Dimension of graphoids of rational vector-functions

DIMENSION OF GRAPHOIDS OF RA TIONAL VECTOR-FUNCTIONS T ARA S BANAKH AND O LES POTY A TY N YK Abstract. Let F ⊂ R ( x, y ) b e a coun table fami ly of rational f unctions of tw o v ariables with real coeﬃcients. Eac h rational f unction f ∈ F can be thought as a con tinuous function f : do m( f ) → ¯ R taking v alues in the pro jec tive li ne ¯ R = R ∪ {∞} and deﬁned on a coﬁnite subset dom( f ) of the torus ¯ R 2 . Then the family F determines a con tin uous v ector-function F : dom( F ) → ¯ R F deﬁned on t he dense G δ -set dom( F ) = T f ∈F dom( F ) of ¯ R 2 . The closure ¯ Γ( F ) of its graph Γ( F ) = { ( x, f ( x )) : x ∈ dom( F ) } in ¯ R 2 × ¯ R F is called the gr aphoid of the family F . W e pr o ve the graphoid ¯ Γ( F ) has topological di mension dim( ¯ Γ( F )) = 2. If the family F contains all linear fractional transformations f ( x, y ) = x − a y − b for ( a, b ) ∈ Q 2 , then the graphoid ¯ Γ( F ) has cohomological dimension dim G ( ¯ Γ( F )) = 1 for an y non-trivial 2-di vi sible abelian group G . Hence the space ¯ Γ( F ) is a natural example of a compact space that is not dimensionally full- v alued and by this prop ert y r esem bles the famous P ont ryagin surface. 1. Introduction Let X, Y be top ological spaces and f : dom( f ) → Y b e a function deﬁned o n a subset dom( f ) ⊂ X . Such a function f will be called a p artial function on X . The closure ¯ Γ( f ) of the gr a ph Γ( f ) = { ( x, f ( x )) : x ∈ do m( f ) } of f in the Cartesian pro duct X × Y will b e called the gr aphoid of f . The gr aphoid ¯ Γ( f ) deter mines a mult i-v a lued function ¯ f : X ⊸ Y ass igning to ea ch p o int x ∈ X the (p ossibly empty) subset ¯ f ( x ) = { y ∈ Y : ( x, y ) ∈ ¯ Γ( f ) } . It is clear that ¯ Γ( f ) co inc ide s with the graph Γ( ¯ f ) = { ( x, y ) ∈ X × Y : y ∈ ¯ f ( x ) } of the m ulti-v a lue d function ¯ f : X ⊸ Y . Also it is clear that f ( x ) ∈ ¯ f ( x ) for ea ch x ∈ dom( f ). The m ulti-v alued function ¯ f is ca lle d the gr aphoi d extension of the partial function f . The set dom( ¯ f ) = { x ∈ X : ¯ f ( x ) 6 = ∅} will b e called the domain of ¯ f . If the spac e Y is compact, then the pro jection pr X : Γ( ¯ f ) → X is a p erfect ma p [9, 3 .7.1], whic h implies that the multi-v alued map ¯ f is upper se mi-contin uo us in the sense that for an y ope n subset U ⊂ Y the preimage ¯ f − 1 ⊂ = { x ∈ X : ¯ f ( x ) ⊂ U } is op en in X . In this pap er we shall study top olog ical prop erties of the graphoids of rational v ector- functions. By a r ational function of k v ariables w e under stand a par tial function f : dom( f ) → ¯ R of the form f ( x 1 , . . . , x k ) = p ( x 1 , . . . , x k ) q ( x 1 , . . . , x k ) where p and q are t wo relatively prime polyno mials o f k v a riables. The rational function f = p q is deﬁned on the op en dense subse t dom( f ) = R k \ ( p − 1 (0) ∩ q − 1 (0)) of ¯ R k and takes its v alues in the pro jective real line ¯ R = R ∪ {∞} (carr y ing the top olog y of o ne-p oint co mpactiﬁcation of the real line R ). By R ( x 1 , . . . , x k ) we denote the ﬁeld of rationa l functions of k v aria bles with co eﬃcients in the ﬁeld R of rea l nu mbers. Each rational function f ∈ R ( x 1 , . . . , x k ) will be thought as a partial function deﬁned on the op en dense subset dom( f ) of the k -dimensio nal torus ¯ R k with v alues in the pro jective line ¯ R . By a r ational ve ctor-function we understand an y fa mily F ⊂ R ( x 1 , . . . , x k ) o f ratio nal functions. If F is c ountable, then the in tersection dom( F ) = T f ∈F dom( f ) is a dense G δ -set in ¯ R k . So, F can b e thought as a par tial function F : dom( F ) → ¯ R F , F : x 7→ ( f ( x )) f ∈F . Its graphoid ¯ Γ( F ) is a closed subset of the compact Hausdor ﬀ space ¯ R k × ¯ R F and its grapho id extension ¯ F : ¯ R k ⊸ ¯ R F is an upper se mi- contin uo us m ulti-v alued function with dom( ¯ F ) = ¯ R k . F or every f ∈ F the comp ositio n pr f ◦ ¯ F : ¯ R k → ¯ R of ¯ F with the pr o jection pr f : ¯ R F → ¯ R , pr f : ( x g ) g ∈F 7→ x f , coincides with the gra phoid extensio n ¯ f o f the r ational function f . F or uncountable families F ⊂ R ( x 1 , . . . , x k ) this approach to deﬁning ¯ F : ¯ R k → ¯ R F do es not w ork pro p erly a s dom( F ) = T f ∈F dom( f ) can be empty . This problem can b e ﬁxed as follows. 