ∀ x ∈ X f L ( x ) = f M ( x ) . W e cla im that ∼ is Borel. T o see this notice tha t the map C × C × X ∋ ( L, M , x ) 7→ | f L ( x ) − f M ( x ) | is Borel (this ca n b e e asily c heck ed arguing as in P ropos ition 27). Moreover, for every ( L, M ) ∈ C × C , the ma p x 7→ | f L ( x ) − f M ( x ) | is Baire-1. Observe that ¬ ( L ∼ M ) ⇔ ∃ x ∈ X ∃ ε > 0 with | f L ( x ) − f M ( x ) | > ε. By the fact that X is compac t metriz able and by the Arsenin- K un ugui theorem we see that ∼ is Bor el. W e set A 1 to b e the ∼ satura tion of A ′′ , i.e. A 1 = { M ∈ C : ∃ L ∈ A ′′ with M ∼ L } . As A ′′ is ana lytic and ∼ is Bore l, we g et that A 1 is ana lytic. As C is coﬁnal, it is easy to chec k that A 1 is coﬁnal in L f , C . Thus, the set A 1 is the desired one. (2) Let C 1 and C 2 be t wo hereditary , Borel subsets of L f and L g coﬁnal in L f and L g resp ectiv ely . By par t (1 ), there exists a hereditary a nd analytic s ubs e t A 1 of L f which is coﬁnal in L f , C . W e s et A ′ 1 = A 1 ∩ C 1 . Consider the following subset S of C 1 × C 2 deﬁned by ( L, M ) ∈ S ⇔ f L = g M ⇔ ∀ x ∈ X f L ( x ) = g M ( x ) where f L denotes the po in t wise limit of the seq uence ( f n ) n ∈ L while g M denotes the po in t wise limit of the sequence ( g n ) n ∈ M . As X is compact metrizable, ar guing a s in par t (1), it is easy to see that S is Borel. W e set A 2 = { M ∈ C 2 : ∃ L ∈ A ′ 1 with ( L, M ) ∈ S } . The se t A 2 is the desired one.  32 S. A. ARGYR OS, P . DODOS AND V. KANELLOPOULOS W e close this subsection with the following prop osition which pr o vides further examples of a nalytic subspaces . Prop osition 29. L et K b e a sep ar able R osenthal c omp act on a Polish sp ac e X . L et also F b e a K σ subset of X . Then the subsp ac e C F = { f ∈ K : f | F = 0 } of K is analytic with r esp e ct to any c ountable dense subset f = { f n } n of K . Pr o of. Let C be a her editary , Bo rel a nd c o ﬁnal subse t o f L f . Let Z be the subset of C × X deﬁned by ( L, x ) ∈ Z ⇔ ( x ∈ F ) and ( ∃ ε > 0 with | f L ( x ) | > ε ) . The set Z is Borel. As F is K σ , w e see that for every L ∈ C the section Z L = { x ∈ X : ( L, x ) ∈ Z } of Z at L is K σ . Thus, setting A = C \ pro j C Z a nd invoking the Arsenin-Kunugui theorem, w e see that the set A witnesses the a nalyticit y of C F with resp ect to { f n } n .  Related to the ab ov e prop ositions a nd the concept of an analytic subspa c e of K , the following q ues tions are op en to us. Pr oblem 1. Is it true that the concept of an ana lytic subspa ce is indep enden t of the choice of the co un table dens e subse t of K ? Mo re precis ely , if C is an a na lytic subspace of a separ a ble Rosenthal co mpact K on a Polish space X and f = { f n } n is a n arbitra r y co un table dense subset of K , do es ther e exists A ⊆ L f analytic with K A, f = C ∩ Acc ( K )? Pr oblem 2. L e t K b e a sepa rable Rosenthal compact on a Polish space X and let B ⊆ X Borel. Is the subspace C B = { f ∈ K : f | B = 0 } a nalytic? 6. Canonical embeddings in anal ytic subsp aces This section is devoted to the ca nonical embedding of the most repres en tative prototype, among the seven minimal families, into a given analytic subspace of a separable Rosenth al compact. The section is divided in to t wo subsections . The ﬁrst subsection c oncerns metr iz a ble Rosenthal compacta a nd the second the non- metrizable ones. W e start with the following deﬁnitions. Deﬁnition 30 . An inje ction φ : 2 < N → N is said to b e c anonic al pr ovi de d that φ ( s ) < φ ( t ) if either | s | < | t | , or | s | = | t | and s ≺ t . By φ 0 we denote the unique c anonic al bije ction b etwe en 2 < N and N . Deﬁnition 31. L et K b e a sep ar able R osenthal c omp act, { f n } n a c ountable dense subset o f K and C a close d subsp ac e of K . L et also { d i t } t ∈ 2 < N (1 ≤ i ≤ 7) b e the c anonic al families describ e d in § 4.3 and let K i (1 ≤ i ≤ 7) b e the c orr esp onding sep ar able R osenthal c omp acta. F or every i ∈ { 1 , ..., 7 } , we say that K i c anonic al ly emb e ds into K with r esp e ct to { f n } n and C if t h er e exists a c anonic al inje ction A CLASSIFICA TION OF S E P ARABLE ROSENTHAL COMP A CT A 33 φ : 2 < N → N such that t he families { d i t } t ∈ 2 < N and { f φ ( t ) } t ∈ 2 < N ar e e quivalent, that is if the map K i ∋ d i t 7→ f φ ( t ) ∈ K is ex tende d t o a home omorph ism b etwe en K i and { f φ ( t ) } p t ∈ 2 < N , and mor e over Acc  { f φ ( t ) : t ∈ 2 < N }  ⊆ C . If C = K , then we simply say that K i c anonic al emb e ds into K with r esp e ct to { f n } n . 