Codings of separable compact subsets of the first Baire class

CODINGS OF SEP ARABLE COMP ACT SUBSETS OF THE FIRST BAIRE CLASS P A NDELIS DODOS Abstract. Let X be a P olish space and K a separable compact subset of the first Baire class on X . F or ev ery sequence f = ( f n ) n dense in K , the descriptiv e set-theoretic prop erties of the set L f = { L ∈ [ N ] : ( f n ) n ∈ L is point wise conv ergen t } are analyzed. It is sho wn that if K is not firs t counta ble, then L f is Π 1 1 - complete. This can also happ en even if K i s a pre-metric compactum of degree at most tw o, in the sense of S. T odorˇ cevi ´ c. How ev er, if K is of degree exactly t wo, then L f is alwa ys Borel. A deep result of G. Debs implies that L f con tains a Borel cofinal set and this gives a tree-represen tation of K . W e sho w that classical ordinal assignments of Baire-1 functions are actually Π 1 1 -ranks on K . W e also pr ov ide an example of a Σ 1 1 Ramsey-nu ll subset A o f [ N ] for whic h there does not exist a Borel set B ⊇ A suc h that the difference B \ A is Ramsey-nu ll. 1. Introduction Let X b e a Polish s pace. A Rosen thal compact on X is a subset o f r eal-v alued Baire-1 functions on X , compact in the p oint wise top olo gy . Standard examples of such compacta include the Helly space (the space of a ll non-decr easing functions from the unit in terv al in to itself ), the split in terv al (the lexicographica l order ed pro duct o f the unit interv al and the tw o-element or dering) and the ball o f the double dual of a separable Banach space not containing ℓ 1 . That the later space is indeed a compact subset of the first Baire class follo ws from the famous Odell- Rosenthal theorem [OR], whic h states that the ball of the double dual o f a separable Banach space with the w eak* top olo gy consis ts only of Baire - 1 functions if and only if the space do es not contain ℓ 1 . Actually this res ult motiv ated H. P . Rosenthal to initiate the study o f compact subsets of the fir st Baire class in [Ro1 ]. He show ed that all such compacta a re sequen tially compact. J. Bourg ain, D. H. F r emlin and M. T alagra nd prov ed that Rosenthal compa c ta ar e F r´ echet spac es [BFT]. W e refer to [A GR], [P1] and [Ro2] for thorough introductions to the theory , as well as, its applications in Analysis . 1 2000 Mathematics Subje ct Classific ation : 03E15, 26A21, 54H05, 05D10. 2 Researc h supp orted by a grant of EPEAEK program ”Pythagoras”. 1 2 P ANDELIS DODO S Separability is the cr ucial prop erty that divides this class in t wo. As S. T o dorˇ cevi´ c has p ointed out in [T o], while non- separable Ro senthal compacta can b e quite pathologica l, the se parable ones are all ”definable” . This is suppor ted by the work of many r esearchers, including G. Go defroy [Go], A. Kr aw c z yk [K r], W. Marciszewski [Ma], R. Pol [P2] and is highlighted in the re ma rk able dic hotomies a nd tric hotomies of [T o]. Our starting p oint of view is how w e can co de separa ble compa ct subsets of the first Baire class by mem b ers o f a standar d Borel space. Sp e cifically , by a c o de of a sepa rable Rosenthal compact K on a Polish s pace X , we mean a standard Bo r el space C and a s ur jection C ∋ c 7→ f c ∈ K such that f or all a ∈ R the relation ( c, x ) ∈ R a ⇔ f c ( x ) > a is Borel in C × X . In other words, inv er se images of s ub-basic op en subsets of K are Borel in C uniformly in X . There is a natural ob ject one asso ciates to ev ery separable Rosenthal compa ct K and ca n serve as a co ding of K . More precisely , fo r every dense sequence f = ( f n ) n in K one defines L f = { L ∈ [ N ] : ( f n ) n ∈ L is p oint wise con vergent } . The Bourgain-F remlin-T alagr and theorem [BFT] implies that L f totally describes the mem b ers o f K , in the sens e that for every accumulation p o int f of K ther e exists L ∈ L f such that f is the point wise limit of the sequence ( f n ) n ∈ L . Moreov er, for every f ∈ K one a ls o defines L f ,f = { L ∈ [ N ] : ( f n ) n ∈ L is p oint wise con vergent to f } . Both L f and L f ,f hav e b een studied in the literature. In [K r], Krawczyk prov ed that L f ,f is Borel if and only if f is a G δ po int o f K . The set L f (more precisely the set L f \ L f ,f ) has been a ls o considered by T o do rˇ cevi´ c in [T o], in his solution o f characters o f points in s eparable Rosenthal compacta. There is an awkward fact concerning L f , namely that L f can b e non-Bor e l. How ever, a deep r esult of G. Debs [De ] implies that L f alwa ys contains a Borel cofinal set and this subset of L f can ser ve a s a c o ding. This leads to the following tree-repr e s entation of separable Rosenthal compacta. Prop ositi o n A. L et K b e a sep ar able R osenthal c omp act. Then ther e exist a c oun t- able t r e e T and a se quenc e ( g t ) t ∈ T in K such that the fol lowing hold. (1) F or every σ ∈ [ T ] t he se quenc e ( g σ | n ) n ∈ N is p ointwise c onver gent. (2) F or every f ∈ K ther e exists σ ∈ [ T ] such that f is the p ointwise limit of the se quenc e ( g σ | n ) n ∈ N . CODINGS OF SEP ARABLE COMP ACT A 3 It is natura l to ask when the set L f is Borel or, equiv alently , when L f can serve itself as a co ding (it is easy to see that L f and L f ,f are alwa y s Π 1 1 ). In this dire c tion, the following is shown. Theorem B. L et K b e a sep ar able R osenthal c omp act. (1) If K is not first c ount able, then for every dense se quenc e f = ( f n ) n in K the set L f is Π 1 1 -c omplete. (2) If K i s pr e-metric of de gr e e ex actly two, then for every de nse se quenc e f = ( f n ) n in K the set L f is Bo r el. Part (1) ab ov e is ba s ed on a r esult of K raw czyk. In par t (2), K is said to b e a pre-metric compactum of degr ee exactly t wo if ther e exist a countable subset D of X and a countable subset D of K such that at mos t tw o functions in K coincide on D and moreov er for every f ∈ K \ D ther e exists g ∈ K with f 6 = g and suc h tha t g coincides with f on D . This is a s ubcla ss o f the class o f pr e-metric c o mpacta o f degree at mos t t wo, as it is defined b y T o dorˇ cevi´ c in [T o]. W e notice that part (2) of Theorem B canno t be lifted to a ll pre-metric compa c ta of degree at most tw o, as there ar e examples of such compacta for which the set L f is Π 1 1 -complete. W e procee d no w to discuss some applications of the abov e appr oach. It is well- known that to every real-v a lued Baire-1 function on a Polish space X one