A property of C_p[0,1]

A prop ert y of C p [0 , 1] Mic hael Levin Abstract W e pro ve that for ev ery finite dimensional compact metric space X there is an op en con tin uou s linear surjection from C p [0 , 1] on to C p ( X ). T he pr o of make s use of em b edd ings int ro duced by Kolmogoro v and Sternf eld in conn ection with Hilb ert’s 13th p roblem. Keyw ords: C p -spaces, b asic embedd ings Math. Sub j. Class.: 54C35, 54F4 5. 1 In tro ductio n All spaces are assumed to b e separable metrizable a nd maps con tinuous . A compactum is a metric compact space. F or a space X , C p ( X ) denotes the space of contin uous r eal- v alued functions equipp ed with the top olo gy of po int wise con v ergence. W e refer to the top ology of p oint wise con v ergence as the C p -top olog y . Let X ⊂ Y 1 × · · · × Y k b e an em b edding of a compactum X into the pro duct of compacta Y 1 , . . . , Y k . Define the space Z as the disjoin t union of Y 1 , . . . , Y k and define the linear transformation L : C ( Z ) − → C ( X ) by L ( g )( x ) = g ( y 1 ) + g ( y 2 ) + · · · + g ( y k ) for g ∈ C ( Z ) and x = ( y 1 , . . . , y k ) ∈ X where y 1 , . . . , y k are the co ordinates of x in the pro duct Y 1 × · · · × Y k . W e will call L the induced tra nsformation of the embedding X ⊂ Y 1 × · · · × Y k . It is obvious that L is con tinuous in b oth the uniform top o logy and the C p -top olog y on the function space s. An em b edding X ⊂ Y 1 × · · · × Y k is called basic if the induced transfor ma t io n L is surjectiv e. Note that, in g eneral, a surjectiv e linear transformation of the function spaces on compacta whic h is con tinuous in b oth the unifor m top ology and the C p -top olog y is not necessarily op en in the C p -top olog y [8]. It w as sho wn in [8] that the transfor ma t io n 1 induced b y a basic em b edding in the pro duct of tw o spaces ( k = 2) is op en in the C p - top ology and it is an op en problem if the similar result holds f o r k > 2. In this pap er w e will giv e a partia l answ er to this pro blem for tw o ty p es of basic em b eddings, na mely , Kolmogorov and Sternfeld-type em b eddings Sternfeld [6 ] c onstructed for ev ery n -dime nsional compactum X a basic em b edding of X in to the pro duct o f n + 1 one-dimensional compacta. W e will call this em b edding a Sternfeld-t yp e em b edding. Sternfeld-t yp e embeddings are defined in Section 2. Adjusting Ko lmogorov ’s solution of Hilb ert’s 13th problem given in Kolmogoro v’s famous sup erp o sition theorem [1], Ostrand [2] defined for ev ery n - dimensional compactum X a basic em b edding X ⊂ [0 , 1] 2 n +1 whic h we will call a Kolmogorov-t ype em b edding. Kolmogorov-t yp e em b eddings are describ ed in Section 3. In Sections 2 and 3 w e will prov e the follow ing theorems. Theorem 1.1 L et X ⊂ Y 1 × · · · × Y n +1 b e a S ternfeld-typ e emb e dding of an n -dimensi o nal c omp actum X into the pr o duct of one-dimensio n al c omp a cta Y 1 , . . . , Y n +1 . Th en the in- duc e d tr ansformation is op en in the C p -top olo gy. Theorem 1.2 L et X ⊂ [0 , 1 ] 3 b e a Ko lmo gor ov-typ e e m b e dding of a 1 -dimensional c om - p actum X into the cub e. Then the induc e d tr a n sformation is op e n in the C p -top olo gy. Let X b e n -dimensional a nd compact. By Theorem 1.1 there is a 1- dimensional com- pactum Z (=the disjoin t union of Y 1 , . . . , Y n +1 ) fo r whic h C p ( Z ) admits an o p en contin uous linear transforma t io n onto C p ( X ). Let ˆ Z =the disjoint union of three copies of [0 , 1 ]. By Theorem 1.2 t here is an op en contin uous linear transformation fro m C p ( ˆ Z ) on to C p ( Z ). Em b ed ˆ Z in to [0 , 1] and tak e the restriction transformation from C p [0 , 1] to C p ( ˆ Z ) which is obv iously surjectiv e, open and con tinuous. Th us w e obtain the main result of the pap er. Theorem 1.3 F or every finite dimensional c omp actum X ther e is an op en c ontinuous line ar tr ans f o rmation fr om C p [0 , 1] onto C p ( X ) . Note that C p [0 , 1] and C p ( X ) for a compactum X are not isomorphic(=linearly home- omorphic) if dim X > 1 and in man y cases when dim X = 1. In con trast to the uniform top ology , the existence of an isomorphism b et w een C p ( X ) and C p ( Y ) for compacta X and Y implies a g r eat deal of similarit y betw een X and Y , in particular, it implies that dim X = dim Y [3]. Theorem 1.3 generalizes some previous results by Leiderman, P esto v, Morris a nd the author [7], [8]. Op en problems and related results are discussed in Section 4. 