math.LO 2010-03-25 0

A new characterization of Baire class 1 functions

We give a new characterization of the Baire class 1 functions (defined on an ultrametric space) by proving that they are exactly the pointwise limits of sequences of full functions (which are particularly simple Lipschitz functions). Moreover we high…

Authors: L. Motto Ros

A NEW CHARA CTERIZA TION OF BAIRE CLASS 1 FUNCTIONS LUCA MOTTO ROS Abstract. W e giv e a new c haracterization of the Baire class 1 functions (de- fined on an ul tr ametric space) by pro ving that they are exactly the point wise limits of sequenc es of full functions (whic h are particularly simple Lipsc hitz functions). Moreov er w e highlight the l ink betw een the t wo classical strati- fications of the Borel functions b y sho wing that the Baire class functions of some lev el are exactly those obtained as unifor m limi ts of sequences of Delta functions (of a corresp onding level). 1. Introduction If X and Y are metriz able spaces, a function f : X → Y is said to b e c ontinuous if the preimage o f a n op en set of Y is op en with resp ect to the top olog y o f X , i.e. if f − 1 ( U ) ∈ Σ 0 1 ( X ) for every U ∈ Σ 0 1 ( Y ). There a re tw o na tural generaliza tions of this definition, namely functions such that f − 1 ( U ) ∈ Σ 0 ξ +1 ( X ) for every U open in Y a nd functions such that f − 1 ( S ) ∈ Σ 0 ξ ( X ) for every S ∈ Σ 0 ξ ( Y ) (for ξ < ω 1 ): the former are ca lled Bair e class functions (of level ξ ) while the latter ar e called Delta functions (of lev el ξ ) . Each generalization provides a stratification of the Borel functions from X to Y , but if we compar e the levels of the tw o hierarchies, that is if we fix some ξ < ω 1 in the definitions ab ove, they are quite different: for example, each level of the Delta functions is clo sed under co mp osition, while no level of the Baire cla ss functions (a part fro m contin uous functions) has such a pro p erty . The Baire class stratification w as introduced by Baire in 1899 (with a s lightly different de finition which, how ever, turns out to b e equiv alent to the one pro p osed here in the relev ant cas es) a nd has bee n extensively studied. Of particular interest are the Bair e class 1 functions, i.e. tho se functions such that the pr eimage of an op en set is a Σ 0 2 set. F or example, if f : [0 , 1] → R is differentiable (at endpoints we take one-side deriv ativ es), then its deriv ative f ′ is of Baire c lass 1. Moreover, Baire class 1 functions (in particular those from the B aire spa ce ω ω or fro m a ny compact space X to R ) have lots of a pplications in the theory of Banach spaces (for more on this sub ject see, for example, [4], [2 ], [3], [6 ], [5] a nd refere nces quo ted there). In this pap er we will g ive a new characterizatio n for the Baire class 1 functions defined fr om an ultrametric spa ce X (such as the Bair e spa ce ω ω or the Cantor space ω 2) to any s eparable metric space Y , by s howing that they a re exa ctly the po int wise limits o f sequences of full functions (which a re particula r Lipschit z func- tions) b etw een X a nd Y . Mo reov er w e will show that the tw o hierar chies presented Date : Nov em b er 17, 2018. 2000 Mathematics Subje ct Classific ation. 03E15, 54H05. Key wor ds and phr ases. Baire class 1 functions, Delta functions, Lipschitz functions. Researc h partially supported b y FWF Grant P 19898-N18. 1 2 LUCA MOTTO ROS befo re are intimately r elated b y proving that a function is of level ξ in the Baire class stratification just in case it is the uniform limit of functions of level ξ + 1 in the Delta stra tification. In particular , this gives another character ization of the Ba ire class 1 functions (taking ξ = 1). The pap er is organized as follows. In Se ction 2 we give so me (old and new) definitions and state the main Theorems o f the pap er. In Sectio n 3 we consider the relations b etw een Bair e class and Delta functions, while in Sectio n 4 we prov e some Theorems a b out zero- dimensional and ultra metric spaces. The results of these t wo Sections a re partia lly implicit