ass o ciates several (equiv a lent) o rdinal rankings measur ing the discont inuities of the function. An extens ive study of them is done b y A. S. K echris and A. Louveau in [KL]. An impo rtant example is the separatio n rank α . W e have the following boundedness result concerning this index. Theorem C. L et X b e a Polish sp ac e and f = ( f n ) n a se quenc e of Bor el r e al-value d functions o n X . L et L 1 f = { L ∈ [ N ] : ( f n ) n ∈ L is p ointwise c onver gent to a Bair e-1 function } . Then f or every C ⊆ L 1 f Bor el, we have sup { α ( f L ) : L ∈ C } < ω 1 wher e, for ev ery L ∈ C , f L denotes the p ointwise limit o f the se quenc e ( f n ) n ∈ L . The proo f of T heo rem C actually is based on the fact that the separa tion rank is a parameterized Π 1 1 -rank. The o rem C, combined with the result of Debs, gives a pro of of the bo undedness r e s ult of [ADK]. Histo r ically the first result of this form is due to J. Bour g ain [Bo]. W e should p o int out that in or der to give a descr iptive set-theoretic pr o of of B ourgain’s re s ult one doe s not nee d to inv oke Debs’ theor e m. Theorem C can als o be used to provide natural counterexamples to the following approximation question in Ramsey theor y . Namely , given a Σ 1 1 subset A of [ N ] can we always find a Bore l set B ⊇ A suc h that the difference B \ A is Ramsey-null? 4 P ANDELIS DODO S A. W. Miller had a lso as ked whether there exists an ana ly tic set which is not equal to Borel mo dulo Ramsey-n ull (see [Mi], P roblem 1 . 6 ∗ ). W e sho w the following. Prop ositi o n D. Ther e exists a Σ 1 1 R amsey-nu l l su bset A of [ N ] for which ther e do es not ex ist a Bor el set B ⊇ A such that t he differ enc e B \ A is R amsey-nul l. Ac knowledgmen ts. Part of this w ork was do ne when I visited the Depar tmen t of Mathematics at Caltech. I would like to thank the department for the hos pitality and the financial supp or t. I w ould also like to thank the anon ymous referee for his thorough rep ort which substantially impro ved the presen tation of the paper. 2. Preliminaries F or a ny Polish spa ce X , by K ( X ) we denote the hype rspace of all co mpa ct subsets o f X , equipp ed with the Vietoris top olog y . By B 1 ( X ) (respectively B ( X )) we denote the space of a ll rea l- v alued Baire- 1 (res pe ctively Bo rel) functions o n X . By N = { 0 , 1 , 2 , ... } we denote the na tural num b ers, while b y [ N ] the set of all infinite subs e ts o f N (whic h is clea r ly a P olish subspa ce of 2 N ). F or ev e r y L ∈ [ N ], by [ L ] w e denote the set of all infinite subsets of L . F or every function f : X → R and every a ∈ R w e set [ f > a ] = { x : f ( x ) > a } . The set [ f < a ] ha s the obvious meaning. Our descriptiv e set-theo retic notation and terminology follows [Ke]. So Σ 1 1 stands for the analy tic sets, while Π 1 1 for the co-a nalytic. A set is said to be Π 1 1 -true if it is co-analytic non-Borel. If X , Y are Polish spaces, A ⊆ X and B ⊆ Y , we s ay that A is W adge (Bor el) reducible to B if there exists a contin uous (Borel) map f : X → Y such that f − 1 ( B ) = A . A set A is sa id to be Π 1 1 -complete if it is Π 1 1 and an y other co- analytic set is Bor el reducible to A . Clearly any Π 1 1 -complete s et is Π 1 1 -true. The conv erse is als o true under large car dina l h yp otheses (see [MK ] or [Mo]). If A is Π 1 1 , then a map φ : A → ω 1 is said to b e a Π 1 1 -rank o n A if there are relations ≤ Σ , ≤ Π in Σ 1 1 and Π 1 1 resp ectively , such that for an y y ∈ A φ ( x ) ≤ φ ( y ) ⇔ x ≤ Σ y ⇔ x ≤ Π y . Notice th at if A is B orel reducible to B via a Borel map f and φ is a Π 1 1 -rank on B , then the map ψ : A → ω 1 defined by ψ ( x ) = φ ( f ( x )) is a Π 1 1 -rank on A . T rees. If A is a non-empty set, by A < N we denote the set o f all finite seq uences of A . W e view A < N as a tree e q uipped with the (strict) partial order ⊏ o f extension. If s ∈ A < N , then the length | s | of s is defined to b e the car dinality o f the s et { t : t ⊏ s } . If s, t ∈ A < N , then by s a t w e deno te their co ncatenation. If A = N and L ∈ [ N ], then b y [ L ] < N we denote the increasing finite sequences in L . F or every x ∈ A N and ev ery n ≥ 1 we let x | n =  x (0) , ..., x ( n − 1)  ∈ A < N while x | 0 = ( ∅ ). CODINGS OF SEP ARABLE COMP ACT A 5 A tree T on A is a do wnw ards closed subset of A < N . The set of all tree s on A is denoted by T r( A ). Hence T ∈ T r( A ) ⇔ ∀ s, t ∈ A < N ( t ⊏ s ∧ s ∈ T ⇒ t ∈ T ) . F or a tre e T on A , the bo dy [ T ] of T is defined to b e the set { x ∈ A N : x | n ∈ T for all n ∈ N } . A tree T is calle d pruned if for every s ∈ T there exists t ∈ T with s ⊏ t . It is ca lled well-founded if for every x ∈ A N there exis ts n such that x | n / ∈ T , e quiv alently if [ T ] = ∅ . The se t o f well-founded trees on A is denoted by WF( A ). If T is a well-founded tree we let T ′ = { t : ∃ s ∈ T with t ⊏ s } . By transfinite r ecursion, one defines the iterated der iv atives T ( ξ ) of T . The order o ( T ) of T is defined to b e the least ordinal ξ such that T ( ξ ) = ∅ . If S, T are well-founded trees, then a map φ : S → T is called mo notone if s 1 ⊏ s 2 in S implies that φ ( s 1 ) ⊏ φ ( s 2 ) in T . Notice that in this case o ( S ) ≤ o ( T ). If A, B are s ets, then we ident ify every tree T on A × B with the set of a ll pairs ( s, t ) ∈ A < N × B < N such that | s | = | t | = k and  ( s (0) , t (0)) , ...., ( s ( k − 1) , t ( k − 1))  ∈ T . If A = N , then w e shall s imply denote by T r and WF the sets of all trees and well-founded trees on N resp ectively . The set WF is Π 1 1 -complete and the map T → o ( T ) is a Π 1 1 -rank on WF. The sa me also holds f or W F( A ) f or ev e r y count able se t A . The s e paration rank. Let X be a Polish space. Given A, B ⊆ X one asso ciates with them a deriv a tive on clo s ed sets, by F ′ A,B = F ∩ A ∩ F ∩ B . By tra nsfinite recursion, we define the iter ated der iv atives F ( ξ ) A,B of F and we s et α ( F , A, B ) to be the least or dinal ξ with F ( ξ ) A,B = ∅ if such an or dinal exists, otherwise we set α ( F, A, B ) = ω 1 . Now let f : X → R b e a function. F or each pair a, b ∈ R with a < b let A = [ f < a ] a nd B = [ f > b ]. F or ev e ry F ⊆ X closed let F ( ξ ) f ,a,b = F ( ξ ) A,B and α ( f , F, a, b ) = α ( F , A, B ). Let a ls o α ( f , a, b ) = α ( f , X , a, b ). The sepa ration rank of f is defined b y α ( f ) = sup { α ( f , a, b ) : a, b ∈ Q , a < b } . The basic fact is the follo wing (see [KL]). Prop ositi o n 1. A function f is Bair e-1 if and only if α ( f ) < ω 1 . 