1991 Mathematics Subje ct Classiﬁc ati on. 54F45, 55M10; 14J80 ; 14P05; 26C15; 55M25; 54C50. Key wor ds and phr ases. Graphoid, graph, r ational v ector-function, top ological di mension, exten sion dimension, cohomological dimen- sion, Pon tryagin sur f ace. 1 2 T ARAS BANAKH AND OLES POTY A TYNYK Let F be a family of partial functions f : do m( f ) → Y d eﬁned on subsets dom( f ) of a topolo gical spa ce X . By the gr aphoid extension of F we under stand the multi-v alue d function ¯ F : X ⊸ Y assigning to each p oint x ∈ X the set ¯ F ( x ) of all p oints y = ( y f ) f ∈F ∈ Y F such that for a ny neighborho o d O ( x ) ⊂ X of the p oint x , any ﬁnite subfamily E ⊂ F , and neighborho ods O ( y f ) ⊂ Y of the p o int s y f , f ∈ E , there is a po int x ′ ∈ O ( x ) ∩ T f ∈E dom( f ) suc h tha t f ( x ′ ) ∈ O ( y f ) fo r all f ∈ E . The graph Γ( ¯ F ) = { ( x, y ) ∈ X × Y F : y ∈ F ( x ) } of the multi-v alued function ¯ F is called the gr aphoid of the family F . The set dom( ¯ F ) = { x ∈ X : ¯ F ( x ) 6 = ∅ } is called the domain of ¯ F . If the family F is empty , then ¯ Y F = Y ∅ is a singleton and the gr a phoid ¯ Γ( F ) coincides with X × Y ∅ . It can b e shown that for any fa mily of rationa l functions F ⊂ R ( x 1 , . . . , x k ) its gr aphoid extension ¯ F : ¯ R k ⊸ ¯ R F has the following pro per ties: (1) ¯ F is upper semi-co nt inuous; (2) dom( ¯ F ) = ¯ R k ; (3) for an y subfamily E ⊂ F and the coo rdinate pro jectio n pr E : ¯ R F → ¯ R E the comp ositio n pr E ◦ ¯ F : R k → ¯ R E coincides with the grapho id extension ¯ E of E ; (4) If dom( F ) = T f ∈F dom( f ) is dense in R k , then ¯ F coincides w ith the graphoid extensio n of the partia l function F : dom( F ) → ¯ R F , F : x 7→ ( f ( x )) f ∈F . In this pap er w e shall co nsider the following problem. Problem 1.1. Given a family of r ational functions F ⊂ R ( x 1 , . . . , x k ) , st udy top olo gic al (and dimensio n) pr op erties of the gr aphoid ¯ Γ( F ) ⊂ ¯ R k × R F of F . A pr e cise question: Has ¯ Γ( F ) the top olo gic al dimension dim(Γ( ¯ F )) = k ? This problem was motiv a ted by the problem o f studying the top olog ic a l structure of the space of r eal places of a ﬁeld of rational functions, posed in [3] and par tly solved in [14], [8]. In this pape r w e shall answer P roblem 1.1 for k ≤ 2. In fact, the case k = 1 is trivial: each rational function f ∈ R ( x ) admits a con tinuous extension to ¯ R and ca n b e thought as a contin uous function f : ¯ R → ¯ R . Then any family F ⊂ R ( x ) can be tho ug ht as a contin uous function F : ¯ R → ¯ R F . Its graphoid extension ¯ F coincides with F . Co nsequently , the graphs ¯ Γ( F ) = Γ( F ) are ho meomorphic to the pro jective real line ¯ R and hence, dim( ¯ Γ( F )) = dim( ¯ R ) = 1 . The cas e of tw o v ariables is muc h more diﬃcult. The following theorem is the main result o f this pap er and has a long and tec hnical pro of that exploits to o ls of Rea l and Complex Analy s is, Algebr aic Geometry , Algebraic T op olo gy , Dimension Theor y , Genera l T op olog y , and Com binatorics. This theorem has b een a pplied in [2] for ev a luating the dimension o f the space of real places of so me function ﬁelds. Theorem 1.2. F or any family of r ational functions F ⊂ R ( x, y ) its gr aphoid ¯ Γ( F ) ⊂ ¯ R 2 × ¯ R F has c overing top olo gic al dimension dim(Γ( ¯ F )) = 2 . This theorem reveals o nly a part of the truth abo ut the dimension o f ¯ Γ( F ). The other part s ays that for suﬃciently rich families F the g raphoid ¯ Γ( F ) has cohomologica l dimension dim G Γ( ¯ F ) = 1 fo r a ny 2- divisible ab elian g roup G ! So, ¯ Γ( F ) is a natural example o