in some cla ssical pr o ofs, but we put them here since we wan t to highlig h t the link b etw een the t wo str atifications o f the Borel-functions and the specia l prop erties of Borel- partitions of completely disconnected spaces. Finally , in Section 5 w e give the pro of of the new characterizatio n of the B aire class 1 functions. All the pro ofs need only a very small fragment o f the Axio m of Choice, na mely Countable Cho ice ov er the Reals ( AC ω ( R ) for short) 1 . It se ems not p os sible to av oid this (very weak) assumption since it is needed even to prov e very ba sic re- sults in Descr iptive Set Theory , e.g. to prov e that Σ 0 2 ( R ) is closed under co unt able unions. Hence w e will alwa ys w ork under ZF + A C ω ( R ). All the metrics d consider ed throughout the pap er are alwa ys assumed to be such that d ≤ 1. This co ndition is needed for the pro ofs of s ome of the results, but it is no t a true limitation. In fact, given any metric d on X , it is easy to see that d ′ = d 1+ d is a metric on X compatible with d suc h that d ′ ≤ 1. Moreov er, d is an ultrametric if and only if d ′ is an ultra metric, and one ca n eas ily chec k tha t a ll the definitions given in this pap er are “ inv ar iant” under such a transformatio n of the metric, e .g. A ⊆ X is a full set with resp ect to d just in case it is a full s et with resp ect to d ′ (although with different constants 2 ). Thus all the res ults ho ld a lso when considering arbitr ary (ul- tra)metrics. Finally , given an y t w o sets A and B , we will denote b y A B the set of all the functions from A to B a nd b y <ω A the set of all the finite seque nces of elemen ts from A . In particula r, ω ω (the s et of all the ω -sequences o f natural num ber s) will denote the Bair e space (endow ed with the usual top ology), while <ω ω will denote the se t o f a ll the finite sequences o f na tural num b ers. F or all the other undefined concepts and symbols w e will alw ays refer the reader to the standard monograph [1]. Finally , it is the author ’s pleasure to a knowledge his debt to S law omir Sole cki for his revie w of the pr esent w ork and for the suggestio n of a further generaliza tion of the characterizatio n pr eviously o btained. 2. P reliminaries and st a tement of the main resul ts W e star t with a few o f definitio ns a nd ba sic results, following closely the presen- tation o f [1]. 1 The fact that we will not use the ful l Axiom of Choice b ecomes relev ant if one w ant s to assume other axioms which con tradict AC (which how eve r are, in general, consisten t wi th AC ω ( R )). F or example, the Axiom of Determinacy AD is nee ded to carr y out the W adge’s analysis of con tinuous reducibility , so it could b e useful to chec k that our results hold also in that conte xt. 2 In particular, one constan t can b e obtained from the other one via the bijection j : R + → (0 , 1) : r 7→ r 1+ r . A NEW CHARA CTERIZA TION OF BAIRE CLASS 1 FUNCTIONS 3 Definition 1 . Let X , Y b e metrizable s paces and ξ < ω 1 a nonzero or dinal. A function f : X → Y is of Bair e class 1 if f − 1 ( U ) ∈ Σ 0 2 ( X ) for every op en s et U ⊆ Y . Recurs ively , for 1 < ξ < ω 1 we define now a function f : X → Y to b e of Bair e class ξ if it is the p oint wise limit of a sequence o f functions f n : X → Y , where f n is of Ba ire cla ss ξ n < ξ . W e denote by B ξ ( X, Y ) the sef of Bair e class ξ functions from X in to Y . A function f which is o f B aire class ξ (for some nonzer o countable o rdinal ξ ) is called a Bair e class function . Definition 2. Let X , Y be metrizable spaces and let Γ b e some collection of subsets of X . W e say that f : X → Y is Γ -me asur able if f − 1 ( U ) ∈ Γ for every open set U ⊆ Y . The link betw een Γ-measurable and Baire class function is giv en b y the following classical The orem. Theorem 2.1 (Lebes gue, Hausdor ff, Banach) . L et X , Y b e met rizable sp ac es, with Y s ep ar able. Then for 1 ≤ ξ < ω 1 , f : X → Y is of Bair e class ξ if and only if f is Σ 0 ξ +1 -me asu r able. By analog y with re sp ect to this Theorem, we say that a