3. Codings of sep arable Rosenthal comp act a Let X be a Polish spa c e and f = ( f n ) n a se q uence of Borel rea l-v alued functions on X . Assume that the c losure K o f { f n } n in R X is a compact subset o f B ( X ). Let us consider the set L f = { L ∈ [ N ] : ( f n ) n ∈ L is p oint wise con vergent } . F or every L ∈ L f , by f L we shall denote the p oint wise limit of the sequence ( f n ) n ∈ L . Notice that L f is Π 1 1 . As the p oint wise top ology is not effected by the top olog y on X , w e ma y (and w e will) assume that each f n is contin uous (and so K is a separable Rosenthal compact). By a r esult of H. P . Rosenthal [Ro1], we get that 6 P ANDELIS DODO S L f is cofina l. T ha t is, for every M ∈ [ N ] there exists L ∈ [ M ] such that L ∈ L f . Also the celebrated Bo urgain-F remlin-T a lagra nd theorem [BFT] implies that L f totally describ es K . How ever, mos t imp ortant for our purp oses is the fact that L f contains a Borel cofinal set. This is a conse q uence of the f ollowing theorem of G. Debs [De] (whic h itself is the classical interpretation o f the effectiv e v er sion of th e Bourga in- F remlin-T alagr and theorem, prov ed by G. Debs in [De]). Theorem 2. L et Y , X b e Po lish sp ac es and ( g n ) n b e a se quenc e of Bo r el fun ctions on Y × X su ch that for every y ∈ Y the se quenc e  g n ( y , · )  n is a se quenc e of c ontinuou s functions r elatively c omp act in B ( X ) . Then ther e ex ists a Bor el map σ : Y → [ N ] su ch that for any y ∈ Y , the se quenc e  g n ( y , · )  n ∈ σ ( y ) is p ointwise c onver gent. Let us show how Theo rem 2 implies the existence of a Bo rel co final subset o f L f . Given L, M ∈ [ N ] with L = { l 0 < l 1 < ... } and M = { m 0 < m 1 < ... } their increasing enumerations, let L ∗ M = { l m 0 < l m 1 < ... } . Clearly L ∗ M ∈ [ L ] and moreo ver the function ( L, M ) 7→ L ∗ M is con tinuous. Let ( f n ) n be as in the beg inning of the sec tion and let Y = [ N ]. F or every n ∈ N define g n : [ N ] × X → R by g n ( L, x ) = f l n ( x ) where l n in the n th element o f the increa sing enumeration o f L . The sequence ( g n ) n satisfies all the hypotheses of Theorem 2. Let σ : [ N ] → [ N ] be the B o rel function such that for ev ery L ∈ [ N ] the sequenc e  g n ( L, · )  n ∈ σ ( L ) = ( f n ) n ∈ L ∗ σ ( L ) is p oint wise con vergent. The function L → L ∗ σ ( L ) is Borel and so the set A = { L ∗ σ ( L ) : L ∈ [ N ] } is an analytic cofinal subset o f L f . By separatio n w e get that there exists a Bor el cofinal subset of L f . The co finality of this set in conjunction with the Bourgain- F remlin-T alagr a nd theor em give us the following corollary . Corollary 3. L et X b e a P olish sp ac e and ( f n ) n a se quenc e of Bor el functions on X which is r elatively c omp act i n B ( X ) . Then ther e ex ists a Bor el set C ⊆ [ N ] such that for every c ∈ C the s e quenc e ( f n ) n ∈ c is p ointwise c onver gent and for every ac cumulation p oint f of ( f n ) n ther e exists c ∈ C with f = lim n ∈ c f n . In the sequel we will say that the set C obtained by Corolla r y 3 is a c o de of ( f n ) n . If K is a separable Rosen thal co mpact and ( f n ) n is a dense sequence in K , then we will say that C is the co de of K . Notice that the co des dep end on the dense sequence. If c ∈ C , then b y f c we shall denote the f unction coded by c . T ha t is f c is the p oint wise limit of the sequence ( f n ) n ∈ c . The fo llowing lemma captures the basic defina bilit y prop erties of the set of co des. Its easy pro o f is left to t he reader. CODINGS OF SEP ARABLE COMP ACT A 7 Lemma 4. L et X and ( f n ) n b e as in Cor ol lary 3 and let C b e a c o de of ( f n ) n . Then f or every a ∈ R the fol lowing r elations (i) ( c, x ) ∈ R a ⇔ f c ( x ) > a , (ii) ( c, x ) ∈ R ′ a ⇔ f c ( x ) ≥ a , (iii) ( c 1 , c 2 , x ) ∈ D a ⇔ | f c 1 ( x ) − f c 2 ( x ) | > a ar e Bor el. The existence of coding s of separ a ble Rosenthal compacta giv es us the following tree-repr e s entation of them. Prop ositi o n 5. L et K b e a sep ar able R osenthal c omp act. Then t her e ex ist a c ount - able t r e e T and a se quenc e ( g t ) t ∈ T in K such that the fol lowing hold. (1) F or every σ ∈ [ T ] t he se quenc e ( g σ | n ) n ∈ N is p ointwise c onver gent. (2) F or every f ∈ K ther e exists σ ∈ [ T ] such that f is the p ointwise limit of the se quenc e ( g σ | n ) n ∈ N . Pr o of. Let ( f n ) n be a dense sequence in K . W e ma y assume that for ev ery n ∈ N the set { m : f m = f n } is infinite. This extra co nditio n guarantees that for every f ∈ K there exis ts L ∈ L f such that f = f L . Let C be the codes o f ( f n ) n . Now w e shall use a common unfolding tric k. As C is Borel in 2 N there exis ts F ⊆ 2 N × N N closed such that C = pro j 2 N F . L e t T be the unique (down wards closed) pruned tree o n 2 × N suc h that F = [ T ]. This will be the desired tree. It remains to define the sequence ( g t ) t ∈ T . Set g ( ∅ , ∅ ) = f 0 . L e t t = ( s, w ) ∈ T and k ≥ 1 with s ∈ 2 < N , w ∈ N < N and | s | = | w | = k . Define n t ∈ N to be n t = max { n < k : s ( n ) = 1 } , if the set { n < k : s ( n ) = 1 } is non-empt y , and n t = 0 otherwise. Finally set g t = f n t . It is easy to chec k that for every σ ∈ [ T ] the sequence ( g σ | n ) n is p oint wis e conv ergent, and so (1) is s atisfied. That (2) is also satisfied follows fr o m the fact that for every f ∈ K ther e exists L ∈ L f with f = f L and the f act that C is cofinal.  