6.1. Metrizable Ros en thal compacta. This subsection is devoted to the pr oof of the following theorem. Theorem 32. L et K b e a sep ar able R osenthal c omp act on a Polish sp ac e X c onsist- ing of b ounde d functions. L et also { f n } n b e a c ountable dense subset of K . Assum e that K is metrizable in the p ointwise top olo gy and non- sep ar able in the s upr emum norm of B 1 ( X ) . Th en ther e exists a c anonic al emb e dding of 2 6 N into K with r e- sp e ct to { f n } n whose ac cumulation p oints ar e ε -sep ar ate d in the supr emum n orm for some ε > 0 . In p articular, its image is n on-sep ar able in the supr emum norm. Pr o of. Fix a compatible metric ρ for the p oin twise top ology of K . Our ass umpt ions on K yield that there ex ist ε > 0 and a family Γ = { f ξ : ξ < ω 1 } ⊆ K suc h that Γ is ε -s e parated in the supremum norm a nd e a c h f ξ is a co ndens ation p oint o f the family Γ in the po in t wise topo logy . By recurs ion on the length of ﬁnite sequences in 2 < N we shall construct the following. (C1) A family ( B t ) t ∈ 2 < N of op en subsets of K , (C2) a family ( x t ) t ∈ 2 < N in X , (C3) tw o families ( r t ) t ∈ 2 < N , ( q t ) t ∈ 2 < N of reals and (C4) a canonica l injection φ : 2 < N → N such tha t for every t ∈ 2 < N the following a re satisﬁed. (P1) B t a 0 ∩ B t a 1 = ∅ , B t a 0 ∪ B t a 1 ⊆ B t and ρ − dia m ( B t ) ≤ 1 | t | +1 . (P2) | B t ∩ Γ | = ℵ 1 . (P3) r t < q t and q t − r t > ε . (P4) F or every f ∈ B t a 0 , f ( x t ) < r t , while for every f ∈ B t a 1 , f ( x t ) > q t . (P5) f φ ( t ) ∈ B t . W e set B ( ∅ ) = K and φ  ( ∅ )  = 0. W e c ho ose f , g ∈ Γ and we pick x ∈ X and r , q ∈ R such tha t f ( x ) < r < q < g ( x ) a nd q − r > ε . W e set x ( ∅ ) = x , r ( ∅ ) = r and q ( ∅ ) = q . W e sele c t B (0) , B (1) op en subsets of K such that f ∈ B (0) ⊆ { h ∈ R X : h ( x ( ∅ ) ) < r ( ∅ ) } , g ∈ B (1) ⊆ { h ∈ R X : h ( x ( ∅ ) ) > q ( ∅ ) } , ρ − diam( B (0) ) < 1 2 and ρ − diam( B (1) ) < 1 2 . Let us observe that x ( ∅ ) , r ( ∅ ) , q ( ∅ ) , B (0) and B (1) satisfy prop erties (P1)-(P4 ) ab o ve. Notice also that B (0) , B (1) are unco untable, hence, they intersect the dense set { f n } n at an inﬁnite set. So, we may se le ct φ  ( ∅ )  < φ  (0)  < φ  (1)  satisfying (P5). The g e neral inductive step pro ceeds in a simila r manner assuming that 34 S. A. ARGYR OS, P . DODOS AND V. KANELLOPOULOS (a) for each t ∈ 2 < N with | t | < n − 1 , x t , r t and q t hav e b een chosen, and (b) for each t ∈ 2 < N with | t | < n , B t and φ ( t ) hav e b een c hosen such tha t (P1)-(P5 ) ar e satisﬁed. This completes the recurs iv e construction. Notice that for every σ ∈ 2 N we hav e T n B σ | n = { f σ } . The map 2 N ∋ σ 7→ f σ ∈ K is a homeomorphic embedding. Moreov er, for e very σ ∈ 2 N , the sequence ( f φ ( σ | n ) ) n is p oin twise co n v erge nt to f σ . W e a lso obser v e the following cons equence of prop erties (P3) and (P4 ). If σ < τ ∈ 2 N , then, setting t = σ ∧ τ , we hav e that f σ ( x t ) ≤ r t < q t ≤ f τ ( x t ) and so k f σ − f τ k ∞ > ε . As ther e ar e at most countable many σ ∈ 2 N with f σ ∈ { f n } n , b y passing to a regular dy adic subtree of 2 < N if necessary , we may assume that for e very t ∈ 2 < N the function f φ ( t ) is isolated in { f φ ( t ) } p t ∈ 2 < N . This easily yields that the fa mily { f φ ( t ) } t ∈ 2 < N is equiv alent to the canonical dense family of 2 6 N . The pro of is completed.  6.2. Non-metrizable separable Rosent hal com pac ta. The main results of this subsection are the following. Theorem 33. L et K b e a sep ar abl e R osenthal c omp act on a Polish sp ac e X and let C b e an analytic subsp ac e of K . L et also { f n } n b e a c ountable dense subset of K and A ⊆ [ N ] analytic, witnessing t he analyticity of C . Assu m e that C is not her e ditarily sep ar able. Then either ˆ A (2 N ) , or ˆ D (2 N ) , or ˆ D  S (2 N )  c anonic al ly emb e ds i nto K with r esp e ct to { f n } n and C . In p articular, if K is ﬁrst c ountable and not her e ditarily sep ar able, then either ˆ D (2 N ) , or ˆ D  S (2 N )  c anonic al ly emb e ds into K with r esp e ct t o every c ountable dense subset { f n } n of K . As it is shown in Cor ollary 45, if K is not ﬁrst co un table, then ˆ A (2 N ) canonica lly embeds into K . Theorem 34. L et K b e a s ep ar able R osenthal c omp act on a Pol ish sp ac e X and { f n } n a c ountable dense subset of K . As s u me t h at K is her e ditarily sep ar ab le and non-metrizable. Then either ˆ S + (2 N ) , or ˆ S − (2 N ) c anonic al ly emb e ds into K with r esp e ct t o { f n } n . 6.2.1. Pr o of of The or em 33. The main goal is to pr o ve the following. Prop osition 35. L et K , C and { f n } n b e as in The or em 33. Then t her e ex i sts a c anonic al inje ction ψ : 2 < N → N su ch that, set ting K σ = { f ψ ( σ | n ) } p n \ { f ψ ( σ | n ) } n for al l σ ∈ 2 N , ther e exists an op en subset V σ ⊆ R X with K σ ⊆ V σ ∩ C and such that K τ ∩ V σ = ∅ for every τ ∈ 2 N with τ 6 = σ . Granting P ropos ition 35, we complete the pro of as follows. Let ψ b e the ca no nical injection obtained b y the ab ov e pr oposition a nd deﬁne f t = f ψ ( t ) for all t ∈ 2 < N . W e apply Theorem 21 and we get a r egular dyadic subtree S = ( s t ) t ∈ 2 < N of 2 < N A CLASSIFICA TION OF S E P ARABLE ROSENTHAL COMP A CT A 35 and i 0 ∈ { 1 , ..., 7 } suc h that { f s t } t ∈ 2 < N is equiv alen t to { d i 0 t } t ∈ 2 < N . By Pro p osition 35, we see that the clo s ure of { f s t } t ∈ 2 < N in R X