f a compact spa ce which is not dimensionally full-v alued. A classical example of this sort is the Pon tryagin surface: a surface with glued M¨ obius bands at each p oint o f a countable dense set, see [1, § 4.7]. The cov ering and co homologica l dimensions are partial cases of the extension dimension [7] deﬁned as follows. W e say that the ext ension dimension of a top ologic a l spa ce X do es not exceed a top ologic a l spa ce Y and wr ite e-dim( X ) ≤ Y if each contin uous map f : A → Y deﬁned o n a clos ed subspace A of X can b e e x tended to a contin uo us map ¯ f : X → Y . By Theo rem 3.2.10 of [10], a compact Hausdorﬀ space X ha s cov ering dimension dim( X ) ≤ n for some n ∈ ω if and only if e-dim( X ) ≤ S n where S n stands for the n -dimensional spher e. On the other hand, for a non-trivial abelia n group G , a compact topo lo gical space X has co homologic a l dimension dim G ( X ) ≤ n if and only if e-dim( X ) ≤ K ( G, n ) where K ( G, n ) is the Eilenberg-Mac L a ne co mplex of G (this is a CW-complex having all homotopy g roups trivial e xcept for the n -th homotopy group π n ( K ( G, n )) which is isomorphic to G , see [11, § 4.2]). It is known [6] that dim G ( X ) ≤ dim( X ) for each abe lian gro up G and dim( X ) = dim Z ( X ) for any ﬁnite-dimensional compact space X . A gr oup G is called 2-divisible if for each x ∈ G there is y ∈ G with y 2 = x . Theorem 1.2 is completed by the following Theorem 1.3. If a famil y of r ational functions F ⊂ R ( x, y ) c ont ains a family of line ar fr actional tra nsformations n x − a y − b : ( a, b ) ∈ D o . for some dense subset D of R 2 , then the gr aphoid ¯ Γ( F ) of F has c ohomolo gic al dimensions dim Z ( ¯ Γ( F )) = dim( ¯ Γ( F )) = 2 and dim G (Γ( ¯ F )) = 1 for any non-t rivial 2-divisible ab elian gr oup G . DIMENSION OF GRAPHOIDS OF RA TIONAL VECTOR-FUNCTIONS 3 Theorems 1.2 and 1.3 will be prov ed in Sections 6 and 7. The main instrument in the pro of of these theore ms is Theo rem 3.1 describing the lo cal structure o f the graphoid extension ¯ F of a ﬁnite family of r ational functions F ⊂ R ( x, y ). Sectio n 2 contains some notation a nd prelimina ry infor mation, nec essary for the pr o of o f The o rem 3 .1. 2. Preliminaries This section has preliminary c haracter and contains no tations and facts necessa ry for understa nding the pro of of Theorem 3.1. 2.1. Notatio n and T erminology. F or tw o p oints a, b ∈ R 2 by [ a, b ] = { (1 − t ) a + tb : t ∈ [0 , 1] } we s hall denote the aﬃne seg ment connecting a and b a nd by α a,b : [0 , 1] → [ a, b ] , α a,b : t 7→ (1 − t ) a + tb, the cor resp onding aﬃne map. Let also ] a, b [ = [ a , b ] \ { a , b } be the op en se g ment with the end-p oints a, b . F o r a subset A ⊂ R 2 and a real n umber t let tA = { ta : a ∈ A } b e a homothetic copy o f A . By 0 = (0 , 0) we denote the origin of the pla ne R 2 . Two p oints a, b of a subs et B ⊂ R 2 \ { 0 } are called n eighb our p oints of B if a 6 = b and a, b ar e unique p oints of the set B that lie in the conv ex cone { ua + v b : u, v ≥ 0 } . By N ( B ) (r e sp. N { B } ) we denote the family of o rdered pair s ( a, b ) ∈ B 2 (resp. unordered pairs { a, b } ⊂ B ) of neighbor points of B . This fa mily will often occ ur in the pr o of o f Theorem 3.1 b elow, so this is an impor tant notion. A subset A of a metric space ( X , d ) is called an ε -net in X if fo r ea ch p oint x ∈ X there is a p oint a ∈ A with d ( x, a ) < ε . F or a p oint z of a metric s pace ( X , d ) and ε > 0 let B ( z , ε ) = { x ∈ X : d ( x, z ) < ε } , ¯ B ( z , ε ) = { x ∈ X : d ( x, z ) ≤ ε } , a nd S ( z , ε ) = { x ∈ X : d ( x, z ) = ε } denote respe c tively the open ε -ball, closed ε -ball and ε -sphere centered at the p oint z . A map f : X → Y b etw een top o logical spa ces X , Y is monotone if f − 1 ( y ) is connected for ea ch y ∈ Y . It is easy to see that for a connected subspace X ⊂ R a function f : X → R is mono tone if and o nly if f is either non-increa sing or non-decreasing. On