2 2 Sternfeld -t yp e em b edding s Let X b e compact and n -dimensional. Sternfeld [6] show ed that there a re a decomp osition X = A 1 ∪ · · · ∪ A n +1 of X into 0-dimensional subsets A 1 , . . . , A n +1 and 1- dimensional compacta Y 1 , . . . , Y n +1 suc h that X admits an em b edding X ⊂ Y 1 × · · · × Y n +1 ha ving the follo wing property: x = p − 1 i ( p i ( x )) for ev ery pro jection p i : X − → Y i and x ∈ A i . W e will call suc h an em b edding of X a Sternfeld-type e mbedding with resp ect t o a decomposition A 1 , . . . , A n +1 of X . The spaces Y i can be c hosen to b e dendrites [6]. A differen t w ay of constructing Sternfeld-t yp e embeddings can b e deriv ed from [9]. Sternfeld [6] prov ed that an y Sternfeld-ty p e embedding is basic. Sternfeld’s pro of is based on Borel measures a nd it is not clear at all if it can be a pplied to prov e Theorem 1.1. In this pap er w e use another more constructiv e appro ac h whic h is describ ed in 2.1 and whic h also show s that Sternfeld-t yp e em b eddings are basic. 2.1 An appr o ximation pr o cedure Let X ⊂ Y 1 × · · · × Y n +1 b e a Sternfeld-type em b edding of a n - dimensional compactum X with resp ect to a decomp osition X = A 1 ∪ · · · ∪ A n +1 , dim A i = 0. Let us describ e an appro ximation pro cedure sho wing that the em b edding o f X is basic. The case n = 0 is trivial. Assume that n > 0. Let f : X − → R b e con tinuous and c > 0 suc h that k f k < c . Fix ǫ > 0 whic h will b e determined later and whic h will dep end only on k f k , c and n . T ak e a disjoin t family V i of o p en s ubsets of Y i suc h that V i co v ers p i ( A i ) and diam f ( p − 1 i ( V )) < ǫ for ev ery V ∈ V i . F or ev ery i c ho o se a finite subfamily U i ⊂ p − 1 i ( V i ) suc h that U = U 1 ∪ · · · ∪ U n +1 co v ers X and the elemen ts of U are non-empt y . By V U , U ∈ U i w e denote the set of V i suc h that U = p − 1 i ( V U ). F or ev ery U ∈ U take a non-empt y subset F U ⊂ X closed in X suc h that for F = { F U : U ∈ U } cov ers X a nd fix a p oint x U ∈ F U . F or ev ery U ∈ U i tak e a contin uous f unction φ U : Y i − → [0 , 1] suc h that φ U ( Y i \ V U ) = 0 and φ U ( p i ( F U )) = 1. Define g ′ i : Y i − → R as g ′ i = P U ∈U i 1 n +1 f ( x U ) φ U . Clearly k g ′ i k ≤ 1 n +1 k f k < 1 n +1 c . Let us sho w that for ev ery x ∈ X w e ha ve ( ∗ ) | f ( x ) − P i g ′ i ( y i ) | < n n +1 c where y i = p i ( x ) ∈ Y i are the co ordinates of x . Assume that | f ( x ) | ≤ ǫ . Then for ev ery U ∈ U suc h that x ∈ U w e hav e | f ( x U ) | < 2 ǫ and hence | g ′ i ( y i ) | < 2 n +1 ǫ for eve ry i . Then | f ( x ) − P i g ′ i ( y i ) | < ǫ + 2 ǫ . Th us taking ǫ < n 3( n +1) c w e get that ( ∗ ) holds. Assume that f ( x ) > ǫ . Then for ev ery U ∈ U suc h that x ∈ U we ha ve 0 < f ( x U ) < f ( x ) + ǫ and hence 0 < g ′ i ( y i ) < 1 n +1 ( f ( x ) + ǫ ) for eve ry i . Note t ha t there is F U , 3 U ∈ U j , suc h that x ∈ F U and hence 1 n +1 ( f ( x ) − ǫ ) < g ′ j ( x ) < 1 n +1 ( f ( x ) + ǫ ). Then 0 < f ( x ) − g ′ j ( x ) < n n +1 ( f ( x ) + ǫ ) and 0 < P i 6 = j g ′ i ( x ) < n n +1 ( f ( x ) + ǫ ). Hence | f ( x ) − P i g ′ i ( y i ) | = | ( f ( x ) − g ′ j ( x )) − P i 6 = j g ′ i ( x ) | < n n +1 ( f ( x ) + ǫ ). Th us taking ǫ < c − k f k w e get that ( ∗ ) ho lds. In a similar w ay w e c hec k the case f ( x ) < − ǫ and g et that in a ll the cases ( ∗ ) holds. Recall that b y Z we denote the disjoin t union of Y 1 , . . . , Y n +1 . Define g ′ : Z − → R b y g ′ | Y i = g ′ i . W e ha ve that k g ′ k < 1 n +1 c and k f − L ( g ′ ) k < n n +1 c . Applying the described ab ov e pro cedure iterativ ely one can construct a se quence of maps g ( t ) : Z − → R suc h that k g ( t ) k < 1 n +1 ( n n +1 ) t − 1 c and k f − L ( P t s =1 g ( s ) ) k < ( n n +1 ) t c . Then for g = P ∞ s =1 g ( s ) w e ha v e f = L ( g ) and hence the em b edding of X is basic. 