function f betw een t wo metrizable spaces is o f Bair e class 0 if and o nly if it is Σ 0 1 -measurable , i.e. if and only if f is co nt inuous. As a consequence of this Theorem, if X and Y are metriza ble space s and Y is separable, then the Ba ire class ξ functions pr ovide a stratification in ω 1 levels o f all the Bore l functions , i.e. functions s uch that f − 1 ( U ) is Bo rel for any U ∈ Σ 0 1 ( Y ) (Borel-meas urable functions). In fact for every nonzero countable ξ and every f ∈ B ξ ( X, Y ), f is clearly Bor el. Co nv er sely , let U n be a countable basis for the top ology of Y and let f b e Bor el. Let µ n be nonzero countable or dinals suc h that f − 1 ( U n ) ∈ Σ 0 µ n and let ξ = s up { µ n | n ∈ ω } (which is a gain a nonzero countable o rdinal). Since Σ 0 ξ is closed under countable unions and f − 1 ( U n ) ∈ Σ 0 ξ for every n ∈ ω , we hav e that f ∈ B ξ ( X, Y ). N ote also that any B ξ ( X, X ) is not closed under c omp osition since, in general, if f ∈ B µ ( X, Y ) and g ∈ B ν ( Y , Z ) then g ◦ f ∈ B µ + ν ( X, Z ). This result follows from the fact that if A ∈ Σ 0 ν ( Y ) and f ∈ B µ ( X, Y ) then f − 1 ( A ) ∈ Σ 0 µ + ν . The following is a nother classical fa ct 3 . Theorem 2. 2 (Leb esg ue, Ha usdorff, Banach) . L et X , Y b e sep ar able metrizable. Mor e over, assume that either X is zer o-dimensional or Y = R n for some n ∈ ω (or even Y = C m or Y = [0 , 1] m for some m ∈ ω ). Then f : X → Y is of Bair e class 1 if and only if f is the p ointwise limit of a se quenc e of c ontinuous fu n ctions. Hence, under the hypo theses of this Theorem, f ∈ B ξ ( X, Y ) if and only if it is the p o int wise limit of a sequence o f functions in S ν <ξ B ν ( X, Y ), for a ll ξ ≥ 1. 3 In gene ral, i f X and Y ar e metri zable with Y s eparable and f : X → Y is the p oint wise lim it of a sequence of contin uous functions then f is of Baire class 1. Nevertheless the conv ers e fails in the general case: for a coun terexample, simply take X = R and Y = { 0 , 1 } (with the di s crete metric) and consider the function such that f (0) = 1 and f ( x ) = 0 f or every x 6 = 0. 4 LUCA MOTTO ROS There is another stra tification of the Borel functions (in the case Y sepa rable) which is imp ortant because , con trar y to the c ase of Baire class functions, every level is a set of functions c losed under comp osition. Definition 3. Let X , Y be metrizable spa ces and ξ < ω 1 be a nonz ero or dinal. A function f : X → Y is a ∆ 0 ξ -function ( ∆ 0 ξ for sho rt) if f − 1 ( A ) ∈ Σ 0 ξ ( X ) for every A ∈ Σ 0 ξ ( Y ). W e denote by D ξ ( X, Y ) the set of such functions. Prop ositi on 2.3. F or ξ > 1 the fol lowing ar e e qu ivalent 4 : i) f is ∆ 0 ξ ; ii) f − 1 ( A ) ∈ Π 0 ξ for every A ∈ Π 0 ξ ; iii) f − 1 ( A ) ∈ ∆ 0 ξ for every A ∈ ∆ 0 ξ ; iv) f − 1 ( A ) ∈ Σ 0 ξ for every ν < ξ and A ∈ Π 0 ν ; v) f − 1 ( A ) ∈ ∆ 0 ξ for every ν < ξ and A ∈ Σ 0 ν . Pr o of. Since Σ 0 ξ is closed under countable union, it is easy to see that i ) ⇐ ⇒ i ii ). i ) ⇐ ⇒ i i ) is obvious, and also ii i ) ⇒ v ) is trivial (since Σ 0 ν ⊆ ∆ 0 ξ for every ν < ξ ). v ) ⇒ iv ) since ∆ 0 ξ is closed under complementation and is contained by definition in Σ 0 ξ . Finally , to s ee that iv ) ⇒ i ) r ecall that, by definition, every Σ 0 ξ set A can be wr itten as a countable union of S ν <ξ Π 0 ν sets.  As in the ca se of Bair e class functions, a function f which is a ∆ 0 ξ -functions (for some no nzero co un table ordinal ξ ) is called a D elta fun ction . T o observe that the Delta functions pr ovide a stratification in ω 1 levels of all the Borel functions it is enough to obser ve that every o p en set of Y is in Σ 0 ξ ( Y ) for every nonzero countable ordinal ξ and every metrizable space Y (and hence every Delta function is Borel) a nd that every Bair e clas s function is a Delta function. T o see this, let f ∈ B ν ( X, Y ) and let ξ b e the first additively clo sed o rdinal ab ov e ν (that is ξ = ν · ω ): we claim tha t f is a ∆ 0 ξ -function. In fact, let S ∈ Σ 0 ξ : b y definition, S = S n P n , where ea ch P n ∈ Π 0 µ n ( Y ) for some µ n < ξ . Since f ∈ B ν ( X, Y ) we hav e that Q n = f − 1 ( P n ) ∈ Π ν + µ n ( X ) and hence f − 1 ( S ) = S n Q n where each Q n is in Π 0 ν + µ n ( X ) . Since ξ is additively clo sed and ν , µ n < ξ we hav e tha t ν + µ n < ξ for every n ∈ ω : therefore f − 1 ( S ) ∈ Σ 0 ξ ( X ) b y definition. Moreov er, using ag ain the fact tha t Σ 0 1 ( Y ) ⊆ Σ 0 ξ ( Y ), it is easy to c heck that D ξ +1 ( X, Y ) ⊆ B ξ ( X, Y ). Definition 4. Let X and Y b e t wo metrizable spaces and let F ⊆ G be tw o sets of functions fro m X to Y . Then F is a b asis for G just in ca se every function in G is the uniform limit of a sequence of functions in F . W e w ill prov e in Section 3 that ea ch lev e l of the Delta functions for ms a basis for a c orresp o nding level of the Baire class functions. This result is e ssentially implicit in the pr o of o f Theorem 2 .1 (see [1]), but w e will reprov e it here for the sa ke of completeness. 4 F or ξ = 1 we ha ve in general that i ) ⇐ ⇒ ii ) ⇒ iii ) ⇐ ⇒ iv ) ⇐ ⇒ v ) but not iii ) ⇒ i ) (in fact i f Y is connect ed we hav e that every function f satisfies iii ), but f is a ∆ 0 1 -function if and only if f is con tinuous) . Neve rtheless the Proposi tion remains true even for ξ = 1 if we r equire that Y is zero-dimensional. A NEW CHARA CTERIZA TION OF BAIRE CLASS 1 FUNCTIONS 5 Theorem 2.4. L et ( X , d X ) , ( Y , d Y ) b e two metric sp ac es and assume that Y is also sep ar able. A function f : X → Y is of Bair e class ξ if and only if it is the uniform limit of a se quen c e of ∆ 0 ξ +1 -functions. Corollary 2.5. L et ( X, d X ) , ( Y , d Y ) b e t wo metric sp ac es and assume that Y is also sep ar able. A fun ct ion f : X → Y is in B 1 ( X, Y ) if and only if it is the uniform limit of a se quenc e of ∆ 0 2 -functions. F ro m Theo rem 2 .4 we can a lso derive the fo llowing Corolla ry . It can b e se en as an extension of Theo rem 2.2: in that case it was prov e d (under stronger hypotheses) that f is of Baire class 1 if and o nly if it is the p oint wis e limit o f a sequence of ∆ 0 1 -functions (i.e. c ontin uo us functions). Her e we prove the sa me r esult for every level different fro m 1 (under weaker hypothes es). Corollary 2.6. L et X , Y b e two metrizable s p ac es and assu me that Y is also sep ar able. Then for every 1 < ξ < ω 1 , f : X → Y is of Bair e class ξ if and only if f is the p ointwise limit of a se quenc e of ∆ 0 ξ -functions. By Theor em 2.2, as previo usly observed, Cor ollary 2.6 remains tr ue in the ca se ξ = 1 if w e requir e that X is separ able a nd either X is zero -dimensional or Y is one o f R n , [0 , 1] n or C n (for so me n ∈ ω ). Finally , we wan t to g ive a new characterizatio n o f the Bair e c lass 1 functions. First r ecall the following Definition. Definition 5. Let ( X , d X ), ( Y , d Y ) b e tw o metric spac es. A function f : X → Y is Lipschitz (with c onstant L ∈ R + ) if ∀ x, x ′ ∈ X ( d Y ( f ( x ) , f ( x ′ )) ≤ L · d X ( x, x ′ )) . W e deno te b y Lip( X , Y ; L ) the set of such functions and put Lip( X , Y ) = S L ∈ R + Lip( X, Y ; L ). Let now ( X , d X ) b e an ultr ametric sp ac e , i.e. a metric s pace s uch that d X is an ultrametric. A set A ⊆ X is ful l (with c onstant r ∈ R + ) if ∀ x ∈ A ( B ( x, r ) ⊆ A ) , were B ( x, r ) = { y ∈ X | d X ( x, y ) < r } is the usual o p en ball. Prop ositi on 2.7. L et ( X , d X ) b e an ultr ametric sp ac e. Then the ful l subsets of X form an algebr a. Mor e over, an arbitr ary union of b al ls with a fixe d r adius is ful l (in p articular, an arbitr ary union of ful l sets with the s ame c onstant is ful l). Pr o of. Let A a nd B be full s ets with constan ts r A and r B resp ectively . Then it is easy to c heck that A ∪ B is full with c onstant r = min { r A , r B } . Mo reov er, let x / ∈ A and assume tow ards a co nt ra diction that y ∈ A for so me y ∈ B ( x, r A ). B y the prop er ties of the ultrametric d X , we hav e that B ( y , r A ) = B ( x, r A ): but since A is full with constant r A , then B ( y , r A ) ⊆ A and hence x ∈ A , a co nt ra diction! Thu s X \ A is full (with constant r A ). The second part follows ag ain from the prop erties o f a n ultra metric.  