the extended rea l line ¯ R = R ∪ {∞} we shall consider the metric d inherited from the complex plane C after the ident iﬁcation of ¯ R with the unit circle T = { z ∈ C : | z | = 1 } with help o f stereogr aphic pro jection that maps T \ { i } onto the real line R . In the metr ic d the extended r eal line ¯ R has diameter 2. O bserve that each (op en o r closed) ball in the metric space ( ¯ R , d ) is connec ted. By an ar c we understand a top olog ical co py of the c lo sed in terv al [0 , 1]. An arc A in ¯ R n is called a monotone ar c if for each i ∈ n the co o rdinate pro jection pr i : A → ¯ R is a monotone map. 2.2. Pusi eux-analytic functions. A function ϕ : A → R deﬁned on a subset A ⊂ R is called analytic if for every a ∈ A ther e are ε > 0 a nd real co eﬃcients ( c n ) n ∈ ω such that P ∞ n =0 | c n | ε n < ∞ a nd for ev ery x ∈ A with | x − a | < ε we get f ( x ) = P ∞ n =0 c n ( x − a ) n . Let ε b e a p ositive real num b er. A function ϕ : [0 , ε ] → R is called Pusieux-analytic if ϕ | (0 , ε ] is analytic and there are m ∈ N , δ ∈ (0 , ε ) and a n analytic function ψ : [0 , m √ δ ) → R such tha t ϕ ( x ) = ψ ( m √ x ) for all x ∈ [0 , δ ). The smallest nu mber m with this pr op erty is called the Pusieux denominator of ϕ . In a neig hborho o d o f z e ro a P usieux-ana ly tic function ϕ ( x ) develops in to a series P ∞ k =0 c k x k m called the Newton-Pusieux series of ϕ , see [4, 8.3]. The interv al [0 , ε ] will be called the domain of the Pusieux analytic function ϕ a nd will b e denoted by dom( ϕ ). The Uniquenes s Theor e m for a na lytic functions (see e.g ., [15]) implies the following Uniqueness Theorem for Pusieux-ana lytic functions. Theorem 2.1. Two Pu s ieux -analytic funct ions f , g : [0 , ε ] → R ar e e qual if and only if the set { x ∈ [0 , ε ] : f ( x ) = g ( x ) } is inﬁnite. The Pus ie ux analycity can b e also introduced for functions deﬁned on an interv al [ − ε , 0]. Namely , we say that a function ϕ : [ − ε, 0] → R is Pusieux -analytic if the function ψ : [0 , ε ] → R , ψ : x 7→ ϕ ( − x ), is Pus ie ux ana ly tic. Two Pusieux analytic function ϕ , ϕ ∗ are called c onjugate if they have the sa me P us iuex denominator m and for some a nalytic function ψ : ( − δ, δ ) we get { ( t m , ψ ( t )) : | t | < δ } = { ( x, ϕ ( x )) : x ∈ dom( ϕ ) ∩ ( − δ m , δ m ) } ∪ { ( x, ϕ ∗ ( x )) : x ∈ dom( ¯ ϕ ) ∩ ( − δ m , δ m ) } . It can be shown that the Pusiuex denomina tor m of tw o conjugate Pusieux analytic functions ϕ, ϕ ∗ is o dd if and only if dom( ϕ ) ∩ dom( ϕ ∗ ) = { 0 } . F or example, the Pusieux analy tic functions ϕ 1 : [ − ε, 0] → R , ϕ 1 : x 7→ x 1 3 , and ϕ ∗ 1 : [0 , ε ] → R , ϕ ∗ 1 : x 7→ x 1 3 , ar e conjugate and hav e P usieux deno minator 3. The Pusie ux analytic functions ϕ 2 : [0 , ε ] → R , ϕ 2 : x 7→ x 3 2 , and ϕ ∗ 2 : [0 , ε ] → R , ϕ ∗ 2 : x 7→ − x 3 2 , ar e conjuga te and hav e Pusieux denominator 2 . 4 T ARAS BANAKH AND OLES POTY A TYNYK ✲ ✻ x y ϕ 1 ϕ ∗ 1 ✲ ✻ x y ϕ 2 ϕ ∗ 2 Lemma 2.2. If ϕ, ϕ ∗ ar e two c onjugate Pusieux analytic functions, then for any r ational fun ction f ∈ R ( x, y ) the limits lim x → 0 f ( x, ϕ ( x )) and lim x → 0 f ( x, ϕ ∗ ( x )) exist and ar e e qual. Pr o of. The lemma is triv ia l if the rationa l function f is constant. If f is not c onstant w e can wr ite it as the fraction f = p q of tw o r e latively prime p oly no mials p and q . Obs erve that for each analytic function ψ : [ − δ, δ ] → R and any m ∈ N the functions p ( t m , ψ ( t )) and q ( t m , ψ ( t )) ar e analytic and hence develop in to Mac la urin series at a neigh b orho o d of zero. This fact c a n b e used to show that a (ﬁnite or inﬁnite) limit lim t → 0 f ( t m , ψ ( t )) = lim t → 0 p ( t m , ψ ( t )) q ( t m , ψ ( t )) exits. Now let m b e the Pusieux denominator of the conjugated Pusieux-analytic functions ϕ a nd ϕ ∗ and ψ : [ − δ, δ ] → R be an analytic function such that { ( t m , ψ ( t )) : | t | < δ } = { ( x, ϕ ( x )) : x ∈ dom( ϕ ) ∩ ( − δ m , δ m ) } ∪ { ( x, ϕ ∗ ( x )) : x ∈ dom( ϕ ∗ ) ∩ ( − δ m , δ m ) } . It fo llows that lim x → 0 f ( x, ϕ ( x )) = lim t → 0 f ( t m , ψ ( t )) = lim x → 0 f ( x, ϕ ∗ ( x )) .  