Remark 1. (1) W e should p oint o ut that Corollary 3, combined with J. H. Silv er’s theorem (see [MK] o r [S2]) on the num b er of equiv alence classes of co-analytic eq uiv - alence r elations, gives an answer to the car dinality problem of s eparable Ro senthal compacta, a well-kno wn fact that can also be der ived b y the results of [T o] (see also [ADK], Remark 3). Indeed, let K be one and let C b e the set of co des o f K . Define the following equiv alence relatio n on C , by c 1 ∼ c 2 ⇔ f c 1 = f c 2 ⇔ ∀ x f c 1 ( x ) = f c 2 ( x ) . Then ∼ is a Π 1 1 equiv alence rela tio n. Hence, by Silver’s dichotom y , either the equiv alence cla sses ar e countable or p erfectly ma ny . The first ca se implies tha t |K| = ℵ 0 , while the s econd one t hat |K | = 2 ℵ 0 . (2) Although the set C o f co des o f a separ able Rosenthal c ompact K is considered to be a Borel set which desc r ib es K efficiently , when it is considered as a subset of [ N ] it can be chosen to hav e rich structur al prop erties. In particular , it can be 8 P ANDELIS DODO S chosen to b e here ditary (i.e. if c ∈ C and c ′ ∈ [ c ], then c ′ ∈ C ) and inv aria nt under finite changes. T o see this, s tart with a co de C 1 of K , i.e. a Bo rel cofinal subset of L f . Let Φ =  ( F, G ) : ( F ⊆ L f ) ∧ ( G ∩ C 1 = ∅ ) ∧ [ ∀ L, M ( L ∈ F ∧ M ⊆ L ⇒ M / ∈ G )] ∧ [ ∀ L, M , s ( L ∈ F ∧ ( L △ M = s ) ⇒ M / ∈ G )]  . Let also A = { N : ∃ L ∈ C 1 ∃ s ∈ [ N ] < N ∃ M ∈ [ L ] with N △ M = s } . Then A is Σ 1 1 and clearly Φ( A, ∼ A ). As Φ is Π 1 1 on Σ 1 1 , hereditary and con tinuous upw ar d in the s econd v ariable, by the dual form of the second reflec tio n theorem (see [Ke], Theorem 35 .16), there ex ists C ⊇ A Borel with Φ( C, ∼ C ). Clear ly C is a s desired. (3) W e notice that the idea of coding subsets of function spaces by conv erg ing se- quences app ear s also in [Be], where a representation result of Σ 1 2 subsets o f C ([0 , 1]) is prov ed. 4. A boundedness resul t 4.1. Determining α ( f ) b y c ompact s ets. Let X b e a Polish space and f : X → R a Baire-1 function. The aim of t his subsection is to show that the v alue α ( f ) is completely determined by the deriv atives tak en ov er compact subsets of X (notice that this is trivial if X is compact metriza ble). Sp ecifically w e hav e the following. Prop ositi o n 6. L et X b e a Polish sp ac e, f : X → R Bair e-1 and a < b r e als. Then α ( f , a, b ) = sup { α ( f , K, a, b ) : K ⊆ X c omp act } . The pro of of P r op osition 6 is an immediate conse q uence of the following lemmas. In what follows, a ll balls in X are taken with resp ect to some compatible complete metric ρ of X . Lemma 7. L et X , f and a < b b e as in Pr op osition 6. L et also F ⊆ X close d, x ∈ X and ξ < ω 1 b e such that x ∈ F ( ξ ) f ,a,b . Then for every ε > 0 , i f we let C = F ∩ B ( x, ε ) , we have x ∈ C ( ξ ) f ,a,b . Pr o of. Fix F and ε as ab ove. F or notational simplicit y let U = B ( x, ε ) and C = F ∩ B ( x, ε ). By induct ion w e shall s how that F ( ξ ) ∩ U ⊆ C ( ξ ) where F ( ξ ) = F ( ξ ) f ,a,b and similarly fo r C . This clearly implies the lemma. F or ξ = 0 is straightforw ard. Su pp ose that the lemma is true for ev ery ξ < ζ . Assume that ζ = ξ + 1 is a s uccessor or dina l. Let y ∈ F ( ξ +1) ∩ U . As U is o p e n, we ha ve y ∈ F ( ξ ) ∩ U ∩ [ f < a ] ∩ F ( ξ ) ∩ U ∩ [ f > b ] . By the inductive assumption we get that y ∈ C ( ξ ) ∩ [ f < a ] ∩ C ( ξ ) ∩ [ f > b ] = C ( ξ +1) CODINGS OF SEP ARABLE COMP ACT A 9 which proves the ca se of success or ordinals. If ζ is limit, then F ( ζ ) ∩ U = \ ξ<ζ F ( ξ ) ∩ U ⊆ \ ξ<ζ C ( ξ ) = C ( ζ ) and the lemma is pr ov ed.  Lemma 8. L et X , f and a < b b e as in Pr op osition 6. L et also F ⊆ X close d, x ∈ X and ξ < ω 1 b e such that x ∈ F ( ξ ) f ,a,b . T hen ther e ex ists K ⊆ F c ount able c omp act such that x ∈ K ( ξ ) f ,a,b . Pr o of. Again fo r notational simplicit y for every C ⊆ X closed and every ξ < ω 1 we let C ( ξ ) = C ( ξ ) f ,a,b . The pro o f is by induction on countable ordinals, as befor e. F or ξ = 0 the lemma is ob viously true. Sup p ose that the lemma has been prov ed for every ξ < ζ . Let F ⊆ X clo sed and x ∈ F ( ζ ) . Notice tha t one of the follo wing alternatives must o ccur. (A1) f ( x ) < a and there exists a sequence ( y n ) n such that y n 6 = y m for n 6 = m , f ( y n ) > b , y n ∈ F ( ξ n ) and y n → x ; (A2) f ( x ) > b a nd there exists a sequence ( z n ) n such that z n 6 = z m for n 6 = m , f ( z n ) < a , z n ∈ F ( ξ n ) and z n → x ; (A3) there exist tw o distinct se quences ( y n ) n and ( z n ) n such that y n 6 = y m and z n 6 = z m for n 6 = m , f ( y n ) < a , f ( z n ) > b , y n , z n ∈ F ( ξ n ) and y n → x , z n → x , where ab ov e the sequence ( ξ n ) n of countable ordinals is as follows. (C1) If ζ = ξ + 1, then ξ n = ξ for every n . (C2) If ζ is limit, then ( ξ n ) n is an incr easing s e quence o f s uccessor ordinals with ξ n ր ζ . W e shall treat the a lternative (A1) (the other o nes a re similar ). Let ( r n ) n be a sequence of p ositive rea ls suc h that B ( y n , r n ) ∩ B ( y m , r m ) = ∅ if n 6 = m a nd x / ∈ B ( y n , r n ) for every n . Let C n = F ∩ B ( y n , r n ). By Lemma 7, we get that y n ∈ C ( ξ n ) n . By the inductive ass umption, there exists K n ⊆ C n ⊆ F n countable compact such tha t y n ∈ K ( ξ n ) n . Finally let K = { x } ∪ ( S n K n ). Then K is coun table compact and it is easy to see that x ∈ K ( ζ ) .  Remark 2. Notice that the pro of of Lemma 8 actually shows that α ( f , a, b ) = sup { α ( f , K, a, b ) : K ⊆ X countable co mpact } . Moreov er observe tha t if α ( f , a, b ) is a successor ordinal, then the ab ove suprem um is attainted. 4.2. The main resul t. This subsection is devoted to the pro of o f the following result. 10 P ANDELIS DODO S Theorem 9. L et X b e a Polish sp ac e and f = ( f n ) n a se quenc e of Bor el r e al-value d functions o n X . L et L 1 f = { L ∈ [ N ] : ( f n ) n ∈ L is p ointwise c onver gent to a Bair e-1 function } . Then f or every C ⊆ L 1 f Bor el, we have sup { α ( f L ) : L ∈ C } < ω 1 wher e, for ev ery L ∈ C , f L denotes the p ointwise limit o f the se quenc e ( f n ) n ∈ L . F or the pro of of Theo r em 9 we will nee d the following theorem, which gives us a wa y of defining para meterized Π 1 1 -ranks (see [Ke], pa ge 275 ). Theorem 10. L et Y b e a standar d Bor el s p ac e, X a Polish sp ac e and D : Y × K ( X ) → K ( X ) b e a Bor el map su ch t hat for every y ∈ Y , D y is a derivative on K ( X ) . Then the set Ω D = { ( y , K ) : D ( ∞ ) y ( K ) = ∅ } is Π 1 1 and the ma p ( y , K ) → | K | D y is a Π 1 1 -r ank on Ω D . W e co ntin ue with t he proof of Theorem 9. Pr o of o f The or em 9. Let C ⊆ L 1 f Borel arbitrary . Fix a , b ∈ R with a < b . D efine D : C × K ( X ) → K ( X ) by D ( L, K ) = K ∩ [ f L < a ] ∩ K ∩ [ f L > b ] where f L is the point wise limit of the sequence ( f n ) n ∈ L . It is clear that for ev e r y L ∈ C the map K → D ( L, K ) is a deriv ative on K ( X ) a nd that α ( f L , K , a, b ) = | K | D L . W e will show that D is Borel. Define A, B ∈ C × K ( X ) × X b y ( L, K , x ) ∈ A ⇔ ( x ∈ K ) ∧ ( f L ( x ) < a ) and ( L, K , x ) ∈ B ⇔ ( x ∈ K ) ∧ ( f L ( x ) > b ) . It is e a sy to chec k that b o th A and B are Borel. Also let ˜ A, ˜ B ⊆ C × K ( X ) × X be defined b y ( L, K , x ) ∈ ˜ A ⇔ x ∈ A ( L,K ) and ( L, K , x ) ∈ ˜ B ⇔ x ∈ B ( L,K ) , where A ( L,K ) = { x : ( L, K , x ) ∈ A } is the section of A (and simila r ly for B ). Notice that for e very ( L, K ) ∈ C × K ( X ) we have D ( L, K ) = ˜ A ( L,K ) ∩ ˜ B ( L,K ) . As ˜ A ( L,K ) and ˜ B ( L,K ) are compact (b eing subsets of K ), by Theorem 28.8 in [Ke], it is enough to sho w that the sets ˜ A and ˜ B ar e Borel. W e will need the follo wing easy consequence of the Arsenin-Kun ugui theorem (the proo f is le ft to the reader ). CODINGS OF SEP ARABLE COMP ACT A 11 Lemma 11 . L et Z b e a standar d Bor el sp ac e, X a Polish sp ac e and F ⊆ Z × X Bor el with K σ se ctions. Then the s et ˜ F define d by ( z , x ) ∈ ˜ F ⇔ x ∈ F z is a Bor el subset of Z × X . By our a s sumptions, for every L ∈ C the function f L is Baire-1 a nd so for every ( L, K ) ∈ C × K ( X ) the sections A ( L,K ) and B ( L,K ) of A and B r esp ectively are K σ . Hence, b y Lemma 1 1, we get that ˜ A and ˜ B are Bor el. By the ab ov e w e conclude that D is a Bor el ma p. By Theorem 10, the map ( L, K ) → | K | D L is a Π 1 1 -rank on Ω D . By Propos ition 1 and the fact that C ⊆ L 1 f , we get that for every ( L, K ) ∈ C × K ( X ) the transfinite sequence  D ( ξ ) L ( K )  ξ<ω 1 m ust be stabilized a t ∅ and so Ω D = C × K ( X ). As Ω D is Borel, by b oundedness we have sup {| K | D L : ( L, K ) ∈ C × K ( X ) } < ω 1 . It follows tha t sup { α ( f L , K , a, b ) : ( L , K ) ∈ C × K ( X ) } < ω 1 . By Prop o sition 6, we ge t sup { α ( f L , a, b ) : L ∈ C } < ω 1 . This completes the pr o of of the theorem.  4.3. Consequences. Let us recall some definitions fro m [ADK]. Let X be a Polish space, ( f n ) n a seq uence of re a l-v alued functions on X and let K be the clo sure of { f n } n in R X . W e will say that K is a (separable) qua s i-Rosenthal if every accumulation p oint of K is a Ba ire-1 function and moreover we will say that K is Bor el separa ble if the sequence ( f n ) n consists of Borel functions. Combining Theorem 9 with Corollary 3 w e get the following result of [AD K]. Theorem 12 . L et X b e a Polish sp ac e and K a Bor el sep ar able qu asi-R osenthal c omp act on X . Then sup { α ( f ) : f ∈ Acc ( K ) } < ω 1 wher e Acc ( K ) denotes the ac cu mulation p oints of K . In p articular, if K is a sep a- r able R osenthal c omp act on X , then sup { α ( f ) : f ∈ K} < ω 1 . Besides b oundedness, the implicatio ns of Theo rem 9 and the rela tion betw een the separation rank a nd the Bo relness of L f are more tr ansparently s een when X is a compact metriza ble space. In particular we ha ve the follo wing. Prop ositi o n 13. L et X b e a c omp act metrizable sp ac e and K a s ep ar able R osenthal c omp act on X . L et f = ( f n ) n b e a dense se quenc e in K and a < b r e als. Then the map L → α ( f L , a, b ) is a Π 1 1 -r ank on L f if a nd only if the set L f is Bo r el. 12 P ANDELIS DODO S Pr o of. First assume tha t L f is not Bo rel. By Theor em 12, we ha ve that sup { α ( f L , a, b ) : L ∈ L f } < ω 1 and so the map L → α ( f L , a, b ) ca nnot b e a Π 1 1 -rank on L f , as L f is Π 1 1 -true. Conv ersely , ass ume that L f is Borel. By t he pro o f of Theorem 9, we ha ve that the map ( L, K ) → | K | D L is a Π 1 1 -rank on L f × K ( X ). It follows that the r elation L 1  L 2 ⇔ α ( f L 1 , a, b ) ≤ α ( f L 2 , a, b ) ⇔ | X | D L 1 ≤ | X | D L 2 is Borel in L f × L f . This implies that the map L → α ( f L , a, b ) is a Π 1 1 -rank on L f , as desired.  Remark 3. Althoug h the map L → α ( f L , a, b ) is not alwa ys a Π 1 1 -rank on L f , it is easy to se e t hat it is a Π 1 1 -rank on the codes C o f K , as the relation c 1  c 2 ⇔ α ( f c 1 , a, b ) ≤ α ( f c 2 , a, b ) ⇔ | X | D c 1 ≤ | X | D c 2 is Borel in C × C for ev ery pair a < b of reals. Hence, when X is compa ct metrizable space, we could say that the separation rank is a Π 1 1 -rank ”in the co des”. W e pro ceed to disc us s ano ther application of Theorem 9 which deals with the following approximation question in Ramsey theory . Rec all that a set N ⊆ [ N ] is called Ra msey-null if for every s ∈ [ N ] < N and every L ∈ [ N ] with s < L , there exists L ′ ∈ [ L ] suc h that [ s, L ′ ] ∩ N = ∅ . As every analytic set is Ramsey [S1], it is natural to ask the follo wing. Is it true that for every analy tic s et A ⊆ [ N ] there exists B ⊇ A Bor el such that B \ A is Ramsey- null? As we will show the answer is no and a coun ter example can b e found whic h is in addition Ramsey-null. T o this end we will need so me notations fro m [A GR]. Let X be a s eparable Banach space. By X ∗∗ B 1 we denote the set of all Bair e-1 elements of the ball of the second dual X ∗∗ of X . W e will say that X is α -univ ersa l if sup { α ( x ∗∗ ) : x ∗∗ ∈ X ∗∗ B 1 } = ω 1 . W e should p oint o ut that there exist non-universal (in the classical sense ) s eparable Banach spa c e s which ar e α -universal (see [AD]). W e hav e the following. Prop ositi o n 14. Ther e exists a Σ 1 1 R amsey-nu l l subset A of [ N ] for which ther e do es not ex ist a Bor el set B ⊇ A such that t he differ enc e B \ A is R amsey-nul l. Pr o of. Let X b e a separa ble α -universal Banach space and fix a norm dense se- quence f = ( x n ) n in the ba ll of X (it will b e conv enient to assume that x n 6 = x m if n 6 = m ). L e t L f = { L ∈ [ N ] : ( x n ) n ∈ L is weak* conv e r gent } . Clearly L f is Π 1 1 . Moreover, notice that L f = L 1 f according to the notation of Theorem 9. CODINGS OF SEP ARABLE COMP ACT A 13 Let x ∗∗ ∈ X ∗∗ B 1 arbitrar y . By the Odell-Rosenthal theor em (see [AGR] or [OR]), there exists L ∈ L f such that x ∗∗ = w ∗ − lim n ∈ L x n . It follows that sup { α ( x ∗∗ ) : x ∗∗ ∈ X ∗∗ B 1 } = sup { α ( x L ) : L ∈ L f } where x L denotes the weak* limit of the sequence ( x n ) n ∈ L . Denote by ( e n ) n the standard basis of ℓ 1 and let Λ = { L ∈ [ N ] : ∃ k such that ( x n ) n ∈ L is ( k + 1 ) − equiv ale nt to ( e n ) n } where, as us ual, if L ∈ [ N ] with L = { l 0 < l 1 < ... } its increas ing en umeratio n, then ( x n ) n ∈ L is ( k + 1)- e quiv alent to ( e n ) n if for every m ∈ N and every a 0 , ..., a m ∈ R we have 1 k + 1 m X n =0 | a n | ≤    m X n =0 a n x l n    X ≤ ( k + 1) m X n =0 | a n | . Then Λ is Σ 0 2 . W e notice that, b y Bourgain’s result [Bo] and o ur assumptions on the space X , the set Λ is non-empt y . Let also Λ 1 = { N ∈ [ N ] : ∃ L ∈ Λ ∃ s ∈ [ N ] < N such that N △ L = s } . Clearly Λ 1 is Σ 0 2 to o. Obs erve t hat both L f and Λ 1 are hereditar y and inv a riant under finite changes. Mo reov er the set L f ∪ Λ 1 is cofina l. This is essentially a consequence of Rosenthal’s dic hotomy (see, f or instance, [L T]). It follo ws that the set A = [ N ] \ ( L f ∪ Λ 1 ) is Σ 1 1 and Ramsey-null. W e claim that A is the desir ed set. Assume not, i.e. there exists a Borel set B ⊇ A suc h that the difference B \ A is Ramsey -null. W e set C = [ N ] \ ( B ∪ Λ 1 ). Then C ⊆ L f is Borel and moreover L f \ C is Ramsey-null. It follows that for every x ∗∗ ∈ X ∗∗ B 1 there exists L ∈ C such that x ∗∗ = x L . As C is Borel, b y Theorem 9 we hav e that sup { α ( x ∗∗ ) : x ∗∗ ∈ X ∗∗ B 1 } = sup { α ( x L ) : L ∈ C } < ω 1 which contradicts the fact that X is α -univ ersa l. The proof is co mpleted.  