contains an unco untable dis crete set. Th us { f s t } t ∈ 2 < N is eq uiv alent to the ca nonical dense family of either ˆ A (2 N ), or ˆ D (2 N ), or ˆ D  S (2 N )  . Setting φ = ψ ◦ i S we see that φ is a cano nical injection impo sing an embedding of either ˆ A (2 N ), or ˆ D (2 N ), or ˆ D  S (2 N )  int o K with res pect to { f n } n and C . W e pro ceed to the pr oof of P ropos ition 35. By enlar ging the topo logy on X if necessary (see [Ke]), we may a ssume that the functions { f n } n are contin uous. W e may als o assume that the s et A is hereditary . It follows by condition (2) of Deﬁnition 23 and the Bourgain-F r e mlin-T alagr and theorem, that for ev ery g ∈ C ∩ Acc ( K ) there exists L ∈ A such that g is the p oint wise limit o f the sequence ( f n ) n ∈ L . W e ﬁx a contin uous map Φ : N N → [ N ] with Φ( N N ) = A . W e will need the following notation. F or e v ery m ∈ N , y = ( x 1 , ..., x m ) ∈ X m , λ = ( λ 1 , ..., λ m ) ∈ R m and ε > 0 we s e t V ( y , λ, ε ) = { g ∈ R X : λ i − ε < g ( x i ) < λ i + ε ∀ i = 1 , ..., m } By V ( y , λ, ε ) we denote the c lo sure of V ( y , λ, ε ) in R X . Using the fact that C is not hereditarily separable, by recurs io n on countable ordinals w e get (1) m ∈ N , λ = ( λ 1 , ..., λ m ) ∈ Q m and p ositive r a tionals ε and δ , (2) a family Γ = { y ξ = ( x ξ 1 , ..., x ξ m ) : ξ < ω 1 } ⊆ X m , (3) a family { f ξ : ξ < ω 1 } ⊆ C , (4) a family { M ξ : ξ < ω 1 } ⊆ [ N ], and (5) a family { b ξ : ξ < ω 1 } ⊆ N N such tha t for every ξ < ω 1 the following a re satisﬁed. (i) f ξ ∈ Acc ( K ). (ii) f ξ ∈ V ( y ξ , λ, ε ), while for every ζ < ξ w e hav e f ζ / ∈ V ( y ξ , λ, ε + δ ). (iii) y ξ is a condensation point of Γ in X m . (iv) Φ( b ξ ) = M ξ and f ξ is the p oin t wise limit of the sequence ( f n ) n ∈ M ξ . Now, by induction on the length of the ﬁnite sequences in 2 < N we shall construct the following. (C1) A canonical injection ψ : 2 < N → N . (C2) A family ( B t ) t ∈ 2 < N of o pen balls in X m , taken with resp ect to a co mpatible complete metric ρ of X m . (C3) A family (∆ t ) t ∈ 2 < N of uncountable subs e ts of ω 1 . The co nstruction is done so that for every t ∈ 2 < N the following a re satisﬁed. (P1) If t 6 = ( ∅ ), then f ψ ( t ) ∈ V ( y , λ, ε ) for all y ∈ B t . (P2) F or all t ′ , t ∈ 2 < N with | t ′ | = | t | and t ′ 6 = t we have f ψ ( t ) / ∈ V ( y , λ, ε + δ ) for every y ∈ B t ′ . (P3) B t a 0 ∩ B t a 1 = ∅ , B t a 0 ∪ B t a 1 ⊆ B t and ρ − dia m ( B t ) ≤ 1 | t | +1 . 36 S. A. ARGYR OS, P . DODOS AND V. KANELLOPOULOS (P4) ∆ t a 0 ∩ ∆ t a 1 = ∅ and ∆ t a 0 ∪ ∆ t a 1 ⊆ ∆ t . (P5) diam  { b ξ : ξ ∈ ∆ t }  ≤ 1 2 | t | . (P6) { y ξ : ξ ∈ ∆ t } ⊆ B t . (P7) If t 6 = ( ∅ ), then ψ ( t ) ∈ M ξ for every ξ ∈ ∆ t . Assume that the construction has b een car ried out. W e set y σ = T n B σ | n and V σ = V ( y σ , λ, ε + δ 2 ) for all σ ∈ 2 N . Using (P1) and (P2), it is easy to see that K σ ⊆ V σ and K σ ∩ V τ = ∅ if σ 6 = τ . W e only need to check that K σ ⊆ C for every σ ∈ 2 N . So, le t σ ∈ 2 N arbitrar y . W e set M = { ψ ( σ | n ) : n ≥ 1 } ∈ [ N ]. It is eno ug h to s ho w that M ∈ A . F or every k ≥ 1 we select ξ k ∈ ∆ σ | k . By pro perties (P4), (P5) and (P7), the s equence ( b ξ k ) k ≥ 1 conv erges to a unique b ∈ N N and, mo reo ver, ψ ( σ | n ) ∈ M ξ k = Φ( b ξ k ) for every 1 ≤ n ≤ k . By the contin uity of Φ we get that M ξ k → Φ( b ), and so, M ⊆ Φ ( b ). As A is hereditar y , we see that M ∈ A , as des ired. W e pro ceed to the constr uction. W e set ψ  ( ∅ )  = 0, B ( ∅ ) = X m and ∆ ( ∅ ) = ω 1 . Assume tha t for some n ≥ 1 and fo r all t ∈ 2 ( q − p ) / 2 } for a ll σ ∈ 2 N , it is eas y to chec k that ψ 1 , ψ 2 and { V σ : σ ∈ 2 N } sa tisfy all requirements of Prop osition 3 6. W e pro ceed to the construction. W e s et ψ 1  ( ∅ )  = ψ 2  ( ∅ )  = 0 and B ( ∅ ) = X . Assume tha t for some n ≥ 1 and for all t ∈ 2 k } . W e will use the following conse q uence of Milliken’s theorem. Theorem 43. F or every b ∈ B and every analytic subset A of B t h er e exists c ∈ [ b ] such that either [ c ] ⊆ A , or [ c ] ∩ A = ∅ . F o r ev ery b = ( b n ) n ∈ B and every n ∈ N we set i n = S n i =0 b i . W e deﬁne C : B → Σ N and A : B → Σ N by C  ( b n ) n  = ( i 0 , ..., i n , ... ) and A  ( b n ) n  = ( i 0 ∪ b 2 , ..., i 3 n ∪ b 3 n +2 , ... ) . W e notice that for every b ∈ B the s equence C ( b ) is a chain of Σ while A ( b ) is a n antic hain of Σ c on verging, in the sense o f Deﬁnition 13, to σ = S n i n ∈ [Σ]. W e also no tice that the functions C and A a re co n tin uous. Lemma 44. L et { f t } t ∈ Σ b e a Kra wczyk t r e e of f with r esp e ct to { f n } n and C . Then ther e exists a blo ck se quenc e b = ( b n ) n such that for every c ∈ [ b ] the se quenc e ( f t ) t ∈ C ( c ) is p ointwise c onver gent to a function b elonging t o C and diﬀer ent t ha n f , while t h e se quenc e ( f t ) t ∈ A ( c ) is p ointwise c onver gent to f . Pr o of. Let C 1 = { c ∈ B : the se quence ( f t ) t ∈ C ( c ) is p oint wise convergen t } . It is easy to see that C 1 is a co-a nalytic