2.3. A lo cal s tructure of a plane algebraic curv e. In this section we recall the known description of the lo cal structure of a n algebra ic curve. By a n algebr aic curve we understa nd a se t o f the form p − 1 (0) = { ( x, y ) ∈ R 2 : p ( x, y ) = 0 } where p ∈ R ( x, y ) is a non-zero poly no mial of t wo v ar iables with real co eﬃcien t. The p olyno mial p in this deﬁnition can be also replaced b y a non-zero ra tional function r = p q where p and q are t wo relatively prime p o lynomials. In this case the symmetric diﬀere nc e of the a lgebraic curves r − 1 (0) = { ( x, y ) ∈ dom( r ) : r ( x, y ) = 0 } and p − 1 (0) lies in the intersection p − 1 (0) ∩ q − 1 (0), whic h is ﬁnite according to the classical B´ ezout Theo r em or [4, 6.1] or [13, 5.7]. W e are going to describ e the structure of a n alg ebraic cur ve A ⊂ R 2 at a neighbor ho o d o f zero 0 = (0 , 0). By K = ( − 1 , 1 ) 2 we shall denote the open sq ua re with side 2 centered at the origin 0 o f the plane and by ¯ K and K ∂ its closure and its bo undary in the plane R 2 . Let ¯ K ◦ = ¯ K \ { 0 } b e the square K with remo ved cen trum and {± 1 } 2 = {− 1 , 1 } 2 be the set of the vertices of the square. Next, de c omp ose the square ¯ K into four triangles: • ¯ K N = { ( x, y ) ∈ ¯ K : | x | ≤ y } , • ¯ K W = { ( x, y ) ∈ ¯ K : | y | ≤ − x } , • ¯ K S = { ( x, y ) ∈ ¯ K : | x | ≤ − y } , • ¯ K E = { ( x, y ) ∈ ¯ K : | y | ≤ x } , whose indice s N , W , S , E corres po nd to the directions: North, W est, South a nd Eas t. A subset C ⊂ R 2 is ca lled an e ast ε -elementary curve if C ⊂ ε ¯ K E and C = { ( x, ϕ ( x )) : x ∈ (0 , ε ] } for a (unique) Pusieux-ana lytic function ϕ : [0 , ε ] → R . The Pusieux denominator m of ϕ will be called the Pusieux denominator o f C . An ea st ε - e le men tary c urve C is drawn o n the following picture: DIMENSION OF GRAPHOIDS OF RA TIONAL VECTOR-FUNCTIONS 5      ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅      ❜ C The deﬁnitions o f north, west, and so uth ε -elementary curves can be obtained b y “rotating” the deﬁnition of an east ε -elementary curve. Namely , let R π 2 : ( x, y ) 7→ ( y , − x ) b e the clo ckwise r otation of the plane on the angle π 2 . Then R π = R π 2 ◦ R π 2 and R 3 π 2 = R π ◦ R π 2 are the clo ckwise rota tio ns of the pla ne b y the angles π and 3 π 2 , res p ectively . A subset C ⊂ R 2 is called north (resp. west , south ) ε -elementary curve if R π 2 ( C ) (r esp. R π ( C ), R 3 π 2 ( C )) is an east ε -elementary c ur ve. A subse t C ⊂ R 2 will b e calle d a n ε -elementary curve if C is an eas t, north, west or south ε -elementary curve. W e sha ll exploit the following fundamen tal fact describing the loca l structure of a plane algebraic curve, see [4, § 8.3] or [13, § 16]. Theorem 2. 3. F or any algebr aic curve A ⊂ R 2 ther e is ε > 0 such t hat t he interse ction A ∩ ε ¯ K ◦ has ﬁnitely many c onne cte d c omp onents and e ach of them is an ε - elementary curve. F or an alg ebraic cur ve A ⊂ R 2 the num b er ε > 0 satisfying the condition of Theo rem 2 .3 will b e ca lle d A -smal l . F or a n A -small n umber ε e a ch co nnected comp onents of A ∩ ε ¯ K is an ε - elementary curve ca lled an ε -br anch of A . Each ε -branch C of A has a conjugated ε -br anch C ∗ of A deﬁned as follows. Assume ﬁrst that the ε -branch C is a n east ε - elementary curve. Then C = { ( x, ϕ ( x )) : x ∈ (0 , ε ] } for so me Pusieux-ana lytic function ϕ : [0 , ε ] → [ − ε , ε ] with Pusieux deno minator m . F o r the function ϕ there exist a p o sitive δ ≤ m √ ε and an analytic function ψ : [ − δ, δ ] → [ − ε , ε ] such that ϕ ( x ) = ψ ( m √ x ) for all x ∈ [0 , δ m ]. If m is o dd, then the formula ϕ ∗ ( x ) = ψ ( m √ x ) determines a Pusieux-analytic function ϕ ∗ : [ − δ m , 0] → R , which is conjugate to ϕ . If m is even, then the c onjugate function ϕ ∗ : [0 , δ m ] → R is deﬁned by the for m ula ϕ ∗ ( x ) = ψ ( − m √ x ). W e claim tha t the gra ph { ( x, ϕ ∗ ( x )) : x ∈ dom( ϕ ∗ ) } lies in some ε -branch C ∗ of the a lgebraic curve A . Find a po lynomial p ∈ R ( x, y ) suc h that A = p − 1 (0). T aking in to acco un t that C ⊂ A , we co nclude that p ( x, ϕ ( x )) = 0 for all x ∈ [0 , ε ] and hence p ( t m , ψ ( t )) = 0 for all t ∈ [0 , δ ]. T aking into acco un t that the formula f ( t ) = p ( t m , ψ ( t )) determines a n analytic function f : [ − δ, δ ] → R , whic h is zero on [0 , δ ], we conclude that f ≡ 0. If m is o dd, then for every x ∈ [ − δ m , 0] = dom( ϕ ∗ ) and t = m √ x , we g et p ( x, ϕ ∗ ( x )) = p ( t m , ψ ( t )) = 0. If m is even, then for every x ∈ [0 , δ m ] = dom( ϕ ∗ ) a nd t = − m √ x , we get p ( x, ϕ ∗ ( x )) = p ( t m , ψ ( t )) = 0 . Ther efore the graph { ( x, ϕ ∗ ( x )) : x ∈ dom( ϕ ∗ ) \ { 0 }} lies in the algebraic curve A a nd b eing a connected subset of A ∩ ε ¯ K ◦ lies in a unique branch C ∗ , which is calle d the c onjugate ε -br anch of the ε -branch C . Observe that the conjugate branch C ∗ is an east ε -elementary curve if m is even and west if m is o dd. By a na logy w e can deﬁne conjugate branches o f north, west and south ε -bra nches of the algebr aic curve A . Since the c o njugate P usieux a nalytic cur ves are not equal, the conjugated ε -branches of A a r e disjoint. So, the in tersection A ∩ ε ¯ K ◦ decomp oses into the unio n of conjugated branches and hence con tains an even num b er of connec ted comp onents. This is a cr ucial o bserv a tion which will b e used in the pro of of the inequality dim ¯ Γ( F ) ≥ 2 in T he o rem 1 .2. 2.4. Degree of m aps b etw een circles. In this section we recall some basic information a bo ut the degree of maps betw een circles. Since the degree ha s top ologica l nature, instead of the cir cle we ca n cons ider the b oundary K ∂ of the square K = ( − 1 , 1) 2 in the plane R 2 . W e assume tha t the rea der knows Elements of Singular Homology Theory with co eﬃcie nts in an ab elian group G at the level of Chapter 2 of Hatcher’s monog r aph [11]. In particula r , we assume tha t the rea der knows the deﬁnition of the ﬁrs t homology group H 1 ( X ; G ) of a top ologica l spa c e X and also that each cont inuous map f : X → Y induces a ho mo morphism f ∗ : H 1 ( X ; G ) → H 1 ( Y ; G ) of the corresp onding ho mology g roups. It is well-kno wn that the ﬁr st homology group H 1 ( K ∂ ; G ) of the (top olog ical) circle K ∂ is isomorphic to the g r oup G , see [11, p.153]. In particular , for the inﬁnite cyclic group G = Z the ﬁrst ho mo logy gr oup H 1 ( K ∂ ; Z ) is isomo rphic to Z . Observe that each homomor phism h : Z → Z is of the form h ( x ) = d · x for some integer num b er d called the de gr e e of the homo morphism h . By the de gr e e o f a contin uous map f : K ∂ → K ∂ we understa nd the degree of the induced homomorphism f ∗ : H 1 ( K ∂ ; Z ) → H 1 ( K ∂ ; Z ). 6 T ARAS BANAKH AND OLES POTY A TYNYK F or the co eﬃcien t gr oup Z 2 = Z / 2 Z the situation simpliﬁes. Ther e are only t wo homomo rphisms from Z 2 to Z 2 : ident ity a nd trivial (or ann ulating). So, we ca ll a map f : X → Y b etw een top olog ical c ir cles Z 2 -trivial if the induced homomorphism f ∗ : H 1 ( X ; Z 2 ) → H 1 ( Y ; Z 2 ) is trivial. It is ea s y to see that a ma p f : K ∂ → K ∂ is Z 2 -trivial if a nd only if it has even degr ee. W e shall need the fo llowing fact whose pro of can be found in [1 1, § 2.2]. Lemma 2.4. A map f : K ∂ → K ∂ has even de gr e e and is Z 2 -trivial if for some p oint y ∈ K ∂ the pr eimage f − 1 ( y ) has ﬁnite even c ar dinality and e ach p oint x ∈ f − 1 ( y ) has a neighb orho o d U x ⊂ K ∂ such that the r estriction f | U x : U x → K ∂ is monotone. Another r esult that will be used in the pro o f of Theorem 1 .2 is the addition formula for degrees, which in Z 2 -case lo oks as follows: Lemma 2.5. L et Z b e a ﬁnite subset of the op en squar e K = ( − 1 , 1) 2 in the plane R 2 endowe d