Remark 4. (1) An exa mple as in Pr op osition 1 4 ca n also be g iven using the conv ergence r a nk γ s tudied by A. S. Kechris a nd A. Louveau [KL]. As the reaso ning is the sa me, we shall brie fly descr ib e the ar gument. Let ( f n ) n be a sequence of contin uo us function on 2 N with k f n k ∞ ≤ 1 for all n ∈ N and such that the se t { f n : n ∈ N } is nor m dense in the ball of C (2 N ). As in Pro p osition 14, consider the sets L f , Λ 1 and A = [ N ] \ ( L f ∪ Λ 1 ). Then the se t A is Σ 1 1 and Ramsey-null. Tha t A cannot b e c overed by a Borel set B suc h that the differ e nce B \ A is Ramsey-null follows essentially by the follo wing facts. (F1) T he map ( g n ) n 7→ γ  ( g n ) n  is a Π 1 1 -rank on the set CN = { ( g n ) n ∈ C (2 N ) N : ( g n ) n is p oint wise con vergent } (see [Ke], page 279). Hence the map L f ∋ L = { l 0 < l 1 < ... } 7→ γ  ( f l n ) n  is a Π 1 1 -rank on L f . 14 P ANDELIS DODO S (F2) F or every ∆ ∈ ∆ 0 2 , there exis ts L ∈ L f such that the se quence ( f n ) n ∈ L is po int wise convergen t to χ ∆ . B y Prop ositio n 1 in [KL], we get that α ( χ ∆ ) ≤ γ  ( f n ) n ∈ L  . It follows that sup { γ  ( f n ) n ∈ L  : L ∈ L f } ≥ sup { α ( χ ∆ ) : ∆ ∈ ∆ 0 2 } = ω 1 . (2) F or the imp or tant sp ecial case of a sepa r able Rosenthal co mpact K defined on a compact metrizable space X and ha ving a dense set of contin uous funct ions, Theorem 12 has originally been prov ed by J. Bourgain [Bo]. W e should point out that in this case one do es no t need Corolla ry 3 in order to carr y out the pr o of. L et us briefly explain how this c a n be done. So assume that X is compact metrizable and f = ( f n ) n is a sequence o f con tinuous functions dens e in K . Fix a, b ∈ Q with a < b and let A n = [ f n ≤ a ] and B n = [ f n ≥ b ]. F o r a given M ∈ [ N ] let as usual lim inf n ∈ M A n = [ n \ k ≥ n,k ∈ M A k and similarly for lim inf n ∈ M B n . Obser ve the following. (O1) F or e very M ∈ [ N ] the sets lim inf n ∈ M A n and lim inf n ∈ M B n are b o th Σ 0 2 . (O2) If L , M ∈ [ N ] are such that L ⊆ M , then lim inf n ∈ M A n ⊆ lim inf n ∈ L A n and similar ly for B n . (O3) If L ∈ L f , then [ f L < a ] ⊆ lim inf n ∈ L A n ⊆ [ f L ≤ a ] and resp ectively [ f L > b ] ⊆ lim inf n ∈ L B n ⊆ [ f L ≥ b ]. Define D : [ N ] × K ( X ) → K ( X ) by D ( M , K ) = K ∩ lim inf n ∈ M A n ∩ K ∩ lim inf n ∈ M B n . By (O1) and using the s ame arguments a s in the pro of of Theorem 9, w e c a n easily verify that D is Bor el. As L f is cofina l, by (O2 ) and (O3) we can also easily verify that Ω D = [ N ] × K ( X ). So b y bo undedness we get s up { | K | D M : ( M , K ) ∈ [ N ] × K ( X ) } < ω 1 . Now us ing (O3 ) again, w e finally g et th at sup { α ( f , a, b ) : f ∈ K} < ω 1 , as des ir ed. 5. On the descriptive set-theoretic proper ties of L f In this sectio n we will s how that cer tain top ologic a l prop erties of a sepa r able Rosenthal compact K imply the Bo relness o f the set L f . T o this end, we recall that K is said to be a pre-metric co mpactum o f deg ree at most t wo if there exists a coun table subset D of X such that at mos t t wo functions in K agree on D (see [T o]). Let us consider the following sub class . Defin tion 15. We wil l say that K is a pr e- m etric c omp actum of de gr e e exactly two, if ther e ex ist a c ountable subset D of X and a c ount able subset D of K such that at most two functions in K c oincide on D and mor e over for every f ∈ K \ D ther e exists g ∈ K with g 6 = f and such that g c oincides with f on D . CODINGS OF SEP ARABLE COMP ACT A 15 An important exa mple o f suc h a compact is the split in ter v al (but it is not the only imp ortant one – see Rema rk 5 below). Under the ab ov e terminology we hav e the following. Theorem 16 . L et X b e a Polish sp ac e and K a sep ar able R osenthal c omp act on X . If K is pr e-metric of de gr e e exactly two, then for every d ense s e quenc e f = ( f n ) n in K the set L f is Bor el. Pr o of. Let f = ( f n ) n be a dense s equence in K and C b e the set of co des of ( f n ) n . Let also D ⊆ X co untable and D ⊆ K count able v erifying that K is pr e-metric of degree exactly tw o. Claim. Ther e exists D ′ ⊆ K c ountable with D ⊆ D ′ and such that for every c ∈ C with f c ∈ K \ D ′ ther e exists c ′ ∈ C su ch that f c ′ 6 = f c and f c ′ c oincides with f c on D . Pr o of of the claim. Let c ∈ C b e such tha t f c ∈ K \ D . L e t g be the (unique) function in K with g 6 = f c and such that g coincides w ith f c on D . If there do es not exist c ′ ∈ C with g = f c ′ , then g is an isolated point of K . W e s et D ′ = D ∪ { f ∈ K : ∃ g ∈ K isolated such that f ( x ) = g ( x ) ∀ x ∈ D } . As the isolated p o in ts o f K a re countable and K is pr e-metric of de g ree at most t wo, we get that D ′ is countable. Clearly D ′ is as desire d. ♦ Let D ′ be the set obtained a bove and put L D ′ = [ f ∈D ′ L f ,f = [ f ∈D ′ { L ∈ [ N ] : ( f n ) n ∈ L is p oint wise conv erg ent to f } . As every point in K is G δ , we see that L D ′ is Borel (actually it is Σ 0 4 ). Consider the following equiv alence relatio n ∼ on C , defined b y c 1 ∼ c 2 ⇔ ∀ x ∈ D f c 1 ( x ) = f c 2 ( x ) . By Lemma 4 , the equiv alence relatio n ∼ is Borel. C o nsider now the re la tion P on C × C × K ( X ) × X defined by ( c 1 , c 2 , K , x ) ∈ P ⇔ ( c 1 ∼ c 2 ) ∧ ( x ∈ K ) ∧ ( | f c 1 ( x ) − f c 2 ( x ) | > 0) . Again P is Bor el. Moreov er notice that for every c 1 , c 2 ∈ C t he function x 7→ | f c 1 ( x ) − f c 2 ( x ) | is Baire-1 , and so, for every ( c 1 , c 2 , K ) ∈ C × C × K ( X ) the section P ( c 1 ,c 2 ,K ) = { x ∈ X : ( c 1 , c 2 , K , x ) ∈ P } of P is K σ . By Theorem 35.4 6 in [Ke], the set S ⊆ C × C × K ( X ) defined by ( c 1 , c 2 , K ) ∈ S ⇔ ∃ x ( c 1 , c 2 , K , x ) ∈ P is Bo rel and there exists