subset of B . By Theorem 43 and the sequential compactness of K , we ﬁnd d ∈ B s uc h that [ d ] is a subset o f C 1 . As we hav e a lr eady rema r k ed, for every blo c k sequence c the sequence C ( c ) is a chain o f A CLASSIFICA TION OF S E P ARABLE ROSENTHAL COMP A CT A 45 Σ. Hence, by Cor ollary 41(1), we s ee that for every c ∈ [ d ] the sequence ( f t ) t ∈ C ( c ) m ust be po in t wise conv ergent to a function b elonging to C a nd diﬀerent than f . Now le t C 2 = { c ∈ [ d ] : the se q uence ( f t ) t ∈ A ( c ) is p oint wise conv ergent to f } . Again b y Millik en’s theorem, there ex ists b = ( b n ) n ∈ [ d ] such that either [ b ] ⊆ C 2 , or [ b ] ∩ C 2 = ∅ . W e claim that [ b ] is subse t of C 2 . It is enough to show that [ b ] ∩ C 2 6 = ∅ . T o this end we argue as follows. Recall that for every l ∈ N we hav e set i l = b 0 ∪ ... ∪ b l . Let A l = { i l ∪ b m : m > l + 1 } ⊆ Σ . As the sequence ( b n ) n is blo c k, b y pro perty (P) ab ov e, we see that the seque nc e ( f t ) t ∈ A l is p oin twise conv ergent to f . By L e mma 42, there exists D ⊆ S l A l such that the sequence ( f t ) t ∈ D is po in t wise conv ergent to f and D ∩ A l 6 = ∅ for inﬁnitely many l . W e may select L = { l 0 < l 1 < ... } , M = { m 0 < m 1 < ... } ∈ [ N ] such that l n + 1 < m n < l n +1 and i l n ∪ b m n ∈ D for all n ∈ N . Now we deﬁne c = ( c n ) n ∈ [ b ] as follows. W e set c 0 = i l 0 , c 1 = b l 0 +1 ∪ ... ∪ b m 0 − 1 and c 2 = b m 0 . F or every n ∈ N with n ≥ 1 let I n = [ m n − 1 + 1 , l n ] a nd J n = [ l n , m n − 1] a nd set c 3 n = [ i ∈ I n b i , c 3 n +1 = [ i ∈ J n b i and c 3 n +2 = b m n . It is easy to see that c ∈ [ b ] and A ( c ) = ( i l n ∪ b m n ) n ⊆ D . Hence, the sequence ( f t ) t ∈ A ( c ) is p oin t wise co nvergen t to f . It follows that [ b ] ∩ C 2 6 = ∅ a nd the pr oof is c o mpleted.  W e a re ready to pro ceed to the pro of o f Theorem 40. Pr o of of The or em 40. Let b = ( b n ) n be the blo ck sequence o bta ined by Le mma 44. If β = ( b n 0 , ..., b n k ) with n 0 < ... < n k is a ﬁnite subsequence of b , then w e let ∪ β = b n 0 ∪ ... ∪ b n k ∈ Σ. Recur siv ely , we shall select a family ( β s ) s ∈ 2 < N such that the following a re satisﬁed. (C1) F o r every s ∈ 2 < N , β s is a ﬁnite subsequence of b . (C2) F o r every s, s ′ ∈ 2 < N we hav e s ⊏ s ′ if and only if β s ⊏ β s ′ . (C3) F o r every s ∈ 2 < N and every c ∈ [ β s a 0 , b ] we hav e ∪ β s a 1 ∈ A ( c ). The construction pro ceeds a s follows. W e set β ( ∅ ) = ( ∅ ). F or e v ery M = { m 0 < m 1 < ... } ∈ [ N ], let b M = ( b m n ) n be the subsequence o f b determined b y M . Assume that for some s ∈ 2 < N the ﬁnite sequence β s has b een deﬁned. W e select M = M s ∈ [ N ] such that β s ⊏ b M . The set A ( b M ) conv erges to the unique branch of Σ determined by the inﬁnite chain C ( b M ). So, we may select a ﬁnite subseq uence β s a 1 with β s ⊏ β s a 1 and such that ∪ β s a 1 ∈ A ( b M ). The function A : [ b ] → Σ N is contin uous. Hence, there exists a ﬁnite subsequence β s a 0 of b with β s a 0 ⊏ b M and such that condition (C3) ab o ve is satis ﬁed. Finally , notice tha t β s a 0 and β s a 1 are inco mparable with res pect to the par tial order ⊏ of extension. 46 S. A. ARGYR OS, P . DODOS AND V. KANELLOPOULOS One can a lso provide a r ecursiv e fo rm ula deﬁning a family ( β s ) s ∈ 2 < N satisfying conditions (C1)-(C3) ab o ve. In particular, set β ( ∅ ) = ( ∅ ), β (0) = ( b 0 , b 1 , b 2 ) and β (1) = ( b 0 , b 2 ). Assume that β s has b e en deﬁned for some s ∈ 2 < N . Let n s = max { n : b n ∈ β s } . If s ends with 0, then w e set β s a 0 = β a s ( b n s +1 , b n s +2 , b n s +3 ) and β s a 1 = β a s ( b n s +1 , b n s +3 ) . If s ends with 1 , then we s et β s a 0 = β a s ( b n s +1 , b n s +2 , b n s +3 , b n s +4 ) and β s a 1 = β a s ( b n s +1 , b n s +2 , b n s +4 ) . It is ea sy to see that, with the ab ove choices, conditions (C1)-(C3) ar e satisﬁed. Having deﬁned the family ( β s ) s ∈ 2 < N for every s ∈ 2 < N we le t t s = ∪ β s ∈ Σ and h s = f t s . Clearly the family { h s } s ∈ 2 < N is a dyadic subtr ee of the Kr a wczyk tree { f t } t ∈ Σ of f with resp ect to { f n } n and C . The bas ic pro perties of the family { h s } s ∈ 2 < N are summarized in the following claim. Claim 1. The fol lowing hold. (1) F or every σ ∈ 2 N the se quenc e ( h σ | n ) n is p ointwise c onver gent to a fun ctio n g σ ∈ C with g σ 6 = f . (2) F or every P ⊆ 2 N p erfe ct the function f b elongs t o t h e closur e of the family { g σ : σ ∈ P } . Pr o of of the claim. (1) Let σ ∈ 2 N and put b σ = S n β σ | n ∈ [ b ]. It is easy to see that the sequence ( t σ | n ) n is a subsequence of the sequence C ( b σ ). So the result follows by Lemma 44. (2) Assume not. Then there exist P ⊆ 2 N per fect and a neig hbo rhoo d V of f in R X such tha t g σ / ∈ V fo r all σ ∈ P . By part (1), for every σ ∈ P there exists n σ ∈ N such that h σ | n / ∈ V for all n ≥ n σ . F o r e v ery n ∈ N le t P n = { σ ∈ P : n σ ≤ n } . Then each P n is a closed subset of P and clear ly P = S n P n . Th us, there exist n 0 ∈ N and Q ⊆ 2 N per fect with Q ⊆ P n 0 . It follows that h σ | n / ∈ V for all σ ∈ Q and n ≥ n 0 . Let τ b e the lexicogra phical minim um of Q . W e may select a sequence ( σ k ) k in Q such that, se tting s k = τ ∧ σ k for all k ∈ N , we have that σ k → τ , τ ≺ σ k and | s k | > n 0 . Notice that s a k 0 ⊏ τ while s a k 1 ⊏ σ k and | s a k 1 | > n 0 . Hence, b y our assumptions o n the set Q and the deﬁnition of { h s } s ∈ 2 < N , we get tha t h s a k 1 = f t s a k 1 / ∈ V for all k ∈ N . (1) W e are ready to derive the contradiction. W e s e t b τ = S n β τ | n ∈ [ b ]. As β s a k 0 ⊏ b τ , by pr operty (C3) in the ab o ve constructio n, we see that t s a k 1 = ∪ β s a k 1 ∈ A ( b τ ) for all k ∈ N . By Lemma 44, the sequence ( f t ) t ∈ A ( b τ ) is p oin twise c o n vergen t to the function f . It follows that the sequence ( f t s a k 1 ) k is also p oin twise conv ergent to f , which cle a rly contradicts (1) ab ov e. The pro of is completed. ♦ A CLASSIFICA TION OF S E P ARABLE ROSENTHAL COMP A CT A 47 W e apply Theorem 21 to the family { h s } s ∈ 2 < N and we get a regula r dyadic subtr ee T = ( s t ) t ∈ 2 < N of 2 < N such that the family { h s t } t ∈ 2 < N is cano nicalized. The main claim is the following. Claim 2. The family { h s t } t ∈ 2 < N is e quivalent to the c anonic al dense family of ˆ A (2 N ) . Pr o of of the claim. In order to prove the claim w e will iso late a proper t y of the whole family { h s } s ∈ 2 < N (pr o perty (Q) b elow). Let S b e an arbitrary regular dyadic subtree of 2 < N . Notice that g σ ∈ { h s } p s ∈ S for every σ ∈ [ ˆ S ]. By pro perty (2) in Claim 1 , we see that the function f b elongs to the p oint wise closur e of { h s } s ∈ S in R X . By the Bourg a in-F remlin-T ala g rand theorem there exists A ⊆ S such that the seq uence ( h s ) s ∈ A is p oin twise co n v ergent to f . By prop erty (1) in Claim 1, we see that A can b e chosen to b e an antic hain conv erg ing to some σ ∈ [ ˆ S ]. As all these facts hold for every r egular dyadic subtree S o f 2 < N we arr iv e to the following prop ert y of the family { h s } s ∈ 2 < N . (Q) F o r every regular dy adic subtree S of 2 < N , there e x ist tw o antic hains A 1 , A 2 of S and σ 1 , σ 2 ∈ [ ˆ S ] with σ 1 6 = σ 2 such that A 1 conv erges to σ 1 , A 2 conv erges to σ 2 while b o th s equences ( h s ) s ∈ A 1 and ( h s ) s ∈ A 2 are p oin twise conv ergent to f . Now let T = ( s t ) t ∈ 2 < N b e the re gular dyadic subtree of 2 < N such that the family { h s t } t ∈ 2 < N is ca nonicalized. Inv oking prop ert y (Q) ab o ve and referring to the de- scription o f the families { d i t : t ∈ 2 < N } (1 ≤ i ≤ 7), we s ee that { h s t } t ∈ 2 < N must be eq uiv alent either to the canonical dense family of A (2 < N ) or the ca nonical dense family of ˆ A (2 N ). By pro perty (1) in Claim 1 , the ﬁr s t c a se is imp ossible. It follows that { h s t } t ∈ 2 < N must b e equiv alen t to the c anonical dense fa mily of ˆ A (2 N ). The claim is proved. ♦ Let T = ( s t ) t ∈ 2 < N and { h s t } t ∈ 2 < N b e as ab ov e. Observe that for every t ∈ 2 < N there exists a unique n t ∈ N with h s t = f n t . Thus, by passing to dyadic subtree of T if necessary a nd inv oking the minimality of the ca nonical dense family o f ˆ A (2 N ), we get that the function 2 < N ∋ t 7→ n t ∈ N is a canonica l injection and that the map ˆ A (2 N ) ∋ v t 7→ f n t ∈ K is e xtended to homeomorphism Φ b etw een ˆ A (2 N ) and { f n t } p t ∈ 2 < N . That this home- omorphism sends 0 to f is an immediate consequence of pro perty (Q) in Cla im 2 ab o ve. Moreov er, b y Claim 1(1), we s ee that Φ( δ σ ) ∈ C for every σ ∈ 2 N . The pro of is c o mpleted.  By Theorem 40 and P ropos ition 24(1) we get the following co r ollary . 48 S. A. ARGYR OS, P . DODOS AND V. KANELLOPOULOS Corollary 45. L et K b e a sep ar able Ro senthal c omp act on a Polish sp ac e X , { f n } n a c ountable dense su bset of K and f ∈ K . If f is a n on- G δ p oi nt of K , then t he r e exists a c anonic al home omorphic emb e dding of ˆ A (2 N ) int o K with r esp e ct to { f n } n which sends 0 to f . After a ﬁr st draft o f the pr esen t pa per, S. T o dorˇ cevi ´ c informed us ([T o3]) that he is aw are of the ab o ve co rollary with a pro of based on his approa c h in [T o1]. W e notice that if K is a non-metriza ble separ able Rosenthal co mpact on a Polish space X , then the constant function 0 is a non- G δ po in t of K − K . Indeed, since K is non-metrizable, fo r every D ⊆ X co un table there exist f , g ∈ K with f 6 = g and such that f | D = g | D . This eas ily yields that 0 is a non- G δ po in t of K − K . By Corolla ry 45, we see that there exists a homeomor phic embedding of ˆ A (2 N ) into K − K with 0 as the unique non- G δ po in t of its image. This fact can be lifted to the class o f analytic subspaces , as follows. Corollary 46. L et K b e a sep ar able R osenthal c omp act and C an analytic subsp ac e of K which is non-metrizable. Le t also D = { f n } n b e a c ountable dense subset of K witnessing the analyticity of C . Then t h er e exists a family { f t } t ∈ 2 < N ⊆ D − D , e quivale nt to the c anonic al dense family of ˆ A (2 N ) , with Acc  f t : t ∈ 2 < N }  ⊆ C − C and su ch that the c onstant funct ion 0 is the u nique non- G δ p oi nt of { f t } p t ∈ 2 < N . Pr o of. Let { g n } n be an en umeration o f the set D − D , which is dense in K − K . It is easy to see that C − C is a na lytic subspace of K − K , witnesse d by the seq uence { g n } n . Moreov er, by the fact that C is non-metrizable, w e get that the consta n t function 0 belong s to C − C and it is a non- G δ po in t of it. By Theo rem 4 0, the result follows.  