Definition 6. Le t ( X, d X ) be a n ultrametric space and Y b e any separable metriz- able space. A function f : X → Y is said to b e ful l if it ha s only finitely man y v alues and the pr eimage of ea ch o f these v alues is a full set. 6 LUCA MOTTO ROS The function f is s aid to b e ω - ful l if it has at most co unt ably many v alues and there is some fixed r ∈ R + such that the preimage of each v alue is a full set with constant r . It is clear tha t every full function is ω -full. More ov er, if f is ω -full and r ∈ R + witnesses this, then f ∈ Lip( X , Y ; r − 1 ) (with resp ect to any metric d Y compatible with the top olog y of Y such that d Y ≤ 1). In fa ct, let d Y be such a metr ic and let x, x ′ ∈ X : if d X ( x, x ′ ) ≥ r then d Y ( f ( x ) , f ( x ′ )) ≤ 1 = r − 1 · r ≤ r − 1 d X ( x, x ′ ) , while if d X ( x, x ′ ) < r then x ′ ∈ f − 1 ( f ( x )) (since x ′ ∈ B ( x, r )) and he nce f ( x ) = f ( x ′ ). Prop ositi on 2.8 . L et ( X , d X ) b e an ult ra metric sp ac e and ( Y , d Y ) , ( Z, d Z ) b e two metric sp ac es. L et f : X → Y b e a ful l function, g ∈ Lip( Y , Z ; L ) and h ∈ Lip( Z, X ; L ) . Then g ◦ f is ful l and, if d Z is an ult ra metric, also f ◦ h is ful l. The same r esult holds if we sistematic al ly r eplac e “ ful l” with “ ω -ful l”. Pr o of. The firs t part is obvious, s ince for every z ∈ Z the set ( g ◦ f ) − 1 ( z ) is either empt y or the union of finitely ma ny full sets (a nd the cardina lity of r ange( g ◦ f ) is less or equa l than the cardinality of range( f )). F or the seco nd part, it is enough to show that the preimag e via h of a full set A ⊆ X (with consta nt r ) is a full set (with cons tant r · L − 1 ). In fact, le t z ∈ Z be such that h ( z ) ∈ A and le t z ′ ∈ Z be such that d Z ( z , z ′ ) < r L − 1 . Then d X ( h ( z ) , h ( z ′ )) ≤ L · d Z ( z , z ′ ) < Lr L − 1 = r , and thus h ( z ′ ) ∈ A . But this implies B ( z , r L − 1 ) ⊆ h − 1 ( A ) and hence we a re done. The cas e in which f is ω - full is proved in a similar wa y .  Now we are rea dy to state the main Theorem of this pap er . Theorem 2.9. L et ( X , d X ) b e an ultr ametric sp ac e and let Y b e any sep ar able metrizable sp ac e. Then f : X → Y is of Bair e class 1 if and only if f is t he p ointwise limit of a se quenc e of ful l functions. By the obser v ations a b ov e and since every Lips chitz function is uniformly con- tin uous, we hav e also the following Corollar y as a simple co nsequence of Theor em 2.9. Corollary 2.10. L et ( X , d X ) , ( Y , d Y ) b e sep ar able metric sp ac es and assume that X is an ultr ametric s p ac e with r esp e ct to d X . F or every f : X → Y the fol lowing ar e e quivalent: i) f is of Bair e class 1 ; ii) f is the p ointwise limit of a se quenc e of ω -ful l funct ions; iii) f is the p ointwise limit of a se quenc e of Lipschitz functions; iv) f is the p ointwise limit of a se quenc e of un iformly c ontinuous functions. The author first prov ed Theor em 2.9 but using Lipschit z (in par ticular ω -full) functions rather than full functions (although the pro of was e ssentially the s ame presented her e in Section 5): the idea to gener alize the r esult to the pr esent form (as well as the definition of fullness) is due to S. Solecki. 3. The link between Baire class and Del t a functions W e first give some basic definitions. A NEW CHARA CTERIZA TION OF BAIRE CLASS 1 FUNCTIONS 7 Definition 7. Let X be a top olo gical space and Γ ⊆ P ( X ) b e any p ointclass. A Γ -p artition of a set C ∈ Γ is a family h C n | n < N i o f no nempt y pairwise disjoint sets o f Γ such that C = S n j i ∧ R j \ S l 1 and f : X → Y is of Bair e class ξ then ther e is a s e quenc e of ∆ 0 ξ -functions p ointwise c onver ging to f . Pr o of. Let d b e a co mpatible metric on Y . Recall that every o p en sphere U o f Y can be written as the union of co untably man y clo sed sphere ea ch of whic h is contained in t he int erio r of the follo wing one. In fact let U = B ( y 0 , ε ) = { y ∈ Y | d ( y , y 0 ) < ε } and let h ε m | m ∈ ω i be a strictly increas