a Bo r el map φ : S → X such that for every ( c 1 , c 2 , K ) ∈ S we ha ve  c 1 , c 2 , K , φ ( c 1 , c 2 , K )  ∈ P . By the above cla im, we hav e that for every c ∈ C \ L D ′ there exist c ′ ∈ C a nd K ∈ K ( X ) such that ( c, c ′ , K ) ∈ S . Mo r eov er, 16 P ANDELIS DODO S observe that the set D ∪  φ ( c, c ′ , K )  determines the neig hborho o d basis of f c . The crucial fact is that this can be done in a Bor el w ay . Now we claim that L ∈ L f ⇔ ( L ∈ L D ′ ) ∨ h ( ∀ x ∈ D  f n ( x )  n ∈ L conv erges) ∧  ∃ s ∈ S with s = ( c 1 , c 2 , K ) suc h that [ ∀ x ∈ D f c 1 ( x ) = lim n ∈ L f n ( x )] ∧ [  f n ( φ ( s ))  n ∈ L conv erges] ∧ [ f c 1 ( φ ( s )) = lim n ∈ L f n ( φ ( s ))]  i . Grating t his, the pro of is completed as the a b ov e expre ssion g ives a Σ 1 1 definition of L f . As L f is also Π 1 1 , this implies t hat L f is Bor el, as desired. It remains to prov e the above equiv alence. Firs t assume that L ∈ L f . W e need to show that L satisfies the expres s ion on the r ight. If L ∈ L D ′ this is clearly true. If L / ∈ L D ′ , then pick a code c ∈ C \ L D ′ such that f c = f L . By the a bove claim and the r emarks of the pr evious pa ragr aph w e ca n easily verify that aga in L satisfies the expression on the right. Conv erse ly , let L fulfil the right side of the equiv alence. If L ∈ L D ′ we are done. If not, then by the Bourgain-F r emlin-T ala grand theor em, it suffices to sho w that a ll conv ergent subsequences o f ( f n ) n ∈ L hav e the same limit. The first tw o conjuncts enclosed in the square brack ets on the right side of the equiv alence g uarantee that each such conv ergent s ubsequence o f ( f n ) n ∈ L conv erges either to f c 1 or to f c 2 . The last tw o conjuncts guar a ntee that it is not f c 2 , s o it is alwa ys f c 1 . Thus L ∈ L f and the pro o f is completed.  Remark 5. (1 ) Let K b e a pre-metric compactum of degree at most t wo and let D ⊆ X countable such tha t at most tw o functions in K coincide on D . No tice that the set C of codes of K is naturally divided in to tw o parts, namely C 2 = { c ∈ C : ∃ c ′ ∈ C with f c 6 = f c ′ and f c ( x ) = f c ′ ( x ) ∀ x ∈ D } and its complemen t C 1 = C \ C 2 . The assumption that K is pre-metric of deg ree exactly tw o, simply reduces to th e assumption that t he functions co ded b y C 1 are at most countable. W e could say that C 1 is the set of metrizable co des, a s it is immediate that the se t { f c : c ∈ C 1 } is a metr izable subspace of K . It is easy to chec k, us ing the set S defined in the pr o of of Theorem 16, that C 2 is always Σ 1 1 . As we shall see, it might ha ppe n that C 1 is Π 1 1 -true. Ho wev er, if the set C 1 of metrizable co des is Bor e l, o r equiv alently if C 2 is Borel, then the set L f is Borel to o. Indeed, let Φ b e the second pa rt of the disjunction of the ex pr ession in the pro of of Theore m 16. Then it is easy to see, using the sa me a rguments as in the pro of of Theor em 16, that L ∈ L f ⇔ ( L ∈ Φ) ∨ ( ∃ c ∈ C 1 ∀ x ∈ D f c ( x ) = lim n ∈ L f n ( x )) . CODINGS OF SEP ARABLE COMP ACT A 17 Clearly the ab ove formula gives a Σ 1 1 definition of L f , provided that C 1 is Borel. (2) Besides the split in terv al, there exists another imp o r tant example of a separ a ble Rosenthal compact which is pre-metric of degree exactly t wo. This is the separa- ble companio n of the Alexa ndroff duplicate o f the Cantor set D (2 N ) (see [T o] for more details). An in ter esting feature of th is compact is that it is not he r editarily separable. Example 1. W e pro ceed to give examples of pre-metric compacta o f degree at most tw o for which Theorem 1 6 is not v alid. Let us reca ll first the split Ca ntor set S (2 N ), whic h is simply the com binatorial a na logue of the split in terv al. In the sequel by 6 we shall denote the lexicogr aphical ordering on 2 N and by < its strict part. F or ev ery x ∈ 2 N let f + x = χ { y : x 6 y } and f − x = χ { y : x 1 d + 1  . CODINGS OF SEP ARABLE COMP ACT A 21 Next w e glue the sequence o f trees ( T d L ) d ∈ N in a natura l wa y and we build a tree T L ∈ T r( N × Fin × N ) defined by the rule ( s, t, w ) ∈ T L ⇔ ∃ d ∃ ( s ′ , t ′ , w ′ ) such that ( s ′ , t ′ , w ′ ) ∈ T d L and s = d a s ′ , t = { d } a t ′ , w = d a w ′ . It is clear that the map [ N ] ∋ L 7→ T L ∈ T r ( N × Fin × N ) is contin uous. Mor eov er the following holds. Lemma 20. L et L ∈ [ N ] . Then L ∈ L f if and only if T L ∈ WF( N × Fin × N ) . Pr o of. First, notice that if L / ∈ L f , then there e x ist L 1 , L 2 ∈ [ L ] such that L 1 ∩ L 2 = ∅ , L 1 , L 2 ∈ L f and f L 1 6 = f L 2 , where as usua l f L 1 and f L 2 are the point wise limits of the s equences ( f n ) n ∈ L 1 and ( f n ) n ∈ L 2 resp ectively . Pic k x ∈ X and d ∈ N such that | f L 1 ( x ) − f L 2 ( x ) | > 1 d +1 . Clear ly we may assume that | f n ( x ) − f m ( x ) | > 1 d +1 for every n ∈ L 1 and every m ∈ L 2 . Let L 1 = { n 0 < n 1 < ... } and L 2 = { m 0 < m 1 < ... } b e the incr easing enum era tions of L 1 and L 2 . Using the co nt inuit y of the functions ( f n ) n , we find w = ( l 0 , ..., l k , ... ) ∈ N N such that w | k is acceptable for all k ∈ N , T k B l k = { x } and | f n k ( z ) − f m k ( z ) | > 1 d +1 for all k ∈ N and z ∈ B l k . Then  ( n 0 , ..., n k ) , ( { m 0 } , ..., { m k } ) , w | k  ∈ T d L for all k ∈ N , which shows that T L / ∈ WF( N × Fin × N ). Conv ersely as s ume that T L is not well-founded. Ther e exists d ∈ N such that T d L is not well-founded to o. Let  ( s k , t k , w k )  k be an infinite branch of T d L . Let N = S k s k = { n 0 < ... < n k < ... } ∈ [ L ], F = S k t k = ( F 0 < ... < F k < ... ) ∈ Fin( L ) N and w = S k w k = ( l 0 , ..., l k , ... ) ∈ N N . By the definition o f T d L , we get that T k B l k = { x } ∈ X a nd that for ev ery k ∈ N there ex ists m k ∈ F k ⊆ L with | f n k ( x ) − f m k ( x ) | > 1 d +1 . As F i < F j for all i < j , we see that m i < m j if i < j . It follows that M = { m 0 < ... < m k < ... } ∈ [ L ]. Thus, the sequence  f n ( x )  n ∈ L is not Cauch y and so L / ∈ L f , as desired.  