A dichotomy for Borel functions

A DICHOTOMY F OR BOREL FUNCTIONS MAR CIN SABOK Abstra t. The di hotom y diso v ered b y Sole ki in [ 3 ℄ states that an y Baire lass 1 funtion is either σ -on tin uous or inludes the P a wlik o wski funtion P . The aim of this pap er is to giv e an argu- men t whi h is simpler than the original pro of of Sole ki and giv es a stronger statemen t: a di hotom y for all Borel funtions. 1. Intr odution An old question of Lusin ask ed whether there exists a Borel funtion whi h annot b e deomp osed in to oun tably man y on tin uous fun- tions. By no w sev eral examples ha v e b een giv en, b y Keldi², A dy an and No vik o v among others. A partiularly simple example, the fun- tion P : ( ω + 1) ω → ω ω , has b een found b y P a wlik o wski (f. [1℄). By denition, P ( x )( n ) =  x ( n ) + 1 if x ( n ) < ω , 0 if x ( n ) = ω . It is pro v ed in [1℄ that if A ⊆ ( ω + 1) ω is su h that P ↾ A is on tin uous then P [ A ] ⊆ ω ω is no where dense. Sine P is a surjetion, it is not σ -on tin uous. In [3℄ Sole ki sho w ed that the ab o v e funtion is, in a sense, the only su h example, at least among Baire lass 1 funtions (in other w ords, it is the initial ob jet in a ertain ategory). Theorem 1 (Sole ki, [3℄) . F or any Bair e lass 1 funtion f : X → Y , wher e X , Y ar e Polish sp a es, either f is σ - ontinuous or ther e exist top olo gi al emb e ddings ϕ and ψ suh that the fol lowing diagr am  ommutes: ω ω ψ − − − → Y x   P x   f ( ω + 1) ω ϕ − − − → X In [4℄ Zapletal generalized Sole ki's di hotom y to all Borel funtions b y pro ving the follo wing theorem. 1 2 MAR CIN SABOK Theorem 2 (Zapletal, [4℄) . If f : X → Y is a Bor el funtion whih is not σ - ontinuous then ther e is a  omp at set C ⊆ X suh that f ↾ C is not σ - ontinuous and of Bair e lass 1. In this pap er w e giv e a new pro of of the ab o v e di hotom y for all Borel funtions, whi h is diret, shorter and more general than the original pro of from [3 ℄. 2. Not a tion W e sa y that a Borel funtion f : X → Y , where X , Y are P olish spaes, is σ -on tin uous if there exist a oun table o v er of the spae X = S n X n (with arbitrary sets X n ) su h that f ↾ X n is on tin uous for ea h n . It follo ws from the Kurato wski extension theorem that w e ma y require that the sets X n b e Borel. If f is a Borel funtion whi h is not σ -on tin uous then the family of sets on whi h it is σ -on tin uous is a prop er σ -ideal in X . W e denote this σ -ideal b y I f . In a metri spae ( X , d ) for A, B ⊆ X let us denote b y h ( A, B ) the Hausdor distane b et w een A and B . The spaes ( ω + 1) ω and ( ω + 1) n for n < ω are endo w ed with the pro dut top ology of order top ologies on ω + 1 . 