ing sequence of real such that ε m < ε for every m ∈ ω and lim m ε m = ε : then U = S m ∈ ω B m ( y 0 , ε m ) = S m ∈ ω { y ∈ Y | d ( y , y 0 ) ≤ ε m } . Moreov er, since ε n < ε n +1 , we hav e a lso B m = B ( y 0 , ε m ) ⊆ B m +1 = B ( y 0 , ε m +1 ), tha t is U = S m ∈ ω B m . Now assume that h f k | k ∈ ω i is a sequence of ∆ 0 ξ -functions point wise conv er ging to f : it is eno ugh to prov e that f − 1 ( U ) ∈ Σ 0 ξ +1 ( X ) for every op en spher e U = S m B m ⊆ Y . Firs t no te that f − 1 ( U ) = [ m ∈ ω [ n ∈ ω \ k ≥ n f − 1 k ( B m ) . In fa ct, if f ( x ) ∈ U then there is an m such that f ( x ) ∈ B m ⊆ B m and hence also f k ( x ) ∈ B m for a ny k large enough (since f k conv erge to f ). F or the other direction, if there is some m s uch that f k ( x ) ∈ B m for a lmost all k , thus also f ( x ) (whic h is the limit of the p oints f k ( x )) must b elong to the sa me B m (since it is closed). Since ea ch f k is a ∆ 0 ξ -function and since Π 0 1 ( Y ) ⊆ Π 0 ξ ( Y ) fo r every nonzero countable ξ , we ha ve that f − 1 ( B m ) ∈ Π 0 ξ ( X ) for every m ∈ ω and hence also T k ≥ n f − 1 k ( B m ) ∈ Π 0 ξ ( X ) for every n ∈ ω (since Π 0 ξ ( X ) is clo sed under co unt able int erse ctions). But then f − 1 ( U ) is a co unt able union of Π 0 ξ ( X ) sets, i.e. it is a Σ 0 ξ +1 ( X ) set and we are done. Conv ersely , if ξ > 1 a nd f is o f Baire class ξ then it is the point wise limit of some sequence f n of functions such that for e very n ∈ ω there is a 1 ≤ ν n < ξ such that f n is of Baire class ν n . Using Theorem 2.4, find for every n ∈ ω a sequence g n,m of ∆ 0 ν n +1 -functions conv erging uniformly to f n . Note that b y the construction ab ov e (Claim 3 .2.1) we ca n as sume that d ( g n,m ( x ) , f n ( x )) ≤ 2 − m for every x ∈ X . Moreov er, since ν n + 1 ≤ ξ we hav e that every g n,m is, in particular, a ∆ 0 ξ -function. T ake a ny diagonal subsequenc e h h n | n ∈ ω i of the g n,m , e.g. h n = g n,n . It remains only to pr ov e that this sequence converges p oint wise to f . T o se e this, fix s ome x ∈ X and k ∈ ω . Let j ∈ ω b e such that ∀ i ≥ j ( d ( f i ( x ) , f ( x )) < 2 − ( k +1) ) 10 LUCA MOTTO ROS and put m = max { j, k + 1 } . Clearly , for every m ′ ≥ m we hav e d ( h m ′ ( x ) , f ( x )) ≤ d ( g m ′ ,m ′ ( x ) , f m ′ ( x )) + d ( f m ′ ( x ) , f ( x )) < < 2 − m ′ + 2 − ( k +1) ≤ 2 · 2 − ( k +1) = 2 − k .  The same Coro llary clearly holds if w e consider functions which are constant (resp ectively , Lipschitz, contin uous) on a fi nite ∆ 0 ξ -partition. 4. Z er o dimensional sp aces W e now prov e some Theorems o n z ero-dimensio nal and ultra metric spaces. In particular, the first is a simple v ariatio n of some classica l results (see [1]). Let s ∈ <ω ω b e a finite sequence of na tural n umbers. W e will deno te the length of s by lh( s ) (for mally , lh( s ) = dom( s )). Theorem 4.1. If ( X , d ) is a metric, sep ar able and zer o-dimensional sp ac e, t hen ther e is some set A ⊆ ω ω and an home omorphism h : A → X such that h ∈ Lip( A, X ; 1) (with r esp e ct to d and the usual metric d ′ that ω ω induc es on A ). If mor e over d is an ultr ametric then h c an b e taken bi-Lipschitz, i.e. h − 1 ∈ Lip( X, A ; 2) (and h ∈ Lip( A, X ; 1) as b efor e). If d is also c omplete t hen the set A c an b e taken to b e a close d set. Pr o of. The firs t part is a standard argument: one can co nstruct a Lusin scheme h C s | s ∈ <ω ω i on X such that i) C ∅ = X ii) C s is clop en iii) C s = S i ∈ ω C s a i iv) diam( C s ) ≤ 2 − lh( s ) . F ro m this one c an conclude that the induced map f is defined on the set A = { y ∈ ω ω | T n C y ↾ n 6 = ∅} and is a n homeomorphism. But condition iv) implies also f ∈ Lip( A, X ; 1). In fa ct, for every x, y ∈ A such tha t x 6 = y , let n ∈ ω be s uch that d ′ ( x, y ) = 2 − n and let s = x ↾ n = y ↾ n . Clear ly we have that h ( x ) ∈ C s and h ( y ) ∈ C s . T h us condition iv) implies that d ( h ( x ) , h ( y )) ≤ 2 − lh( s ) = 2 − n = d ′ ( x, y ). If we now as sume that d is a n ultrametric on X then we can construct a Lusin scheme h C s | s ∈ <ω ω i on X such tha t