By Lemma 20, the reduction of L f to WF ( N × Fin × N ) is constructed. Notice that for every L ∈ L f and every d 1 ≤ d 2 we ha ve o ( T d 1 L ) ≤ o ( T d 2 L ) and mo reov er o ( T L ) = sup { o ( T d L ) : d ∈ N } + 1. Remark 7. W e should p o int out that the reason wh y in the definition of T d L the no de t is a finite se q uence of finite sets rather than natura l num ber s, is to get the estimate in Pro po sition 22 b elow. Having natural num b ers ins tead o f finite sets would a ls o lead to a ca nonical ra nk. 2. The r e duction of L f ,f to WF( N × N ). The reduction is similar to that of the previous step, a nd so, we shall indicate o nly the necessary changes. Let d ∈ N . As 22 P ANDELIS DODO S befo re, for every L ∈ [ N ] w e a sso ciate a tree S d L ∈ T r( N × N ) a s follows. W e let S d L =  ( s, w ) : ∃ k ∈ N with | s | = | w | = k, s = ( n 0 < ... < n k − 1 ) ∈ [ L ] < N , w = ( l 0 , ..., l k − 1 ) ∈ N < N is acceptable and ∀ 0 ≤ i ≤ k − 1 ∀ z ∈ B l i we have | f n i ( z ) − f ( z ) | > 1 d + 1  . Next we glue the sequence o f trees ( S d L ) d ∈ N as we did with the sequence ( T d L ) d ∈ N and we build a tree S L ∈ T r( N × N ) de fined b y the rule ( s, w ) ∈ S L ⇔ ∃ d ∃ ( s ′ , w ′ ) such that ( s ′ , w ′ ) ∈ T d L and s = d a s ′ , w = d a w ′ . Again it is easy to c heck that the map [ N ] ∋ L 7→ S L ∈ T r( N × N ) is contin uous. Moreov er we hav e the following analogue of Lemma 20. The pr o of is identical and is left to t he reader. Lemma 21. L et L ∈ [ N ] . Then L ∈ L f ,f if and only if S L ∈ WF( N × N ) . This giv es us the reduction of L f ,f to WF( N × N ). As b efore we hav e o ( S L ) = sup { o ( S d L ) : d ∈ N } + 1 for ev ery L ∈ L f ,f . W e pro ce ed now to discuss the question whether for a given L ∈ L f ,f we can bo und the order of the tree S L by the order of T L . The following example sho ws that this is not in general po ssible. Example 2 . Let A (2 N ) = { δ σ : σ ∈ 2 N } ∪ { 0 } b e the o ne p oint co mpactification of 2 N . This is not a separable Rosen thal compact, but it ca n be s upplement ed t o o ne in a standa rd wa y (see [P1], [Ma ], [T o]). Specifica lly , let ( s n ) n be the en umera tio n of the Cantor tree 2 < N as in Exa mple 1. F or ev ery n ∈ N , let f n = χ V s n , where V s n = { σ ∈ 2 N : s n ⊏ σ } . Then A (2 N ) ∪ { f n } n is a separable Rosen thal compact. Now let A b e a Σ 1 1 non-Borel subset of 2 N . F ollowing [P2] (see a lso [Ma]), let K A be the se pa rable Rosenthal compact o btained by re stricting every f unction in A (2 N ) ∪ { f n } n on A . The s e q uence f A = ( f n | A ) n is a countable dense subset o f K A consisting of contin uous functions and 0 ∈ K A is a non- G δ po int (a nd obviously contin uo us ). Consider the sets L A f = { L ∈ [ N ] : ( f n | A ) n ∈ L is p oint wise conv erg ent on A } and L A f , 0 = { L ∈ [ N ] : ( f n | A ) n ∈ L is p oint wise con vergent to 0 on A } . Let φ be a Π 1 1 -rank o n L A f and ψ a Π 1 1 -rank on L A f , 0 . W e claim that there do es no t exist a map Φ : ω 1 → ω 1 such that ψ ( L ) ≤ Φ  φ ( L )  for a ll L ∈ L A f , 0 . Assume not. Let R = { L ∈ [ N ] : ∃ σ ∈ 2 N with s n ⊏ σ ∀ n ∈ L } . CODINGS OF SEP ARABLE COMP ACT A 23 Then R is a clos ed subset of L A f . F or every L ∈ R , le t σ L = S n ∈ L s n ∈ 2 N . The map R ∋ L 7→ σ L ∈ 2 N is clearly contin uo us. O bserve that for every L ∈ R w e hav e that L ∈ L A f , 0 if and only if σ L / ∈ A . As R is a Bo rel subset of L A f , by b oundedness we get tha t sup { φ ( L ) : L ∈ R } = ξ < ω 1 . Let ζ = sup { Φ( λ ) : λ ≤ ξ } . The set B = R ∩ { L ∈ L A f , 0 : ψ ( L ) ≤ ζ } is Bo rel and B = R ∩ L A f , 0 . Hence, the set Σ B = { σ L : L ∈ B } is an analytic subse t of 2 N \ A . As 2 N \ A is Π 1 1 -true, there exists σ 0 ∈ 2 N \ A with σ 0 / ∈ Σ B . P ick L ∈ R with σ L = σ 0 . Then L ∈ B yet σ L / ∈ Σ B , a co ntradiction. Although w e cannot, in general, bo und the order of t he tree S L by that of T L , the following propo sition shows tha t this is po ssible for an important spec ial ca s e. Prop ositi o n 22. L et X b e lo c al ly c omp act, K a sep ar able R osenthal c omp act on X , f = ( f n ) n a dense se quenc e in K c onsisting of c ont inuous fun ctions and f ∈ K . If f is c ontinuous, then o ( S L ) ≤ o ( T L ) fo r al l L ∈ L f ,f . In p articular, ther e exists a Π 1 1 -r ank φ on L f and a Π 1 1 -r ank ψ on L f ,f with ψ ( L ) ≤ φ ( L ) for al l L ∈ L f ,f . Pr o of. W e will show that for every d ∈ N we hav e o ( S d L ) ≤ o ( T d L ) for e very L ∈ L f ,f . This clear ly completes the pro of. So fix d ∈ N and L ∈ L f ,f . W e shall construct a monotone ma p Φ : S d L → [Fin( L )] < N such that for every ( s, w ) ∈ S d L the follo wing hold. (i) | ( s, w ) | = | Φ  ( s, w )  | . (ii) If s = ( n 0 < ... < n k ), w = ( l 0 , ..., l k ) and Φ  ( s, w )  = ( F 0 < ... < F k ), then for every i ∈ { 0 , ..., k } a nd every z ∈ B l i there exists m i ∈ F i with | f n i ( z ) − f m i ( z ) | > 1 d +1 . Assuming that Φ has been constructed, let M : S d L → T d L be defined b y M  ( s, w )  =  s, Φ  ( s, w )  , w  . Then it is easy to see that M is a well-defined monotone map, and so, o ( S d L ) ≤ o ( T d L ) as desired. W e pr o ceed to the constructio n of Φ. It will b e c onstructed by recursion on the length of ( s, w ). W e set Φ  ( ∅ , ∅ )  = ( ∅ ). Let k ∈ N and ass ume that Φ  ( s, w )  has bee n defined for every ( s, w ) ∈ S d L with | ( s, w ) | ≤ k . L e t ( s ′ , w ′ ) = ( s a n k , w a l k ) ∈ S d L with | s ′ | = | w ′ | = k + 1 . By the definition of S d L , we hav e that | f n k ( z ) − f ( z ) | > 1 d for every z ∈ B l k . Put p = max  n : n ∈ F and F ∈ Φ  ( s, w )  ∈ N . F o r every z ∈ B l k we ma y sele c t m z ∈ L with m z > p and suc h that | f n k ( z ) − f m z ( z ) | > 1 d +1 . As the functions ( f n ) n are con tinuous, we pick an o p e n neigh b orho o d U z of z suc h that | f n k ( y ) − f m z ( y ) | > 1 d +1 for a ll y ∈ U z . By the compactness o f B l k , there e x ists z 0 , ..., z j k ∈ B l k such th at U z 0 ∪ ... ∪ U z j k ⊇ B l k . Let F k = { m z i : i = 0 , ..., j k } ∈ Fin( L ) and notice that F ≤ p < F k for every F ∈ Φ  ( s, w )  . W e set Φ  ( s ′ , w ′ )  = Φ  ( s, w )  a F k ∈ [Fin( L )] < N . 24 P ANDELIS DODO S It is ea sy to chec k that Φ  ( s ′ , w ′ )  satisfies (i) and (ii) ab ove. The pro o f is c om- pleted.  References [AD] S. A. Argyros and P . Do dos, Generic ity and amalgamation of classes of Banach sp ac es , Adv ances in Math. (to appear). [ADK] S. A. Ar gyros, P . Dodos and V. Kanellop oulos, T r e e structur es asso ciated to a family o f functions , Journal Sym b. Logic, 70 (2005), 681-695. 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Na tional Technical University of A thens, F acul ty of Applied Sciences, Dep a r tmen t of Ma thema tics, Zografou Campus, 157 80, A thens, Greece E-mail addr e ss : pdodos@math.nt ua.gr