3. The Zaplet al 's game In [4℄ Zapletal in tro dued a t w o-pla y er game, whi h turnes out to b e v ery useful in examining σ -on tin uit y of Borel funtions. Let B ⊆ ω ω b e a Borel set and f : B → 2 ω b e a Borel funtion. Let ρ : ω → ω × 2 <ω × ω b e a bijetion. The game G f ( B ) is pla y ed b y A dam and Ev e. They tak e turns pla ying natural n um b ers. In his n -th mo v e, A dam pi ks x n ∈ ω . In her n -th mo v e, Ev e  ho oses y n ∈ 2 . A t the end of the game w e ha v e x ∈ ω ω and y ∈ 2 ω formed b y the n um b ers pi k ed b y A dam and Ev e, resp etiv ely . Next, y ∈ 2 ω is used to dene a sequene of partial on tin uous funtions (with domains of t yp e G δ in ω ω ) in the follo wing w a y . F or n < ω let f n b e a partial funtion from ω ω to 2 ω su h that for t ∈ ω ω and σ ∈ 2 <ω f n ( t ) ⊇ σ iff ∃ k ∈ ω y ( ρ ( n, σ , k )) = 1 and dom ( f n ) = { t ∈ ω ω : ∀ n < ω ∃ ! σ ∈ 2 n f n ( t ) ⊇ σ } . Ev e wins the game G f ( B ) if x 6∈ B or ∃ n f ( x ) = f n ( x ) . Otherwise A dam wins the game. It is easy to see that if f is a Borel funtion then G f is a Borel game. The k ey feature of the game G f is that it detets σ -on tin uit y of the funtion f . A DICHOTOMY F OR BOREL FUNCTIONS 3 Theorem 3 (Zapletal,[4 ℄) . F or B ⊆ ω ω and f : B → 2 ω Eve has a winning str ate gy in the game G f ( B ) if and only if f is σ - ontinuous on B . Note that if A dam has a winning strategy then the image of his strategy (treated as a on tin uous funtion from 2 ω to B ) is a ompat set on whi h f is also not σ -on tin uous. This observ ation and the Borel determinay giv es the follo wing orollary . Corollary 1 (Zapletal,[4 ℄) . If B is a Bor el set and f : B → 2 ω is a Bor el funtion whih is not σ - ontinuous then ther e is a  omp at set C ⊆ B suh that f ↾ C is also not σ - ontinuous. 4. Pr oof of the dihotomy In the statemen t of Theorem 1 b oth funtions ϕ and ψ are to b e top ologial em b eddings. Ho w ev er, as w e will see b elo w, for the di-  hotom y it is enough that they b oth are injetiv e, ϕ on tin uous and ψ op en. W e are going to pro v e rst this v ersion of the di hotom y . Theorem 4. L et X b e a Polish sp a e and f : X → 2 ω b e a Bor el funtion. Then pr e isely one of the fol lowing  onditions holds: (1) either f is σ - ontinuous (2) or ther e ar e an op en inje tion ψ and a  ontinuous inje tion ϕ suh that the fol lowing diagr am  ommutes: ω ω ψ − − − → 2 ω x   P x   f ( ω + 1) ω ϕ − − − → X Notie that ompatness of ( ω + 1) ω implies that the ψ ab o v e m ust b e a top ologial em b edding. Pr o of. It is straigh tforw ard that ( 2) implies that f is not σ -on tin uous. Let us assume that f is not σ -on tin uous and pro v e that ( 2 ) holds. By Corollary 1 w e ma y assume that X is ompat. Notation. First w e in tro due some notation. F or a xed n and 0 ≤ k ≤ n let S n k b e the set of p oin ts in ( ω + 1) n of Can tor-Bendixson rank ≥ n − k . F or ea h