i) C ∅ = X ii) either C s = ∅ or C s is a sphere iii) C s = S i ∈ ω C s a i iv) diam( C s ) ≤ 2 − lh( s ) . In fact every nonempty C s (with s 6 = ∅ ) will b e defined as C s = B  x, 2 − lh( s )  for s ome x ∈ X . Let D b e c ountable and dense in X : we cons truct the scheme by induction on lh( s ). First put C ∅ = X . Suppo se to have constructed C s with prope rties i)- iv) . If C s = ∅ then put C s a i = ∅ fo r every i ∈ ω , otherwise fix a n enumeration h x k | k ∈ ω i of C s ∩ D . Then define C s a 0 = B  x 0 , 2 − (lh( s )+1)  and either C s a i +1 = B  x k i +1 , 2 − (lh( s )+1)  , where k i +1 is the smallest k > k i such that x k / ∈ S j ≤ i C s a j , or C s a i +1 = ∅ if such a k do es not exist. A NEW CHARA CTERIZA TION OF BAIRE CLASS 1 FUNCTIONS 11 Clearly C s a i ⊆ C s (since d is an ultra metric), diam( C s a i ) ≤ 2 − (lh( s )+1) and C s = S i ∈ ω C s a i bec ause D is dense , hence we a re done. Arguing as b efore, h is a bijection defined o n a set A ⊆ ω ω and h ∈ Lip( A, X ; 1). Now w e want to show that d ′ ( h − 1 ( x ) , h − 1 ( y )) ≤ 2 d ( x, y ) for e very dis tinct x, y ∈ X . Put S x,y = { s ∈ <ω ω | x ∈ C s ∧ y ∈ C s } . Clearly S x,y is linearly ordered and a dmits an element t of maxima l length (otherwise x = y ). Thus d ′ ( h − 1 ( x ) , h − 1 ( y )) = 2 − lh( t ) . If d ( x, y ) < 2 − (lh( t )+1) then, by the construc tion a b ov e and the fact that d is an ultrametric, there w ould be an i ∈ ω such that x ∈ C t a i and y ∈ C t a i , contradicting the ma ximality o f t . Hence d ( x, y ) ≥ 2 − (lh( t )+1) and d ′ ( h − 1 ( x ) , h − 1 ( y )) = 2 − lh( t ) = 2 · 2 − (lh( t )+1) ≤ 2 d ( x, y ) as r equired. Finally it is not hard to c heck that the completeness of d implies that A is a closed set.  Note th at, in particular, this Theorem pro vides also that every separ able, metriz- able and zero- dimensional space is ultr ametrizable (i.e. it admits a co mpatible ul- trametric d ): let h b e the homeomo rphism given by the Theo rem and simply put d ( x, x ′ ) = d ′ ( h − 1 ( x ) , h − 1 ( x ′ )) for every x, x ′ ∈ X , where d ′ is the usual (ultra )metric on ω ω . Clea rly , if X is also Polish w e have that the ultrametric d is also complete. F or notational s implicit y w e put Σ 0 0 = Π 0 0 = ∆ 0 0 = ∆ 0 1 . More ov er, for every countable ordinal ξ , we denote by Π 0 <ξ (resp ectively , Σ 0 <ξ and ∆ 0 <ξ ) the p ointclass S ν <ξ Π 0 ν (resp. S ν <ξ Σ 0 ν and S ν <ξ ∆ 0 ν ). Theorem 4.2. L et X b e a sep ar able, metrizable, zer o-dimensional sp ac e and let A b e a subset of X . F or every nonzer o ξ < ω 1 the fol lowing ar e e quivalent: i) A ∈ Σ 0 ξ ; ii) ther e is a ∆ 0 ξ -p artition of A , i.e. t her e is h C n | n < N i such that for every n, m < N we have C n ∈ ∆ 0 ξ , n 6 = m ⇒ C n ∩ C m = ∅ , and A = S n 1 and S ∈ Σ 0 ξ , b y definition there are some se ts P n ∈ Π 0 ν n such that S = S n P n and ν n < ξ for all n ∈ ω . Fir st define inductively P ′ 0 = P 0 and P ′ n +1 = P n +1 \ S i ≤ n P i and note that they form a partition of S . C learly eac h P ′ n can b e seen a s the difference of tw o Π 0 ν sets where ν = max { ν 0 , . . . , ν n } < ξ (since ν ′ ≤ ν ⇒ Π 0 ν ′ ⊆ Π 0 ν and Π 0 ν is clos ed under finite unions) and hence w e hav e only to prov e that for all ν < ξ , every set o f the fo rm Q ∩ R with Q ∈ Π 0 ν and 12 LUCA MOTTO ROS R ∈ Σ 0 ν admits a Π 0 ν partition. Using the inductive hypothesis, find a par tition h R n | n ∈ ω i of R such that R n ∈ Π 0 µ n for so me µ n < ν and note that R n ∈ Π 0 ν for every n ∈ ω . The n it is easy to chec k that the sets Q n = Q ∩ R n are in Π 0 ν and that they form a partition of Q ∩ R , hence we are do ne.  In particular, every Σ 0 ξ +1 set admits a Π 0 ξ -partition. This implies that every Σ 0 ξ +1 -partition of X ca n be refined to a Π 0 ξ -partition and, more genera lly , every Σ 0 ξ - partition can b e refined to a Π 0 <ξ -partition. Ther efore, in the case X is sepa rable, metrizable a nd ze ro-dimensiona l, we have the following improv ement o f Theo rem 3.2. Corollary 4.3. L et ( X , d X ) and ( Y , d Y ) b e two met ric sep ar able sp ac es and assume that X is also zer o-dimensional. Then a function f : X → Y i s of Bair e class ξ if and only if ther e is a se qu en c e of functions c onver ging uniformly t o it and su ch that e ach of t hem is lo c al ly c onstant (r esp e ctively, Lipschitz, c ontinuous) on a Π 0 ξ - p artition of X . 