with t he max-norm, and ε > 0 b e a num b er su ch t hat K ∂ ∩ ¯ B ( Z , ε ) = ∅ and ¯ B ( z , ε ) ∩ ¯ B ( z ′ , ε ) = ∅ for distinct p oints z , z ′ ∈ Z . L et f : ¯ K \ B ( Z, ε ) → K ∂ b e a c ontinuous map. If for every z ∈ Z the r estriction f | S ( z , ε ) : S ( z , ε ) → K ∂ is a Z 2 -trivial map, then the r estriction f | K ∂ : K ∂ → K ∂ also is Z 2 -trivial. Pr o of. Let σ : [0 , 1] → K ∂ be a contin uous map such that σ (0 ) = σ (1) and σ | [0 , 1) : [0 , 1 ) → K ∂ is bijective. By [11, 2.23], its homolo gy class [ σ ] is a gener ator of the ho mology gro up H 1 ( K ∂ ; Z 2 ), which is isomor phic to Z 2 . Let X = ¯ K \ B ( Z, ε ) and f ∗ : H 1 ( X ; Z 2 ) → H 1 ( K ∂ ; Z 2 ) denote the homomor phis m betw een the ﬁrst homolog y groups, induced by the map f : X → K ∂ . Let i : K ∂ → X denote the identit y embedding. F or every z ∈ Z consider the singular simplex σ z : [0 , 1] → S ( z , ε ), σ z : t 7→ z + εσ ( t ), whos e homolog y class is a generator o f the homology gro up H 1 ( S ( x, ε ); Z 2 ) which is isomorphic to Z 2 . Since the comp ositio n f | S ( z , ε ) : S ( z , ε ) → K ∂ is Z 2 -trivial, f ∗ ([ σ z ]) = 0. It is easy to show tha t the 1-cycle σ − P z ∈ Z σ z is equal to the b oundary o f some singular 2-chain in X . Consequen tly , f ∗ ([ σ ]) = P z ∈ Z f ∗ ([ σ z ]) = 0 and f ∗ ◦ i ∗ = 0, whic h means that the ma p f | K ∂ = f ◦ i is Z 2 -trivial.  3. Resol ving the singularity of a ra tional vector-function In this section given a ﬁnite non-empty family F ⊂ R ( x, y ) thought as a r a tional vector-function, w e study the lo c al structure of its ca nonical multi-v alued extension ¯ F : R 2 ⊸ ¯ R F at a neighborho o d of an a rbitrary p oint ( a, b ) ∈ R 2 . W e lose no genera lit y ass uming the p oint ( a, b ) co incides with the origin 0 = (0 , 0) of the plane R 2 . The principa l result of this sec tion is the following structure theor em. Theorem 3.1. L et F ⊂ R ( x, y ) b e a non-empty ﬁnite family of r ational functions and ¯ F : R 2 ⊸ ¯ R F b e its gr aphoid extension. Ther e is ˜ ε > 0 su ch that for every ε ∈ (0 , ˜ ε ] ther e is a home omorphi sm h : ε ¯ K \ ε 2 ¯ K → ε ¯ K ◦ such that (1) ε ¯ K ◦ ⊂ dom( F ) . (2) h | εK ∂ = id . (3) F or every f ∈ F the c omp osition f ◦ h : ε ¯ K \ ε 2 ¯ K → ¯ R has a c ont inuous extension ¯ f h : ε ¯ K \ ε 2 K → ¯ R . (4) The fun ctions ¯ f h , f ∈ F , c omp ose a c ontinuous extens ion ¯ F h = ( ¯ f h ) f ∈F : ε ¯ K \ ε 2 K → ¯ R F of F ◦ h such that ¯ F h ( ε 2 K ∂ ) = ¯ F ( 0 ) . (5) Ther e is a ﬁnite subset B 0 of εK ∂ c ontaining the set {− ε, ε } 2 of vertic es of εK s u ch t hat for any n eighb or p oints a, b of 1 2 B 0 and every f ∈ F t he r estriction ¯ f h | [ a, b ] : [ a, b ] → ¯ R is monotone and the image ¯ f h ([ a, b ]) lies in one of the se gments [0 , 1 ] , [ − 1 , 0] , [1 , ∞ ] , [ ∞ , − 1 ] c omp osing t he cir cle ¯ R . (6) The set ¯ F ( 0 ) is either a singleton or a ﬁn ite union of monotone ar cs in ¯ R F . (7) F or any c ontinuous map g : ¯ F ( 0 ) → K ∂ the c omp osition g ◦ ¯ F h | ε 2 K ∂ : ε 2 K ∂ → K ∂ is Z 2 -trivial. Pr o of. W e lose no generality assuming that all functions f ∈ F are not constant. Obs erve that for each ra tional function f = p q ∈ F ⊂ R ( x, y ) the set R 2 \ dom( f ) ⊂ p − 1 (0) ∩ q − 1 (0) is ﬁnite accor ding to the c la ssical theo r em of B´ ezout [4, 6 .1] (which sa ys that for tw o r elatively prime polynomia ls p, q ∈ R ( x, y ) the algebra ic curves p − 1 (0) and q − 1 (0) hav e ﬁnite intersection). This implies tha t the set dom( F ) = T f ∈F dom( f ) is coﬁnite in R 2 (i.e., has ﬁnite complement in R 2 ). In the family F consider the subfamilies: • F x of ra tional functions f ∈ F with non-zero partial deriv ative f x = ∂ f ∂ x ; • F y of ra tional functions f ∈ F with non-zero partial deriv ative f y = ∂ f ∂ y . Let C 0 = { 0 , 1 , − 1 , ∞} , X = { ( x, y ) ∈ R 2 : x 2 = y 2 } , and conside r the algebraic curve A 0 = X ∪ [ f ∈F x f − 1 x (0) ∪ [ f ∈F y f − 1 y (0) ∪ [ f ∈F f − 1 ( C 0 ) . DIMENSION OF