8. Connections with Banach sp ace Theor y This section is devoted to a pplications, motiv ated by the results obtained in [ADK2], of the embedding of ˆ A (2 N ) in analytic subspaces of separa ble Ros e n thal compacta containing 0 a s a non- G δ po in t. The ﬁrst one concer ns the ex istence of unconditional families. The second deals with spreading and level unco nditional tree bases. 8.1. Existence of unconditi onal fami lies. W e recall that a family { x i } i ∈ I in a Banach spac e X is said to b e 1-unconditiona l, if for every F ⊆ G ⊆ I and every ( a i ) i ∈ G ∈ R G we have    X i ∈ F a i x i    ≤    X i ∈ G a i x i    . W e will need the following refo rm ulation of Theorem 4 in [ADK2], where we also refer the r eader for a pro of. Theorem 47 . L et X b e a Polish sp ac e and { f σ : σ ∈ 2 N } b e a b ounde d family of r e al-va lue d functions on X which is p ointwise discr ete and having the c onstant A CLASSIFICA TION OF S E P ARABLE ROSENTHAL COMP A CT A 49 function 0 as the unique ac cumulation p oint in R X . Assume that the map Φ : 2 N × X → R deﬁne d by Φ( σ, x ) = f σ ( x ) is Bor el. Then ther e exists a p erfe ct subset P of 2 N such that the family { f σ : σ ∈ P } is 1-unc onditional in the supr emum norm. In [ADK2] it is shown that if X is a separ able Banach space not containing ℓ 1 and with non-separ able dual, then X ∗∗ contains an 1-unco nditional fa mily of the size of the contin uum. This result ca n be lifted to the fr a me of separa ble Rosenthal compacta, a s follows. Theorem 4 8. L et K b e a sep ar able Rosenthal c omp act on a Polish sp ac e X . L et also C b e an analytic su bsp ac e of K c onsisting of b ounde d functions. (a) If C c ontains the funct io n 0 as a non- G δ p oi nt, then ther e exists a fam- ily { f σ : σ ∈ 2 N } in C which is 1-u nc onditional in t h e supr emum norm, p oi ntwise discr ete and having 0 as unique ac cumulation p oint. (b) If C is non-metrizable, t he n ther e exists a family { f σ − g σ : σ ∈ 2 N } , wher e f σ , g σ ∈ C for al l σ ∈ 2 N , which is 1-unc onditional in the supr emum norm. Pr o of. (a) Let D = { f n } n be a countable dense subset of K witnessing the ana- lyticity of C . As 0 is a non- G δ po in t o f C , by Theo rem 40 there e x ists a family { f t } t ∈ 2 < N ⊆ D , equiv alen t to the canonical dense family of ˆ A (2 N ), with Acc  { f t : t ∈ 2 < N }  ⊆ C and such that the constant function 0 is the uniq ue non- G δ po in t of { f t } p t ∈ 2 < N . F o r every σ ∈ 2 N let f σ be the po in t wise limit o f the sequence ( f σ | n ) n . Clearly the family { f σ : σ ∈ 2 N } is p oin twise discr ete, having 0 as the unique a ccu- m ulation p oin t. Mo reo ver, it is easy to see that the map Φ : 2 N × X → R deﬁned by Φ( σ, x ) = f σ ( x ) is Borel. By Theorem 47, the result follows. (b) It fo llo ws by Co rollary 46 and Theorem 47.  Actually we can strengthen the prop erties of the family { f σ : σ ∈ 2 N } obta ined by pa rt (a) of Theorem 48 as follows. Theorem 49. L et K b e a s ep ar able R osenthal c omp act on a Pol ish sp ac e X and C b e an analytic subsp ac e of K c onsisting of b ounde d funct io ns. Assu me that C c ontains the function 0 as a non- G δ p oi nt. Then ther e exist a family { ( g σ , x σ ) : σ ∈ 2 N } ⊆ C × X and ε > 0 satisfying | g σ ( x σ ) | > ε , g σ ( x τ ) = 0 if σ 6 = τ and such that the family { g σ : σ ∈ 2 N } is 1-un c onditional in the supr emum norm and having 0 as unique ac cumulating p oint. Pr o of. Let { f σ : σ ∈ 2 N } ⊆ C b e the fa mily obtained by Theor e m 48(a). W e notice that, by the pro of of Theorem 48, w e also have tha t the map Φ : 2 N × X → R deﬁned by Φ( σ, x ) = f σ ( x ) is Borel. Using this and by passing to a p erfect subset of 2 N if necessar y , we may ﬁnd ε > 0 such that k f σ k ∞ > ε for all σ ∈ 2 N . Deﬁne N ⊆ 2 N × X b y ( σ , z ) ∈ N ⇔ | f σ ( z ) | > ε. 50 S. A. ARGYR OS, P . DODOS AND V. KANELLOPOULOS As the map Φ is B orel, we s e e that the set N is B orel. Mo reo ver, by the choice of ε , we hav e that for ev ery σ ∈ 2 N the section N σ = { z : ( σ , z ) ∈ N } of N at σ is non-empty . By the Y ankov-V on Neumann Uniformization theorem (see [Ke], Theorem 1 8.1), there exists a map 2 N ∋ σ 7→ z σ ∈ X which is measura ble with r espect to the σ -algebra g enerated by the a na lytic sets and s uc h that ( σ, z σ ) ∈ N for ev ery σ ∈ 2 N . In v oking the classica