2.2 Pro of of Theorem 1.1 Supp ose that X ⊂ Y 1 × · · · × Y n +1 is a St ernfeld-type em b edding of a n n -dimensional compactum X with resp ect to a decomp osition X = A 1 ∪ · · · ∪ A n +1 , dim A i = 0. Let Z ′ b e a finite subset of Z =the disjoin t union of Y i ’s. Denote Y ′ i = Y i ∩ Z ′ , X ′ i = p − 1 i ( Y ′ i ) ∩ A i and X ′ = X ′ 1 ∪ · · · ∪ X ′ n +1 . It is clear that each X ′ i is finite and therefore X ′ is finite as w ell. T ak e any map f : X − → R suc h that f ( x ) = 0 for ev ery x ∈ X ′ . W e will sho w that there is a map g : Z − → R suc h that g ( z ) = 0 f o r eve ry z ∈ Z ′ and L ( g ) = f . This prop ert y together with t he fact that L is op en in the unifo r m top o lo gy implies that L is op en in the C p -top olog y at the zero-map on Z and hence, b y the linearit y of L , L is op en in the C p -top olog y eve rywhere and Theorem 1.1 follo ws. The construction of g follow s the pro cedure describ ed in 2.1 with the f ollo wing ad- ditional requiremen ts. W e assume that no p oin t of Y ′ i \ p i ( A i ) is co ve red b y V i , p − 1 i ( V ) con tains at most one p oin t of X ′ for eve ry V ∈ V i and if U ∈ U contains a p oin t of X ′ then this p o int is also con tained in F U and x U = F U ∩ X ′ . It is easy to see that under these assumptions we hav e that g ′ i ( y ) = 0 for ev ery y ∈ Y ′ i and g ′ i ( p i ( x )) = 0 for ev ery x ∈ X ′ . Th us g ′ ( z ) = 0 for ev ery z ∈ Z ′ and f ( x ) − L ( g ′ )( x ) = 0 for ev ery x ∈ X ′ . Hence t he approximation pro cedure can b e rep eated iterativ ely and w e can construct a map g : Z − → R with the required prop erties. The theorem is pro ve d. 3 Kolmogoro v-t yp e em b e ddings Let X b e an n -dimensional compactum. A co ver of X is said to cov er X at least n + 1 times if ev ery p oint of X b elong s to at least n + 1 elemen ts of the co v er. Generalizing 4 Kolmogorov’s pap er [1 ], Ostrand [2] sho we d that there is a coun ta ble fa mily Ω of closed finite co v ers of X suc h that inf { mesh F : F ∈ Ω } = 0, eac h F ∈ Ω co v ers X at least n + 1 times and eac h F ∈ Ω splits into the union F = F 1 ∪ · · · ∪ F 2 n +1 of 2 n + 1 families of disjoin t sets. Suc h a family of cov ers Ω with a fixed splitting F = F 1 ∪ · · · ∪ F 2 n +1 for eac h F ∈ Ω will b e called a Kolmogo ro v f amily o f co v ers (Kolmogoro v constructed a Kolmogorov family of co vers for X = [0 , 1] n and this family can b e easily transferred to an arbitrary n -dimensional compactum using a 0 -dimensional map to [0 , 1] n ). In the case when a Kolmogorov family con tains a co v er of mesh= 0 (that may happen only if X is finite) w e assume that this co ve r app ears in the family infinitely man y times. W e will also a ssume that a Ko lmo g oro v family contains only finitely many cov ers of mesh > ǫ for ev ery ǫ > 0. Th us w e alw ays assume that a Kolmogo ro v family is infinite and an y infinite subfamily o f a Kolmog o ro v family is Kolmog o ro v as w ell. A map from X to [0 , 1] is said to separate a family o f disjoin t sets in X if the images of the sets are disjoin t in [0 , 1]. Let Ω b e a Kolmogorov family of cov ers of X . An em b edding X ⊂ [0 , 1] 2 n +1 is said to separate a cov er F ∈ Ω if the pro j ection p i : X − → [0 , 1] separates F i for ev ery i . An em b edding X ⊂ [0 , 1] 2 n +1 will b e called a Kolmogoro v-t yp e embedding with resp ect to Ω if for ev ery ǫ > 0 there is F ∈ Ω with mesh F < ǫ suc h that the em b edding of X separates F . Note that almost all em b eddings o f X in to [0 , 1] 2 n +1 are of Kolmog oro v-type with resp ect to Ω, see [4]. In the next subsection w e presen t an appro ximation pro cedure whic h can b e deriv ed from Kolmogorov’s pap er [1] and whic h sho ws that Kolmogoro v-type em b eddings are basic. This fact w as observ ed b y Ostrand [2], see also [4]. Note t ha t for a Kolmogo r ov-t yp e em b edding X ⊂ [0 , 1] 2 n +1 with respect to Ω w e can replace Ω b y an y infinite subfamily of Ω and the em b edding of X will remain of Kolmogorov-t yp e with r esp ect to the replaced Ω. (Th us, in particular, w e ma y assume that a Kolmogorov -type embedding with r esp ect to Ω separates ev ery co v er in Ω.) 3.1 An appr o ximation pro cedure Let X ⊂ [0 , 1] 2 n +1 b e a Kolmogo ro v-type embedding with respect to a family of cov ers Ω. Here we describ e an appro ximation pro cedure sho wing that the em b edding of X is basic . The case n = 0 is trivial. Assume that n > 0. Let f : X − → R b e con tinuous and c > 0 suc h t ha t k f k < c . Fix ǫ > 0 which will b e