n < ω and 1 ≤ k ≤ n let us pi k a funtion π n k : S n k → S n k − 1 su h that • on S n k − 1 π n k is the iden tit y , 4 MAR CIN SABOK • if τ ∈ S n k \ S n k − 1 then w e pi k one i ∈ n su h that τ ( i ) < ω and τ ( i ) is maximal su h and dene π n k ( τ )( i ) = ω , π n k ( τ )( j ) = τ ( j ) for j 6 = i. This denition learly dep ends on the  hoie of the index i ab o v e. Note, ho w ev er, that w e ma y pi k the funtions π n k so that they are oheren t, in the sense that for τ ∈ ( ω + 1) n +1 , unless τ ( n ) is the biggest nite v alue of τ , w e ha v e π n +1 k +1 ( τ ) = π n k ( τ ↾ n ) a τ ( n ) . In partiular π n +1 k +1 ( σ a ω ) = π n k ( σ ) a ω for an y σ ∈ ( ω + 1) n . The funtions π n k will b e alled pro jetions. Lemma 1. F or e ah n and 1 ≤ k ≤ n the pr oje tion π n k : S n k → S n k − 1 is  ontinuous. Pr o of. Note that an y p oin t in S n k exept ( ω , . . . , ω ) ( k times ω ) has a neigh b orho o d in whi h pro jetion is unam bigous and hene on tin uous. But it is easy to see that at the p oin t ( ω , . . . , ω ) an y pro jetion is on tin uous.  F or ea h n < ω let us also in tro due the funtion r n : ( ω + 1) n → ( ω + 1) n dened as r n ( τ a a ) = τ a ω . T o mak e the ab o v e notation more readable w e will usually drop sub- sripts and sup ersripts in π n k and r n . W e pi k a w ell-ordering ≤ of ( ω + 1) <ω in to t yp e ω su h that for ea h p oin t τ ∈ ( ω + 1) <ω all elemen ts of the transitiv e losure of τ with resp et to π , r and restritions (i.e. funtions of the form ( ω + 1) n ∋ τ 7→ τ ↾ m ∈ ( ω + 1) m for m < n ) are ≤ τ . F or a set B ∈ Bor ( X ) \ I f let B ∗ denote the set B shrunk b y all basi lop ens C whi h ha v e I -small in tersetion with B . Strategy of the onstrution. In order to dene funtions ϕ and ψ , w e will onstrut for ea h τ ∈ ( ω + 1) <ω a lop en set C τ ⊆ 2 ω and a ompat set X τ ⊆ X su h that if σ ⊆ τ then C τ ⊆ C σ and X τ ⊆ X σ . The sets C τ will b e disjoin t, whi h means that for τ 6 = τ ′ , | τ | = | τ ′ | C τ ∩ C τ ′ = ∅ . W e will also need X τ ⊆ f − 1 [ C τ ] and diam ( X τ ) < 1 / | τ | . The onstrution of the sets X τ , C τ will b e done b y indution along the ordering ≤ on ( ω + 1) <ω . In fat, w e will do something more: at ea h step n if τ is the n -th elemen t of ( ω + 1) <ω w e will onstrut not only a ompat set X τ but also I f -p ositiv e Borel sets X n σ for σ ≤ τ su h that: • X τ ⊆ X n − 1 τ ↾ ( | τ |− 1) , • X n σ ⊆ X n − 1 σ if σ < τ , • X n σ a a ⊆ X n σ if σ , σ a a < τ , • X n σ ∩ f − 1 [ C σ a a ] = ∅ if σ , σ a a ≤ τ , A DICHOTOMY F OR BOREL FUNCTIONS 5 • X n σ a ω ⊆ l ( X n σ ) if σ , σ a ω ≤ τ . The set X n σ is to b e understo o d as the spae for further onstrution of sets X ρ for ρ ⊇ σ and ρ > τ , as an b e seen in the rst ondition ab o v e. The last ondition, as w e will see