l fa ct that analytic sets hav e the Baire prop erty , b y T he o rem 8.38 in [Ke] and by pa ssing to a further per fect subset of 2 N if necessa ry , we may assume that the map σ 7→ z σ is actua lly contin uous. F o r every m ∈ N deﬁne A m ⊆ 2 N × 2 N by ( σ , τ ) ∈ A m ⇔ | f τ ( z σ ) | > 1 m + 1 . Notice that the set A m is Bo r el. Since the family { f σ : σ ∈ 2 N } ac cum ulates to 0, we get that for every σ ∈ 2 N the s ection ( A m ) σ = { τ : ( σ, τ ) ∈ A m } of A m at σ is ﬁnite, hence mea g er in 2 N . By the Kur ato wski-Ulam theorem (see [Ke], Theorem 8.41), we have that the set A m is meager in 2 N × 2 N . Hence so is the set A = [ m ∈ N A m . By a r esult of J. Mycielski (see [K e], Theor em 19 .1 ) there exists P ⊆ 2 N per fect such that for every σ, τ ∈ P with σ 6 = τ we hav e that ( σ, τ ) / ∈ A and ( τ , σ ) / ∈ A . This implies that f τ ( z σ ) = 0 and f σ ( z τ ) = 0. W e ﬁx a homeo morphism h : 2 N → P and we s et g σ = f h ( σ ) and x σ = z h ( σ ) for every σ ∈ 2 N . Clea rly the family { ( g σ , x σ ) : σ ∈ 2 N } is as desir ed.  The pr oof o f the cor responding r esult in [ADK2] is based on Ra msey and Ba nac h space to ols, av oiding the embedding of ˆ A (2 N ) in to ( B X ∗∗ , w ∗ ). W e recall that a Ba nac h space X is said to be repre sen table if X isomorphic to a subspace of ℓ ∞ ( N ) which is a na lytic in the weak* top ology (see [GT], [GL] and [A GR]). W e close this subsection with the following. Theorem 50. L et X b e a non-s ep ar able r epr esentable Banach sp ac e. Then X ∗ c ontains an u nc onditional family of size | X ∗ | . Pr o of. Ident ify X with its isomorphic c o p y in ℓ ∞ ( N ). T hen B X is an a nalytic s ubset of ( B ℓ ∞ , w ∗ ). Let f : N N → B X be an onto contin uous map. Let { x n } n be a norm dense subset of ℓ 1 ( N ). Viewing ℓ 1 as a subspac e of ℓ ∗ ∞ , w e deﬁne f n : N N → R by f n = x n ◦ f . Then { f n } n is a uniformly b ounded sequence of cont inuous rea l-v alued functions on N N . Notice tha t { f n } p n = { x ∗ ◦ f : x ∗ ∈ B X ∗ } whic h can b e naturally ident iﬁed with { x ∗ | B X : x ∗ ∈ B X ∗ } . By the non- eﬀectiv e version of Debs’ theo rem (see [AGR]) we hav e the following a lter nativ es. A CLASSIFICA TION OF S E P ARABLE ROSENTHAL COMP A CT A 51 Case 1. There exist an incr easing sequence ( n k ) k , a contin uous map φ : 2 N → N N and rea l num b ers a < b s uch that for every σ ∈ 2 N and every k ∈ N , if σ ( k ) = 0 then f n k  φ ( σ )  < a , while if σ ( k ) = 1, then f n k  φ ( σ )  > b . In this ca s e, for every p ∈ β N we set g p = p − lim k f n k . Then g p = x ∗ p | B X for some x ∗ p ∈ X ∗ . W e claim that { x ∗ p : p ∈ β N } is equiv alent to the natural basis of ℓ 1 (2 c ). T o s ee this, observe that g p  φ ( σ )  ≤ a if and only if { k : σ ( k ) = 0 } ∈ p and g p  φ ( σ )  ≥ b if and only if { k : σ ( k ) = 1 } ∈ p . Setting A p = [ g p ≤ a ] and B p = [ g p ≥ b ] for a ll p ∈ β N , we see that the family ( A p , B p ) p ∈ β N is an independent family of disjoint pairs. By Rosenthal’s cr iterion, the family { g p : p ∈ β N } is e quiv alent to ℓ 1 (2 c ). Thus, so is { x ∗ p : p ∈ β N } . Case 2. The seq ue nc e { f n } n is r elativ ely co mpact in B 1 ( N N ). In this case, as X is non-separa ble, 0 ∈ { f n } p n is a non- G δ po in t. Thus, by Theorem 48(a), there e x ists an 1 -unconditional family in X ∗ of the size of the co n tin uum.  It can b e also shown that every r epresen table Banach space has a s eparable quotient (see Theorem 15 in [ADK2]). F o r further applicatio ns of the existence of unconditional families w e refer the reader to [ADK2]. 8.2. Spreading and lev el unconditional tree bases. W e start with the follow- ing de ﬁnitio n. Deﬁnition 51. L et X b e a Banach sp ac e. (1) A tr e e b asis is a b ounde d family { x t } t ∈ 2 < N in X which is Schauder b asic when it is enumer ate d ac c or ding t o the c anonic al bije ction φ 0 b etwe en 2 < N and N . (2) A tr e e b asis { x t } t ∈ 2 < N is said to b e s p r e adi ng if ther e exists ( ε n ) n ↓ 0 su ch that for every n, m ∈ N w ith n < m , every 0 ≤ d < 2 n and every p air { s i } d i =0 ⊆ 2 n and { t i } d i =0 ⊆ 2 m with s i ⊏ t i for al l i ∈ { 0 , ..., d } , we have that k T k · k T − 1 k < 1 + ε n wher e T : span { x s i : i = 0 , ..., d } → s pan { x t i : i = 0 , ..., d } is t he natur al 1-1 and onto line ar op er ator. (3) A tr e e b asis { x t } t ∈ 2 < N is said t o b e level unc onditional if ther e exists ( ε n ) n ↓ 0 such t ha t for every n ∈ N , t h e family { x t : t ∈ 2 n } is (1+ ε n ) -unc onditional. In [ADK2] the ex istence of spr eading and level unconditional tree bas e s was established for every separable Ba nac h spa ce X no t containing ℓ 1 and with no n- separable dual. This result can b e extended in the frame of separ able Rosenthal co mpacta, a s follows. Theorem 52. L et K b e a uniformly b ounde d sep ar able R osenthal c omp act on a c omp act metrizable sp ac e X and having a c ountable dense subset D of c ontinuous functions. L et also ( ε n ) n b e a de cr e asing se quenc e of p ositive r e als with ε n → 0 . 