determined later and which will dep end only on k f k , c and n . Clearly w e ma y assume that eac h F ∈ Ω consists of no n- empt y sets. Cho ose an y cov er F ∈ Ω suc h t ha t mesh f ( F ) = { f ( F ) : F ∈ F } < ǫ . F or ev ery no n-empt y F ∈ F fix a p oin t x F ∈ X suc h that f ( x F ) is at distance < ǫ from f ( F ). (F or sho wing that X ⊂ [0 , 1] 2 n +1 is basic is 5 enough to c ho ose x F in F , ho wev er in the pro of of Theorem 1.2 we may need to choose x F outside F .) Define g ′ i : [0 , 1] − → R , 1 ≤ i ≤ 2 n + 1, suc h that g ′ i ( F ) = 1 2 n +1 f ( x F ) for ev ery non-empt y F ∈ F i and k g ′ i k ≤ 1 2 n +1 k f k < 1 2 n +1 c . Let us sho w that for ev ery x ∈ X w e ha v e ( ∗ ) | f ( x ) − P i g ′ i ( y i ) | < 2 n 2 n +1 c where y i = p i ( x ) ∈ [0 , 1] are the co ordinates of x . Indeed, recall that F co ve rs x at least n + 1 times. Cho ose a set I + ⊂ { 1 , 2 , . . . , 2 n + 1 } con taining exactly n + 1 indices suc h tha t x is co v ered by F i for ev ery i ∈ I + and denote I − = { 1 , 2 , . . . , 2 n + 1 } \ I + . F or ev ery i ∈ I + there is F ∈ F i con taining x and hence | 1 2 n +1 f ( x ) − g ′ i ( y i ) | = 1 2 n +1 | f ( x ) − f ( x F ) | < 1 2 n +1 2 ǫ . Then | f ( x ) − P i g ′ i ( y i ) | = | P i ∈ I + ( 1 2 n +1 f ( x ) − g ′ i ( y i )) + P i ∈ I − ( 1 2 n +1 f ( x ) − g ′ i ( y i )) | < n +1 2 n +1 2 ǫ + n ( 1 2 n +1 k f k + 1 2 n +1 k f k ) = 2( n +1) 2 n +1 ǫ + 2 n 2 n +1 k f k . Thus taking ǫ < n n +1 ( c − k f k ) we get that ( ∗ ) holds. Denote b y Z the disjoin t union of 2 n + 1 copies Y i = [0 , 1], 1 ≤ i ≤ 2 n + 1, of the in terv al [0 , 1]. Define g ′ : Z − → R b y g ′ | Y i = g ′ i . W e hav e that k g ′ k < 1 2 n +1 c and k f − L ( g ′ ) k < 2 n 2 n +1 c where L is the linear transformatio n induced b y the em b edding X ⊂ [0 , 1] 2 n +1 . Applying the describ ed abov e procedure iterativ ely one can construct a se quence of maps g ( t ) : Z − → R suc h that k g ( t ) k < 1 2 n +1 ( 2 n 2 n +1 ) t − 1 c and k f − L ( P t s =1 g ( s ) ) k < ( 2 n 2 n +1 ) t c . Then for g = P ∞ s =1 g ( s ) w e ha ve f = L ( g ) and hence the em b edding of X is basic. 3.2 Em b eddings of 1 -dimensional compacta Let X b e a one-dimensional compactum and X ⊂ [0 , 1 ] 3 a Kolmog oro v-type em b edding with resp ect to a Kolmogo r o v family Ω of co ve rs of X . Denote Y i = [0 , 1], 1 ≤ i ≤ 3 and Z =the disjoint union o f Y i ’s. Recall that b y p i w e denote the pro jection p i : X − → Y i . Reserv ed and free p oints . W e say that z ∈ Y i ⊂ Z is a reserv ed p oin t of Z with resp ect to Ω if f o r all but finitely many co vers F in Ω, F i in tersects p − 1 i ( z ), that is there is F ∈ F i suc h that F in tersects p − 1 i ( z ). Note that, since p i separates F i , a t most one elemen t of F i can in tersect p − 1 i ( z ). A p o in t z ∈ Z is said to b e strongly res erv ed with resp ect to Ω if F i in tersects p − 1 i ( z ) for ev ery F ∈ Ω and the collection { F : F ∩ p − 1 i ( z ) 6 = ∅ , F ∈ F i , F ∈ Ω } con verges to a p oin t x ∈ X , that is ev ery ne ighbor ho o d of x in X con ta ins all but finitely man y elemen ts of the collection. W e will sa y that the p oint x witnessing the reserv ation of z or say that z is reserv ed by x . A p oint z ∈ Y i ⊂ Z whic h is not reserv ed with resp ect to Ω is said to b e free with 6 resp ect to Ω. A p oin t z ∈ Y i ⊂ Z is said to b e fully free with resp ect to Ω if F i do es not in tersect p − 1 i ( z ) f or ev ery F ∈ Ω. It is obv ious that if z ∈ Z is reserv ed (strongly reserv ed, reserv ed b y x ∈ X , fully free) with respect to Ω then z remains to b e reserv ed (strongly reserv ed, reserv ed b y x , fully free resp ective ly) with resp ect to any infinite subfamily of Ω. Note that if z ∈ Z is reserv ed (free) with resp ect to Ω then replacing Ω by its infinite subfamily we can get that z is strongly reserv ed (fully free) with respect to Ω. Also note that, since eac h F ∈ Ω co v ers X at least tw ice, ev ery p oin t x ∈ X has at least t w o co ordinates rese rved and if ev ery co ordinate of x is either strongly reserv ed or fully free with res p ect Ω then at least t w o co ordinates of x are reserv ed by x . Chains. A c hain χ of length m with respect to Ω is a couple χ = ( A, B ) suc h that A = { z 0 , z 1 , . . . , z 2 m } is a sequence of 2 m + 1 elemen ts of