later, will b e used to guaran tee on tin uit y of the family of sets X τ . F or te hnial reasons w e will also mak e sure that X n σ = ( X n σ ) ∗ . W e are going to ensure disjoin tness of C τ 's b y satisfying the follo wing onditions: • C τ a a ⊆ C τ , • C τ a a ∩ C τ a b = ∅ for a 6 = b . The fat that diam ( X τ ) < 1 / | τ | will follo w from the follo wing indu- tiv e onditions (reall that π ( τ ) ≤ τ for an y τ ): • diam ( X τ ) < 3 diam ( X π ( τ ) ) , • diam ( X τ a ω ) < 1 / (3 | τ | +1 ( | τ | + 1) ) , b eause iterating pro jetions in ( ω + 1) n stabilizes b efore n + 1 steps. The ruial feature of the sets X τ is that this family should b e on- tin uous. Namely , w e will require that if τ and π ( τ ) o ur b y the n -th step then (1) h ( X n τ , X n π ( τ ) ) < 3 | τ | d ( τ , π ( τ )) This ondition is the most diuult. T o fulll it w e will onstrut y et another kind of ob jets. Notie rst that if h ( A, B ) < ε for t w o nonempt y sets in X then there are t w o nite families (w e will refer to them as to an hors) A i and B i ( i ∈ I 0 ) of subsets of A and B resp etiv ely su h that for an y A ′ i ⊆ A i , B ′ i ⊆ B i still h ( S i A ′ i , S i B ′ i ) < ε . Similarly , if h ( A, B ) < ε and C ⊆ A then there exist a nite family D i ( i ∈ I 0 ) of subsets of B su h that for an y D ′ i ⊆ D i h ( S i D ′ i , C ) < ε . A t ea h step n if τ is the n -the elemen t of ( ω + 1) <ω w e will addi- tionally onstrut an hors • for ea h pair X n σ and X n π ( σ ) su h that σ , π ( σ ) ≤ τ • and for ea h tripple X n σ , X n π ( σ ) , X n σ a a su h that a ∈ ω + 1 , π ( σ a a ) ⊆ π ( σ ) and σ , π ( σ ) , σ a a ≤ τ . Completing the diagram. As w e no w ha v e a lear piture of what should b e onstruted let us argue that this is enough to nish the pro of. F or ea h t ∈ ( ω + 1) ω the in tersetion T n X t ↾ n has preisely one p oin t so let us dene ϕ ( t ) to b e this p oin t. The other funtion, ψ is dened as f ◦ ϕ ◦ P − 1 . Let us  he k that this w orks. Both funtions ψ and ϕ are injetiv e thanks to the disjoitness of the sets C τ and to the fat that X τ ⊆ f − 1 [ C τ ] . The funtion ψ is op en b eause C τ are lop ens. 6 MAR CIN SABOK T o see on tin uit y of ϕ notie rst that sine the sets X τ ha v e diam- eters v anishing to 0 , it sues to  he k that ϕ is on tin uous on ea h ( ω + 1) n (whi h are treated as subsets of ( ω + 1) ω via the em b edding e : τ 7→ τ a ( ω , ω , . . . ) ). Con tin uit y on ( ω + 1) n is  he k ed indutiv ely on the sets S n k for 0 ≤ k ≤ n . The set S n 0 onsists of one p oin t, so there is nothing to  he k. Sup- p ose that τ i → τ , τ , τ i ∈ S n k , i ∈ ω . Then either the sequene is ev en tu- ally onstan t or τ ∈ S n k − 1 . Let us assume the latter. By the indutiv e assumption and on tin uit y of pro jetion ϕ ( π ( τ i )) → ϕ ( τ ) . No w pi