52 S. A. ARGYR OS, P . DODOS AND V. KANELLOPOULOS Assume that the c onstant function 0 is a n o n- G δ p oi nt o f K . Then t her e exists a family { u t } t ∈ 2 < N ⊆ conv( D ) e quivalent to the c anonic al dense family of ˆ A (2 N ) such that, set t ing g σ = lim n u σ | n for al l σ ∈ 2 N , t he fol lowing ar e satisﬁe d. (1) The function 0 is the unique non- G δ p oi nt of { u t } p t ∈ 2 < N . (2) The family { u t } t ∈ 2 < N is a t r e e b asis with r esp e ct to the supr emum norm. (3) The family { g σ : σ ∈ 2 N } is a su bset of K and 1-unc onditional. (4) F or every n ∈ N , if { t 0 ≺ ... ≺ t 2 n − 1 } is the ≺ - incr e asing enumera tion of 2 n , then for every { σ 0 , ..., σ 2 n − 1 } ⊆ 2 N with t i ⊏ σ i for al l i ∈ { 0 , ..., 2 n − 1 } we have that ( g σ i ) 2 n − 1 i =0 is (1 + ε n ) -e quivalent to ( u t i ) 2 n − 1 i =0 . The pr oof o f the ab o ve result is a slig ht mo diﬁcation of The o rem 1 7 in [ADK2], where we a lso r efer the reader for more information. W e close this subsection with the following r esult whose pro of is based on Ste- gall’s co nstruction [St]. Theorem 53. L et X b e a Banach sp ac e such that X ∗ is sep ar able and X ∗∗ is non-sep ar able. L et also ε > 0 . Then ther e exists a family { u t } t ∈ 2 < N ⊆ B X such that the fol lowing ar e satisﬁe d. (i) The family { u t } t ∈ 2 < N is e quivalent to the c anonic al dense family of 2 6 N . (ii) F or every σ ∈ 2 N , if y ∗∗ σ is the we ak* limit of ( u σ | n ) n , then ther e ex- ists y ∗∗∗ σ ∈ X ∗∗∗ with k y ∗∗∗ σ k ≤ 1 + ε and such t h at y ∗∗∗ σ ( y ∗∗ σ ) = 1 while y ∗∗∗ σ ( y ∗∗ τ ) = 0 for al l τ 6 = σ . (iii) F or every n ∈ N , i f { t 0 ≺ ... ≺ t 2 n − 1 } is the ≺ - incr e asing enumera tion of 2 n , then for every { σ 0 , ..., σ 2 n − 1 } ⊆ 2 N with t i ⊏ σ i for al l i ∈ { 0 , ..., 2 n − 1 } , we have that ( y ∗∗ σ i ) 2 n − 1 i =0 is (1 + 1 n ) -e quivalent to ( u t i ) 2 n − 1 i =0 . Pr o of. Since X ∗ is separa ble, we have that ( B X ∗∗ , w ∗ ) is compa c t metrizable. Fix a compatible metr ic ρ for ( B X ∗∗ , w ∗ ). Using Stegall’s construction [St], we get the following. (C1) A family { x ∗ t } t ∈ 2 < N ⊆ X ∗ , and (C2) a family { B t } t ∈ 2 < N of op en subse ts of ( B X ∗∗ , w ∗ ) such tha t for all t ∈ 2 < N the following a re satisﬁed. (P1) 1 < k x ∗ t k < 1 + ε . (P2) B t a 0 ∩ B t a 1 = ∅ , B t a 0 ∪ B t a 1 ⊆ B t and ρ − dia m ( B t ) ≤ 1 | t | +1 . (P3) F or all x ∗∗ ∈ B t , | x ∗∗ ( x ∗ t ) − 1 | < 1 | t | +1 . (P4) F or all t ′ 6 = t with | t | = | t ′ | and for all x ∗∗ ∈ B t ′ , | x ∗∗ ( x ∗ t ) | < 1 | t | +1 . By prop erty (P2), for every σ ∈ 2 N we have that T n B σ | n = { x ∗∗ } . Moreover, the map 2 N ∋ σ 7→ x ∗∗ σ ∈ ( B X ∗∗ , w ∗ ) is a homeomor phic embedding. By Goldstine’s theorem, for e v ery t ∈ 2 < N we choose x t ∈ B t ∩ X . Notice that w ∗ − lim n x σ | n = x ∗∗ σ for all σ ∈ 2 N . F o r e v ery σ ∈ 2 N we choo se x ∗∗∗ σ ∈ T n { x ∗ σ | k : k ≥ n } w ∗ . By (P3) we see that x ∗∗∗ σ ( x ∗∗ σ ) = 1 while, by (P4), x ∗∗∗ σ ( x ∗∗ τ ) = 0 for all τ 6 = σ . Moreover, A CLASSIFICA TION OF S E P ARABLE ROSENTHAL COMP A CT A 53 we have that sup {| λ i | : i = 0 , ..., n } ≤ (1 + ε )    n X i =0 λ i x ∗∗ σ i    for ev ery n ∈ N , ev ery { σ 0 , ..., σ n } ⊆ 2 N and ev ery ( λ i ) n i =0 ∈ R n +1 . Arguing as in the proo f of Theor em 17 in [ADK2], we may construct a family { u t } t ∈ 2 < N ⊆ conv { x t : t ∈ 2 < N } and a re g ular dy adic s ubtree S = ( s t ) t ∈ 2 < N of 2 < N such that the following a re satisﬁed. (1) F o r all σ ∈ 2 N , the sequence ( u σ | n ) n is weak* conv ergent to y ∗∗ σ , wher e y ∗∗ σ = lim n x s σ | n . (2) F o r every n ∈ N , if { t 0 ≺ ... ≺ t 2 n − 1 } is the ≺ -increa sing enumeration of 2 n , then for every { σ 0 , ..., σ 2 n − 1 } ⊆ 2 N with t i ⊏ σ i for all i ∈ { 0 , ..., 2 n − 1 } we hav e that ( y ∗∗ σ i ) 2 n − 1 i =0 is (1 + 1 n )-equiv alen t to ( u t i ) 2 n − 1 i =0 . F o r a ll σ ∈ 2 N , let ¯ σ = S n s σ | n ∈ 2 N . Setting y ∗∗∗ σ = x ∗∗∗ ¯ σ for all σ ∈ 2 N , we see that prop erties (ii) and (iii) in the statement of the theorem ar e satisﬁed. Finally , by passing to a regular dyadic s ubtree if necessary , w e also have that the family { u t } t ∈ 2 < N is equiv alen t to the canonical dense family of 2 6 N , i.e. pr operty (i) is satisﬁed. The pro of is completed.  Remark 4. (1) W e do not know if the family { u t } t ∈ 2 < N o btained in Theorem 53 can be chosen to be Schauder ba sic or an FDD. It seems a ls o to b e unknown whether for every Bana c h space X with X ∗ separable a nd X ∗∗ non-separa ble, there exis ts a subspace Y of X with a Schauder basis such tha t Y ∗∗ is no n-separable. (2) The family { y ∗∗ σ : σ ∈ 2 N } obtained in Theorem 53 cannot b e chosen to b e unconditional, a s the examples of non- separable HI spaces show (see [AA T], [A T]). 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A classification of separable Rosenthal compacta and its applications

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