Z , B = { x 1 , x 2 , . . . x m } is a sequence if m elemen ts of X suc h that z 2 j − 2 , z 2 j − 1 and z 2 j are the co ordinates of x j , the p oints { z 0 , . . . , z 2 m − 1 } ar e strongly reserv ed with resp ect to Ω, z 2 j − 2 and z 2 j − 1 are reserv ed by x j and the p oint z 2 m is strongly reserv ed or fully free with respect to Ω. Note the co ordinates z 2 j − 2 , z 2 j − 1 and z 2 j of x j do not necessarily go in the order corre- sp onding to the order of the co ordinates x j = ( y 1 , y 2 , y 3 ) of x j in the pro duct of Y 1 , Y 2 and Y 3 (for example it may happ en that z 2 j − 2 = y 2 ), t he only thing that w e assume is that { z 2 j − 2 , z 2 j − 1 , z 2 j } = { y 1 , y 2 , y 3 } as subsets of Z . The c hain χ is of length 0 if A con tains only one p oin t z 0 and B = ∅ . The p oints z 0 and z 2 m ( x 1 and x m if m > 0) a r e called the initial and the terminal Z -p oints ( X -p o in ts resp ectiv ely) of the chain χ . The sequence s A and B are called the Z -seque nce and the X -se quence o f χ respective ly . A c hain χ ′ = ( A ′ , B ′ ) is said to b e an extens ion of the c hain χ if the sequences A ′ and B ′ starts with A and B resp ectiv ely . A chain χ ′ is said to b e a contin uation of the chain χ if the terminal Z -p oin t of χ is the initial Z -p oin t o f χ ′ . If χ ′ = ( A ′ , B ′ ) is a contin ua t ion of length m ′ of the c hain χ ( A, B ) then w e can define the chain χ ′′ = χ + χ ′ = ( A ′′ , B ′′ ) of length= m + m ′ b y letting B ′′ b e the sequen ce B follo wed b y the sequence B ′ and A ′′ b e the seq uence A follo we d by the sequence A ′ when the last eleme nt of A is identified w ith the first o ne of A ′ . W e will call χ and χ ′ the head o f length m and the ta il of length m ′ resp ectiv ely of the chain χ ′′ and also write χ = χ ′′ − χ ′ and χ ′ = χ ′′ − χ . Almost free, p erio dic and non-per io dic p oin ts. If the terminal Z -p oin t of the c hain χ is not free then χ c an b e exte nded in the following w ay . D efine x m +1 = the p oin t of X witnessing the reserv ation of z 2 m , replace Ω b y its infinite subfamily in or der to get that ev ery co ordinate of x m +1 is either strongly r eserv ed or fully free and define z 2 m +1 and z 2 m +1 as the t w o other co ordinates of x m +1 (in addition to z 2 m ) suc h t ha t z 2 m +1 is 7 reserv ed by x m +1 (recall that x m +1 has at least t w o co ordinates reserv ed b y x m +1 ). Note that constructing chains w e will constantly replace by Ω its infinite s ubfamily . Thus talk- ing ab out t w o (or finitely man y) chains built one a fter another w e alwa ys r efer to the smallest subfamily obtained in the last replacemen t. Let Z ′ b e a finite s ubset of Z . En umerate the p oin ts of Z ′ and for ev ery z ∈ Z ′ define the c hain χ ( z , 0) a s the c hain o f length 0 with the initial Z-p oin t z . F or eve ry m w e will replace Ω by its infinite subfamily and construct f or ev ery z ∈ Z ′ a c hain χ ( z , m ) pro ceeding fro m m to m + 1 as follow s. Define χ ( z , m + 1) = χ ( z , m ) if the terminal Z -p oin t of χ ( z , m ) is free. According to our en umeration of Z ′ go o v er the p oints z of Z ′ suc h that the terminal Z -p oint of χ ( z , m ) is not free a nd replacing (if necessary) eac h time Ω b y its infinite subfamily extend χ ( z , m ) to a c hain of χ ( z , m + 1) as desc rib ed ab o v e (b y adding one elemen t to the X -sequence and t wo elemen ts to Z -sequence of χ ( z , m )). Th us the length of χ ( z , m ) ≤ m and the length of χ ( z , m ) = m if the terminal Z - p oin t of χ ( z , m ) is not free. Let us call z ∈ Z ′ almost free if there is m suc h that the terminal Z -p oint of χ ( z , m ) is free. If z ∈ Z ′ is not almost free then w e will sa y that z is perio dic if there is m suc h that the X -sequence of χ ( z , m ) con tains t w o equal elemen ts, and z is non-p erio dic otherwise (that is for ev ery m all the elemen ts of the X -sequence of χ ( z , m ) are distinct). Define the X -supp ort ( Z -supp ort) of z ∈ Z ′ as t he subset of X ( Z ) consisting of all the elemen ts of the X -sequences ( Z -sequences ) of χ ( z , m ) for all m . The Z -supp ort of z is t he union of the co or dinates of the p oin ts of the X -support of z . It is ob vious that the X -supp ort and the Z -supp o rt of an almost free p oint in Z ′ are finite. Note tha t if fo r tw o c hains χ and χ ′ of length m and m ′ resp ectiv ely w e ha v e tha t x j = x ′ j ′ for the elemen ts x j , x ′ j ′ , j < m and j ′ < m ′ in the X -sequences of χ and χ ′ resp ectiv ely then for the elemen ts x j +1 and x ′ j ′ +1 follo wing x j and x ′ j ′ in the X - sequences w e also ha v e x j +1 = x ′ j ′ +1 . Indeed, if x j = x j +1 then all the co ordinates of x j are reserv ed b y x j and therefore x ′ j ′ +1 = x j . If x j +1 6 = x j then one of the co ordinates of x j is reserv ed b y x j +1 and therefore x ′ j ′ +1 = x j +1 . Also note that if we assume in addition that j + 1 < m , j ′ + 1 < m ′ and the elemen ts o f the X -seque nce of χ are distinct then not only x j +1 = x ′ j ′ +1 but also z 2 j = z ′ 2 j ′ , z 2 j +1 = z ′ 2 j ′ +1 and z 2 j +2 = z ′ 2 j ′ +2 for the corresp onding elemen ts of the Z -sequences of χ and χ ′ resp ectiv ely . Indeed, z 2 j , z 2 j +1 , z 2 j +2 are the co ordinates of x j +1 whic h are c haracterized b y the pro p erties: z 2 j +2 is the only co ordinate of x j +1 not reserv ed by x j +1 , and z 2 j is the only co ordinate o f x j +1 whic h is also a co ordinate of x j (since otherwise either z 2 j +1 or z 2 j +1 w ould b e reserv ed b y x j and by x j +1 or x j +2 whic h c ontradicts the assumption that x j , x j +1 and x j +2 are distinct). Then the required equalities follow b ecause the similar c haracterization holds f o r the co ordinates of x ′ j ′ +1 8 and x j = x ′ j ′ x j +1 = x ′ j ′ +1 , x j +2 = x ′ j ′ +2 . Th us we get the follo wing fa cts for t he po in ts in Z ′ . The X -support and the Z -supp ort of a p erio dic p oint are finite. The X - supp ort of a non-p erio dic p oin t z is infinite and t he elemen ts of the X -sequence of χ ( z , m ) are distinct for ev ery m . The X -supp orts of a non-p erio dic p oin t and a p erio dic p oin t are disjoint. The X -supp orts of a non-p erio dic p oin t z and an almost free point are disjoint because otherwise the X -sequence of χ ( z , m ) w ould con tain an elemen t with a fully free co ordinate for some m . Since eac h elemen t o f the Z -sequence of χ ( z, m ) of a non-p erio dic or perio dic po int z is re serv ed by an elemen t of the X - sequenc e of χ ( z , m + 1) w e get that the elemen ts of the Z - sequence o f χ ( z , m ) of a non-p erio dic po in t z are distinct fo r ev ery m and the Z - suppo rts o f a non-p erio dic p oin t and a p erio dic p o in t are disjoint. Because every p oin t of the Z -suppo r t of an almost free p oint in Z ′ is either fully f r ee or res erv ed by a point in it s X -supp ort w e get that the Z -supp o rts of a non-p erio dic p oint and an almost free point are disjoin t. It also fo llo ws from what w e said b efore that if fo r t w o non-p erio dic p oin ts z 1 and z 2 in Z ′ the c hains χ ( z 1 , m 1 ) and χ ( z 2 , m 2 ), m 1 , m 2 > 0, ha v e the same terminal Z -p oin ts then χ ( z 1 , m 1 + k ) − χ ( z 1 , m 1 ) = χ ( z 2 , m 2 + k ) − χ ( z 2 , m 2 ) for ev ery k (the tails of length k of χ ( z 1 , m 1 + k ) and χ ( z 2 , m 2 + k ) are equal). Let us sa y that t w o non-p erio dic p oin ts in Z ′ are equiv alen t if their X -supp ort s in tersect. Then fo r ev ery non-p erio dic p oin t z ∈ Z ′ w e can find m ( z ) > 0 suc h that if z 1 , z 2 are equiv alen t non-p erio dic p oints in Z ′ w e hav e that the chains χ ( z 1 , m ( z 1 )) and χ ( z 2 , m ( z 2 )) hav e the same t erminal Z -p oin ts. D enote b y Z ′ − a collection non-p erio dic p oints having exactly one represen tative in eac h equiv alence class. F or eve ry z ∈ Z ′ − denote χ − ( z , k ) = χ ( z , m ( z ) + k ) − χ ( z , m ( z )) a nd call χ − ( z , k ) the reduced c hain of length k g enerated by z . Note that for ev ery k all the elemen ts of b oth the X -sequence and Z -sequence of χ − ( z , k ), z ∈ Z ′ − are distinct and for z 1 , z 2 ∈ Z ′ − , z 1 6 = z 2 , w e hav e that the elemen ts o f the X -sequence and Z -sequence of χ − ( z 1 , k ) are distinct from the elemen ts of the X -sequence and Z -sequenc e resp ectiv ely of χ − ( z 2 , k ) Define X ′ ⊂ X as the set consisting of the X - supp orts o f a lmo st free and p erio dic p oin ts o f Z ′ and the elemen ts of the X - sequences of the c hains χ ( z , m ( z )) for t he non- p erio dic p oints z ∈ Z ′ . Similarly define Z ′ + ⊂ Z as the set consisting of the Z -supp orts of almost free and p erio dic p oints of Z ′ and the eleme nts of the Z -sequences of the c hains χ ( z , m ( z )) fo r the non-p erio dic p oints z ∈ Z ′ . Clearly X ′ and Z ′ + are finite and Z ′ − ⊂ Z ′ ⊂ Z ′ + . Fix an integer k and let 0 ≤ j ≤ k . Define X ( j ) ⊂ X as the union o f X ′ and the elemen t s of the X -sequences of χ − ( z , j ) for all z ∈ Z ′ − and define Z ( j ) ⊂ Z a s the union of Z ′ + and the elemen ts of the Z -sequences of χ − ( z , j ) for all z ∈ Z ′ − . It follows from our construction that the elemen ts of the X - sequence of χ − ( z , k ), z ∈ Z ′ − , do not lie in X (0) = X ′ , the initial Z -p oin t of the c hain χ − ( z , k ), z ∈ Z ′ − , is 9 the only e lemen t o f the Z -sequence of χ − ( z , k ) lying in Z (0) = Z ′ + , the coordinates of the p oin ts of X ( j ) are con tained in Z ( j ) and ev ery p oint of Z ( j ) is either fully f r ee or reserv ed b y a p oint in X ( j +1) . 3.3 Pro of of Theorem 1.2 Let X ⊂ [0 , 1] 3 b e a K o lmogorov -type em b edding with respect to Ω. W e will sho w that the induced transformation is op en at the zero-map on Z and this prov es Theorem 1.2. T ak e a finite subset Z ′ of Z . F ollo wing 3.2 replace Ω b y its infinite subfamily and define the finite sets X ′ ⊂ X a nd Z ′ − ⊂ Z ′ ⊂ Z ′ + ⊂ Z . Let a map φ : X − → R b e suc h that φ ( x ) = 0 for ev ery x ∈ X ′ and let δ > 0. W e will construct a map g : Z − → R suc h that k φ − L ( g ) k < δ and g ( z ) = 0 for eve ry z ∈ Z ′ . This sho ws that L is op en at the zero-map on Z b ecause L is op en in the uniform top ology and Theorem 1.2 follo ws. The case k φ k = 0 is trivial so w e can a ssume that k φ k > 0. Fix an in teger k ≥ 0 wh ich will be dete rmined later a nd whic h will dep end only on k φ k and δ . Again replace Ω b y its infinite subfamily and fo llo wing 3.2 construct the chains χ − ( z , k ) , z ∈ Z ′ − , and the sets X ( j ) and Z ( j ) , 0 ≤ j ≤ k . Define h : Z ( k ) − → R suc h that h ( z ) = 0 for ev ery z ∈ Z ′ + and for ev ery c hain χ − ( z , k ) = ( A, B ), z ∈ Z ′ − , with A = { x 1 , . . . , x k } and B = { z 0 , . . . , z 2 k } define h ( z 2 j − 2 ) = 0 and h ( z 2 j − 1 ) = φ ( x j ) in order to get φ ( x j ) = h ( z 2 j − 2 ) + h ( z 2 j − 1 ) + h ( z 2 j − 2 ) for 1 ≤ j ≤ k . Extend h ov er Z suc h that k h k ≤ k f k . Then k L ( h ) k ≤ 3 k φ k and φ ( x ) = L ( h )( x ) f or ev ery x ∈ X ( k ) . Th us for f = φ − L ( h ) w e ha v e k f k ≤ 4 k φ k and f ( x ) = 0 for ev ery x ∈ X ( k ) . No w w e will apply the appro ximation pro cedure 3 .1 for f and c = 5 k φ k to construct the map g ′ : Z − → R . The appro ximation pro cedure will b e applied with the following additional requiremen ts. W e assume tha t the cov er F ∈ Ω is c hosen so that each elemen t of F con tains a t most one p oint of X ( k ) and w e a ssume that for ev ery 1 ≤ i ≤ 3 we ha v e that the set p i ( F ) contains at most one point of Z ( k ) ∩ Y i for every F ∈ F i , g ′ i ( z ) = 0 for ev ery fully free z ∈ Z ( k ) ∩ Y i and, finally , if p i ( F ), F ∈ F i , con tains a strong ly rese rved point of z ∈ Z ( k ) ∩ Y i then F is so c lose to the p oint x ∈ X witness ing the reserv ation of z that we can set x F = x (recall that ev ery p oin t of Z ( k ) is either fully fr ee or strongly reserv ed). One can easily ve rify that suc h a cov er F ∈ Ω satisfying the requiremen ts of 3 .1 along with those w e just listed can b e chosen indeed. The conditions we imp ose on F imply that that if a p oint z ∈ Y i ∩ Z ( k ) is reserv ed b y a p oint x ∈ X then g ′ i ( z ) = 1 2 n +1 f ( x ). Th us w e get that g ′ ( z ) = 0 if z ∈ Z ( k ) is fully f ree and g ′ ( z ) = 0 if z ∈ Z ( k ) is rese rved b y a p oin t x ∈ X with f ( x ) = 0. 