k an y ε > 0 . Let m b e su h that diam ( X σ ) < ε for σ ∈ ( ω + 1) m and j ∈ ω su h that d ( τ j , π ( τ j )) < 3 − m ε . Let us write ρ a ω l for ρ extended b y l man y ω 's. By (1 ) and oherene of pro jetions w e ha v e h ( X τ j a ω m − n , X π ( τ j ) a ω m − n ) < ε, whi h implies that ϕ ( τ j ) and ϕ ( π ( τ j )) are loser than 3 ε . This sho ws that ϕ ( τ j ) → ϕ ( τ ) and pro v es on tin uit y of ϕ . Key lemma. No w w e state the k ey lemma, whi h will b e used to guaran tee on tin uit y of the family of sets X τ . Lemma 2. L et X b e a Bor el set, f : X → ω ω a Bor el, not σ - ontinuous funtion. Ther e exist a b asi lop en C ω ⊆ f [ X ] and a  omp at set X ω ⊆ f − 1 [ C ω ] suh that • f ↾ X ω is not σ - ontinuous, • X ω ⊆ l  ( f − 1 [ ω ω \ C ω ]) ∗  . The  omp at set X ω  an b e hosen of arbitr arily smal l diameter. Pr o of. Without loss of generalit y assume that f − 1 [ C ] = ( f − 1 [ C ]) ∗ for all lop en sets C ⊆ ω ω . Let us onsider the follo wing tree of op en sets, indexed b y ω <ω U τ = in t  f − 1 [[ τ ]]  . Let G = T n S | τ | = n U τ and Z τ = f − 1 [[ τ ]] \ U τ . Notie that f ↾ G is on tin uous and sine X = G ∪ S τ Z τ there is τ ∈ ω <ω su h that Z τ 6∈ I f . Observ e that Z τ ⊆ l  S τ ′ 6 = τ , | τ ′ | = | τ | f − 1 [[ τ ′ ]]  b eause if an op en set U ⊆ f − 1 [[ τ ]] is disjoin t from S τ ′ 6 = τ , | τ ′ | = | τ | f − 1 [[ τ ′ ]] then U ⊆ U τ . No w put C ω = [ τ ] and pi k an y ompat set with small diameter X ω ⊆ Z τ su h that X ω 6∈ I f .  The onstrution. W e b egin with X ∅ = X 0 ∅ = X and C ∅ = ω ω . Without loss of generalit y assume that X = X ∗ . Supp ose w e ha v e done n − 1 steps of the indutiv e onstrution up τ ∈ ( ω + 1) <ω . Let | τ | = l and σ = τ ↾ ( l − 1) . There are three ases. A DICHOTOMY F OR BOREL FUNCTIONS 7 Case 1. The four p oin ts τ , π ( τ ) , r ( τ ) and r ( π ( τ )) are equal. So τ = ( ω , . . . , ω ) and C τ ↾ n − 1 and X n − 1 σ are already onstruted. In this ase w e use Lemma 2 to nd a lop en set C τ and a ompat set X τ ⊆ X n − 1 σ of diameter < | τ | / 3 n +1 small enough so that no elemen t of the an hors onstruted so far is on tained in X τ . W e put X n τ = X ∗ τ , X n σ = ( X n − 1 σ \ f − 1 [ C τ ]) ∗ and X n ρ = X n − 1 ρ for other ρ < τ . By the assertion of Lemma 2 w e still ha v e X n τ ⊆ l ( X n σ ) . In this ase w e do not need to onstrut an y new an hors. Case 2. The t w o p oin ts π ( τ ) and r ( τ ) are equal but distint from τ . Let δ = d ( τ , r ( τ )) . Sine X n − 1 r ( τ ) ⊆ l ( X n − 1 σ ) b y the indutiv e assump- tion, w e ma y nd nitely man y sets B i ⊆ X n − 1 σ , i ≤ k su h that • h ( S i B ′ i , X r ( τ ) ) < δ for an y B ′ i ⊆ B i , • B i 6∈ I f . The seond ondition follo ws from X n − 1 σ = ( X n − 1 σ ) ∗ . W e ma y assume that for ea h lop en set C ⊆ 2 ω the set B i ∩ f − 1 [ C ] is either empt y or outside of the ideal I f . W e are going to nd lop ens C i ⊆ C σ , for i ≤ k su h that C i ∩ C r ( τ ) = ∅ and then put C τ = S i ≤ k C i , X n σ = ( X n − 1 σ \ S i f − 1 [ C i ]) ∗ and nd X τ ⊆ S i ≤ k B i ∩ f − 1 [ C i ] . W e will ha v e to arefully dene X n r ( τ ) so that X n r ( τ ) ⊆ l ( X n σ ) . It is easy to see that for an y A ⊆ X n − 1 σ X n − 1 r ( τ ) = X n − 1 r ( τ ) ∩ l  ( X n − 1 σ ∩ A ) ∗  ∪ X n − 1 r ( τ ) ∩ l  ( X n − 1 σ ∩ A c ) ∗  so (putting A = f − 1 [ C σ ∩ [( m, 0)]] for m < ω ) w e ma y indutiv ely on m pi k binary sequenes β m i ∈ 2 m , i ≤ k su h that f − 1 [[ β m i ]] ∩ B i 6 = ∅ and X n − 1 r ( τ ) ∩ l  ( X n − 1 σ \ f − 1 [ [ i ≤ k [ β m i ]]) ∗  6∈ I f . W e are going to arry on this onstrution up to some m < ω and put X n ρ =  X n − 1 ρ \ S i ≤ k f − 1 [[ β m i ]]  ∗ for ρ < τ , ρ 6⊇ r ( τ ) and X n ρ =  X n − 1 ρ ∩ l ( X n σ )  ∗ for ρ < τ , ρ ⊇ r ( τ ) . W e m ust, ho w ev er, tak e are that this do es not destro y the existing an hors. Sine f − 1 [ { x } ] ∈ I f for an y x ∈ 2 ω and there are only nitely man y elemen ts of the existing an hors, w e ma y pi k m < ω and onstrut the sequenes β m i so that for an y elemen t A of an an hor b elo w X n − 1 r ( τ ) it is the ase that A ∩ l ( f − 1 [ C σ \ S i ≤ k [ β m i ]] ∩ X n − 1 σ ) 6∈ I f and for an y elemen t A of other an hors A \  S i ≤ k f − 1 [ β m i ]  6∈ I f . One w e ha v e onstruted the sequenes β m i for i ≤ k w e put C i = [ β m i ] and C τ = S i [ β m i ] . Next w e nd I f -p ositiv e ompat sets X i inside B i ∩ f − 1 [[ β i ]] , ea h of diameter < 1 / (3 n +1 | τ | ) . 8 MAR CIN SABOK If δ > diam ( X n − 1 π ( τ ) ) then w e an pi k one X i as X τ and then h ( X τ , X n − 1 π ( τ ) ) ≤ 3 h ( X n − 1 σ , X n − 1 π ( σ ) ) < 3 | τ | δ . Otherwise, let X τ = S i ≤ k X i and then diam ( X τ ) < 3 diam ( X n − 1 π ( τ ) ) ≤ 3 diam ( X π ( τ ) ) . Dene X n τ = X ∗ τ . A t this step w e reate an hors for the pair X n τ and X n r ( τ ) as w ell as for the tripples X n σ , X n ρ , X n τ for ρ < τ . Case 3. The t w o p oin ts π ( τ ) , r ( τ ) are distint. Let δ = d ( τ , π ( τ )) . By oherene of the pro jetions π ( τ ) ⊇ π ( σ ) . By the indutiv e assump- tion w e ha v e h ( X n − 1 σ , X n − 1 π ( σ ) ) < 3 | σ | δ . Using the existing an hor for the tripple X n − 1 σ , X n − 1 π ( σ ) , X n − 1 π ( τ ) let us nd nitely man y sets B i , i ≤ k in X σ su h that • h ( S i B ′ i , X π ( τ ) ) < 3 | σ | δ for an y B ′ i ⊆ B i , • B i 6∈ I f . As b efore, w e assume assume that for ea h lop en set C ⊆ 2 ω if B i ∩ f − 1 [ C ] ∈ I f then it is empt y . W e ha v e no w t w o sub ases, in analogy to the t w o previous ases. Sub ase 3.1. Supp ose τ = r ( τ ) . Similarly as in Case 1, w e use Lemma 2 to nd X i ⊆ B i and C i for i ≤ k . Put C τ = S i ≤ k C i . If δ > diam ( X n − 1 π ( τ ) ) then w e an pi k one X i as X τ and then h ( X τ , X n − 1 π ( τ ) ) ≤ 3 h ( X n − 1 σ , X n − 1 π ( σ ) ) < 3 | τ | δ . Otherwise, let X τ = S i ≤ k X i and then diam ( X τ ) < 3 diam ( X n − 1 π ( τ ) ) ≤ 3 diam ( X π ( τ ) ) . Again, similarly as in Case 1, w e put X n τ = ( X n τ ) ∗ , X n σ = ( X n − 1 σ \ f − 1 [ C τ ]) ∗ , X n ρ = X n − 1 ρ for other ρ < τ . Sub ase 3.2. Supp ose τ 6 = r ( τ ) . Similarly as in Case 2, w e nd lop ens C i in ω ω su h that X n − 1 r ( τ ) ∩ l  ( f − 1 [ C σ \ S i ≤ k C i ]) ∗  6∈ I f and no existing an hor is destro y ed when w e put X n ρ = ( X n − 1 ρ \ S i ≤ k f − 1 [ C i ]) ∗ for ρ < τ , ρ 6⊇ r ( τ ) and X n ρ = X n − 1 ρ ∩ l ( X n σ ) for ρ < τ , ρ ⊇ r ( τ ) . Next w e nd I f -p ositiv e ompat sets X i ⊆ B i ∩ f − 1 [ C i ] ea h of diameter < 1 / (3 | τ | +1 | τ | ) . As previously , if δ > diam ( X n − 1 π ( τ ) ) then w e an pi k one X i as X τ and then h ( X τ , X n − 1 π ( τ ) ) ≤ 3 h ( X n − 1 σ , X n − 1 π ( σ ) ) < 3 | τ | δ . Otherwise, let X τ = S i ≤ k X i and then diam ( X τ ) < 3 diam ( X n − 1 π ( τ ) ) ≤ 3 diam ( X π ( τ ) ) . Again, w e put X n τ = ( X n τ ) ∗ , In Case 3 w e onstrut the same an hors as in Case 2. This ends the onstrution and the en tire pro of.  A DICHOTOMY F OR BOREL FUNCTIONS 9 Theorem 5. If f : X → ω ω is not σ - ontinuous then ther e exist top o- lo gi al emb e ddings ϕ and ψ suh that the fol lowing diagr am  ommutes: ω ω ψ − − − → ω ω x   P x   f ( ω + 1) ω ϕ − − − → X Pr o of. By Theorem 4 w e ha v e ψ and ϕ su h that ψ is 1 - 1 op en. But as a Borel funtion it on tin uous on a dense G δ set G ⊆ ω ω . On the other hand b y the prop erties of the funtion P X ∈ I P implies P [ X ] is meager. So P − 1 [ G ] 6∈ I P and the problem redues to the restrition of the funtion P . This, ho w ev er, has b een pro v ed in [ 2 ℄ (Corollary 2). So w e get the follo wing diagram: ω ω ψ ′ − − − → G ψ − − − → ω ω x   P x   P ↾ G x   f ( ω + 1) ω ϕ ′ − − − → P − 1 [ G ] ϕ − − − → X whi h ends the pro of.  Referenes [1℄ Ci ho« J., Mora yne M., P a wlik o wski J. and Sole ki S., De  omp osing Bair e fun- tions , Journal of Sym b oli Logi, V ol. 56, Issue 4, 1991, pp. 1273-1283 [2℄ Sab ok M., σ - ontinuity and r elate d for ings , submitted, [3℄ Sole ki S., De  omp osing Bor el sets and funtions and the strutur e of Bair e lass 1 funtions , Journal of the Amerian Mathematial So iet y , V ol. 11, No. 3, 1998, pp. 521-550 [4℄ Zapletal J., Desriptive Set The ory and Denable F or ing , Memoirs of the Amer- ian Mathematial So iet y , 2004 [5℄ Zapletal J., F or ing Ide alize d , Cam bridge T rats in Mathematis 174, 2008 Ma thema tial Institute, Wr oªa w University, pl. Gr unw aldzki 2 / 4 , 50 - 384 Wr oªa w, Poland E-mail addr ess : sabokmath.uni. wro . pl