5. Baire class 1 and full functions Let Γ ⊆ P ( ω ω ) b e a b oldface p ointclass, i.e. a collection of subsetes of ω ω clo sed under co n tinuous preimage. W e s ay that a set A ∈ Γ is Γ -c omplete if for every B ∈ Γ there is a contin uous function f : ω ω → ω ω suc h that B = f − 1 ( A ) (such a function will b e ca lled a r e duct ion of B in A ). Recall also that a contin uous function from ω ω to ω ω can b e viewed as the function arising from s ome particula r function ϕ : <ω ω → <ω ω . W e say that ϕ : <ω ω → <ω ω is c ontinuous if s ⊆ t ⇒ ϕ ( s ) ⊆ ϕ ( t ) for every s, t ∈ <ω ω and for every x ∈ ω ω lim n ∈ ω (lh( ϕ ( x ↾ n ))) = ∞ . If ϕ is contin uous it induces in a canonical wa y the unique function f ϕ : ω ω → ω ω : x 7→ [ n ∈ ω ϕ ( x ↾ n ) , and it is not hard to see that f ϕ is a contin uous function. Conv ersely , supp ose f : ω ω → ω ω is contin uous. F or every s ∈ <ω ω consider the set Σ s = { t ∈ <ω ω | f ( N s ) ⊆ N t } . C learly Σ s is linearly or dered (beca use if t and t ′ are inc ompatible then N t ∩ N t ′ = ∅ ), and hence we ca n define ϕ ( s ) = t s where t s ∈ Σ s is such that lh( t s ) = ma x { lh( t ) | lh( t ) ≤ lh( s ) ∧ t ∈ Σ s } . It is not difficult to chec k that ϕ : <ω ω → <ω ω is contin uous and that f ϕ = f . By ana logy with the previous definitio ns, if A, B ⊆ ω ω and ϕ : <ω ω → <ω ω is a contin uous function such that f − 1 ϕ ( A ) = B , we call ϕ a r e duction of B into A and we say that ϕ r e duc es B to A . F r om the obser v ation a bove, it is clear that if A is Γ -complete for some pointclass Γ ⊆ P ( ω ω ) then for every B ∈ Γ there is a reduction ϕ : <ω ω → <ω ω of B in A . F or every t, s ∈ <ω ω define t − s = ∅ if lh( t ) < lh( s ), and t − s = u ∈ <ω ω , where u is such that t = ( t ↾ lh( s )) a u , o therwise. Let ~ ϕ = h ϕ n | n < N i b e a sequence of contin uous functions ϕ n : <ω ω → <ω ω . Moreov er, let h n k | k ∈ ω i b e an enumeration of N with infinite rep etitions such that 7 n k 6 = n k +1 for every k ∈ ω . Define ( ~ ϕ ) ∗ : <ω ω → <ω ω and σ : <ω ω → N 7 Clearly this last condition is required only i f N > 1. A NEW CHARA CTERIZA TION OF BAIRE CLASS 1 FUNCTIONS 13 in the following way: first put ( ~ ϕ ) ∗ ( ∅ ) = ∅ a nd σ ( ∅ ) = n 0 . Then supp os e to hav e defined ( ~ ϕ ) ∗ ( s ) a nd σ ( s ) = n k and inductively put ( ~ ϕ ) ∗ ( s a i ) = ( ~ ϕ ) ∗ ( s ) a 1 if ϕ σ ( s ) ( s a i ) − ( ~ ϕ ) ∗ ( s ) do es not contain 0, and ( ~ ϕ ) ∗ ( s a i ) = ( ~ ϕ ) ∗ ( s ) a 0 otherwise. Finally put σ ( s a i ) = σ ( s ) = n k in the firs t ca se a nd σ ( s a i ) = n k +1 in the s econd o ne. The function ( ~ ϕ ) ∗ is clear ly contin uous (since it is cons tructed extending a t each step the pr evious v alue and is such that lh(( ~ ϕ ) ∗ ( s )) = lh( s ) for every s ∈ <ω ω ) and is calle d the Σ 0 2 -c ont ro l funct ion 8 of the sequence ~ ϕ , while the function σ is the state function asso ciated to it. Mo reov er we will s ay that σ ( s ) ∈ N is the state of s with r esp e ct to ( ~ ϕ ) ∗ . Consider no w a family A n ⊆ ω ω of Σ 0 2 sets (for n < N ) and S = { x ∈ ω ω | ∃ n ∀ m ≥ n ( x ( m ) 6 = 0 ) } . Since S is Σ 0 2 -complete there a re cont inuous functions ϕ n : <ω ω → <ω ω which reduce A n to S , i.e. such that f − 1 ϕ n ( S ) = A n . Define ~ ϕ = h ϕ n | n < N i and let ( ~ ϕ ) ∗ and σ be co nstructed a s above. F or notational simplicity we put φ = ( ~ ϕ ) ∗ . W e want to prove the following Claim 5.0.1 . The function f φ : ω ω → ω ω is a reduction of S n m s uch that ϕ n ( x ↾ m ′ ) − ϕ n ( x ↾ m ) contains some 0. This implies that for every m ∈ ω there is an m ′ > m such that σ ( x ↾ m ′ ) 6 = σ ( x ↾ m ) a nd thus h σ ( x ↾ m ) | m ∈ ω i is not eventually co nstant.  Claim With the notation a b ov e, if x ∈ S n

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