10 Recall that ev ery p oin t of Z ( k − 1) is either fully free or reserv ed by a p oin t of X ( k ) . Then, since f ( X ( k ) ) = 0 (tha t is f ( x ) = 0 for ev ery x ∈ X ( k ) ) w e get that g ′ ( Z k − 1) ) = 0. Since the co ordinates of the p oin ts in X ( k − 1) are contained in Z ( k − 1) w e also get that L ( g ′ )( X ( k − 1) ) = 0. Th us applying the approx imatio n pro cedure iteratively w e can construct the maps g ( t ) : Z − → R , 1 ≤ t ≤ k , suc h tha t L ( g ( t ) )( X ( k − t ) ) = 0 , g ( t ) ( Z ( k − t ) ) = 0, k g ( t ) k < 1 2 n +1 ( 2 n 2 n +1 ) t − 1 c and k f − L ( P t s =1 g ( s ) ) k < ( 2 n 2 n +1 ) t c . Then for g = h + P k t =1 g ( t ) w e hav e that g ( Z (0) ) = 0 and k φ − L ( g ) k < ( 2 n 2 n +1 ) k c . No w assum e that k is tak en such that ( 2 n 2 n +1 ) k c = ( 2 n 2 n +1 ) k (5 k φ k ) < δ . Th us w e get that g ( Z ′ ) = 0 and k φ − L ( g ) k < δ , and the theorem follo ws. 4 Problems As w e already men tioned in Section 1, Theorems 1.1 and 1.2 are partial p ositiv e solutions of the follo wing op en problem whic h was posed in [8]. Problem 4.1 L et X ⊂ Y 1 × · · · × Y k b e a b asic emb e dding of a c o mp actum X i n to the pr o duct of c omp a c ta Y 1 , . . . , Y k . Is the induc e d tr ansforma tion always op en in the C p - top olo gy? Problem 4.1 in its full generalit y seems to b e diffic ult, therefore it is justified to discuss some cases of this problem related to the t yp es of basic em b edding considered in this pap er. It w as already mentioned in Section 1 that Pro blem 4.1 has the affirmativ e answ er if k = 2 [8]. The cas e k = 2 and Theorem 1.1 can b e considere d in the following generalizing con text. Let X and Y 1 , . . . , Y k b e compact, X ⊂ Y 1 × · · · × Y k and p i : X − → Y i the pro j ections. Define S i ( X ) = { x ∈ X : p − 1 i ( p i ( x )) = x } , E i ( X ) = X \ S i ( X ), E ( X ) = E 1 ( X ) ∩ · · · ∩ E k ( X ), E 1 ( X ) = E ( X ) and b y induction E t ( X ) = E ( E t − 1 ( X )). Let us call the em b edding of X a Sternfeld e mbedding of general t yp e if the re is t suc h that E t ( X ) = ∅ . Sternfeld show ed that a ny basic em b edding in to the pro duct o f t w o spaces is of Sternfeld’s general t yp e and a n y Sternfeld em b edding of general type is basic [5 ]. Theorem 1.1 admits a relatively simple generalization f or embeddings with E ( X ) = ∅ . It w o uld be in teresting to answ er Problem 4.2 Is the induc e d tr ans f o rmation of any Sternfeld em b e dding of gener al typ e op en in the C p -top olo gy? 11 Note tha t not ev ery basic em b edding is of Sternfeld’s general ty p e (for example no em b edding of a circle S 1 in to [0 , 1] k is of Sternfeld’s general t yp e). In connection to The orem 1 .2 it se ems v ery in teresting to address the case of Kolmogoro v- t yp e em b eddings of compacta of dim > 1. Problem 4.3 Is the induc e d tr ansformation of an y Kolmo gor ov - typ e em b e dding op en in the C p -top olo gy? Note that the [0 , 1] in terv al in Theorem 1.3 cannot b e replaced by a 0-dimensional compactum [8]. Also note Theorem 1.3 do es not hold if X is not strongly coun table dimensional [7]. This suggests the fo llowing problem. Problem 4.4 Char a c terize c omp a cta X admitting a line ar op en c ontinuous tr ansform a- tion fr om C p [0 , 1] onto C p ( X ) . Problem 4.4 is also unsettled for not necessarily op en transformations [7]. References [1] Kolmogorov , A. N. On the represen tation of contin uous functions of man y v ariables b y sup erp osition of con tinuous functions of one v ariable and addition. Amer. Math. So c. T ransl. (2) 28 1963 55– 5 9. [2] Ostrand, Phillip A. Dimension of metric spaces and Hilb ert’s problem 13. Bull. Amer. Math. So c. 71 1965 619–62 2. [3] P a vlov ski ˘ ı, D . S. Spaces of contin uo us functions. Sov iet Math. Dokl. 22 (1980), no . 1, 34–37 (1981). [4] Sternfeld, Y. Sup erp ositions of con tin uous functions. J. Approx . Theory 25 (1979), no. 4, 360–368 . [5] Sternfeld, Y aki. Hilb ert’s 13th problem and dimension. Geometric asp ects of func- tional a na lysis (1987– 88), 1–49, Lecture Notes in Math., 1376, Springer, Berlin, 1989. [6] Sternfeld, Y aki. Mappings in dendrites and dimension. Houston J. Math. 19 (19 93), no. 3, 483–497 . 12 [7] Leiderman, Ark ady; Morris, Sidney A.; P esto v, Vladimir The free abelian top ological group and the free lo cally conv ex space on the unit in terv al. J. London Math. So c. (2) 56 (1997), no. 3, 529–5 38. [8] Leiderman, A.; Levin, M.; P esto v, V. On linear contin uous op en surjections of the spaces C p ( X ). T op ology Appl. 81 (199 7), no. 3, 269–279. [9] Levin, Mic hael. Certain finite-dimensional maps and their application to h yp erspaces. Israel J. Math. 105 (1998), 257–26 2 . Departmen t of Mathematics Ben Gurion Univ ersit y of the Negev P .O.B. 653 Be’er Shev a 84105, ISRAEL e-mail: mlevine@math.bgu.ac.il 13