GRAPHOIDS OF RA TIONAL VECTOR-FUNCTIONS 7 Using Theo rem 2.3, choose an A 0 -small n umber ˜ ε ∈ (0 , 1 ) such that ˜ ε ¯ K ◦ ⊂ dom( F ). F or this n umber ˜ ε the int ersectio n A 0 ∩ ˜ ε ¯ K ◦ decomp oses into even num be r of pairwise disjoint ˜ ε -elementary curves. Since the set R 2 \ dom( F ) is ﬁnite, we can assume that ˜ ε is s o small that ˜ εK ◦ ⊂ dom( F ). Now, given any real num ber ε ∈ (0 , ˜ ε ] we shall construct a homeo morphism h : ε ¯ K \ ε 2 K → εK ◦ that satisﬁes the conditions (1)–(7) o f Theorem 3.1. Let us re call that d s ta nds for the metric on the extended rea l line ¯ R identiﬁ ed with the unit circle in the complex plane via the stereogr a phic pro jection. This metric induced the max- metr ic d F  ( x f ) , ( y f )  = max f ∈F d ( x f , y f ) on the F -tor us ¯ R F . Using Theore m 2.3, by induction we can constr uct a sequence of algebra ic curves ( A n ) ∞ n =1 , a sequence o f rea l nu mbers ( ε n ) ∞ n =1 and a sequence of ﬁnite subs ets ( C n ) n ∈ ω of ¯ R such that for ev ery n ∈ N the following conditions hold: (1) 0 < ε n < min { ε n − 1 , 2 − n } ; (2) the num b er ε n is A n -small; (3) the set C n +1 contains C n and is a ﬁnite 2 − n -net in ( ¯ R , d ); (4) A n +1 = A n ∪ S f ∈F f − 1 ( C n +1 ). Let ε 0 = ε . Now w e a re ready to c onstruct a ho meo morphism h : ε ¯ K \ ε 2 ¯ K → ε ¯ K ◦ required in Theorem 3.1. This homeomorphism will b e deﬁned recursively with help of the alg ebraic cur ves A n , n ∈ ω . F or every n ∈ ω consider the ﬁnite set B n = A n ∩ ε n K ∂ in the b oundar y of the square ε n K . It follows from X ⊂ A 0 that the set B 0 contains the set {− ε n , ε n } 2 of v ertices of the squar e ε n K . F or each p oint b ∈ B n there is a unique ε n -elementary branch C b of the alg ebraic curve A n such that { b } = C b ∩ A n . F or every n ∈ ω let δ n = ε 2 + ε n 2 and observe that lim n →∞ δ n = ε 2 . Let h − 1 be the identit y map of εK ∂ . By induction, for every n ∈ ω we shall deﬁne a subset B ′ n ∈ δ n K ∂ and a homeomorphism h n : δ n ¯ K \ δ n +1 K → ε n ¯ K \ ε n +1 K such that: (5) h n  ( ε 2 + t 2 ) K ∂  = tK ∂ for ea ch t ∈ [ ε n +1 , ε n ]; (6) B ′ n = h − 1 n − 1 ( B n ) ⊂ δ n K ∂ ; (7) for any b ′ ∈ B ′ n we get h n ([ δ n +1 δ n , 1] b ′ ) = C b \ ε n +1 K wher e b = h n − 1 ( b ′ ) ∈ B n ; (8) for a ny neighbor po ints a, b ∈ B ′ n and any t ∈ [ δ n +1 δ n , 1] the map h n | [ ta, tb ] is aﬃne, whic h means that h n ((1 − u ) ta + utb ) = (1 − u ) h n ( ta ) + u h n ( tb ) for all u ∈ [0 , 1]. The conditions (6)–(8) imply that for every n ∈ ω we get h n − 1 | δ n K ∂ = h n | δ n K ∂ . So , we can deﬁne a homeomor - phism h : ε ¯ K \ ε 2 ¯ K → ε ¯ K ◦ letting h | δ n ¯ K \ δ n +1 K = h n for a ll n ∈ ω . The pr op erties (5)–(8) of the homeo morphisms h n imply that the homeomorphism h has the following prope rties for every n ∈ ω : (9) h n  ε 2 + t 2 ) K ∂ ) = tK ∂ for ea ch t ∈ (0 , ε ]; (10) B ′ n = h − 1 ( B n ); (11) for any b ′ ∈ B ′ n ⊂ δ n K ∂ we get h (( ε 2 δ n , 1] b ′ ) = C b where b = h ( b ′ ) ∈ B n ; (12) for any neighbor p oints a, b of B ′ n and any t ∈ [ δ n +1 δ n , 1] the map h | [ ta, tb ] is aﬃne. Moreov er the c hoice of the algebraic curve A 0 guarantees that for any neighbor point a, b ∈ B 0 = B ′ 0 , any t ∈ (1 / 2 , 1], and any function f ∈ F (13) the restriction f ◦ h | [ ta, tb ] is either constant or injective (this follows from f − 1 x (0) ∪ f − 1 y (0) ⊂ A 0 ) a nd (14) the image f ◦ h ([ ta, tb ]) lies in one of se gments [0 , 1], [ − 1 , 0 ], [1 , ∞ ], [ ∞ , − 1] comp osing the pro jective line ¯ R (this fo llows from f − 1 ( { 0 , 1 , − 1 , ∞} ) ⊂ A 0 ). Now we shall pro ve the statements (1)–(7) of T he o rem 3.1. In fact, the statements (1) and (2 ) follow from the choice of ε = ε 0 and the deﬁnition of h | εK ∂ = h − 1 . The other sta temen ts will b e pr ov ed in a series of claims and lemmas. In the following claim (that proves the statement (3) of T heo rem 3 .1) o n the plane R 2 we consider the metric ρ  ( x, y ) , ( x ′ , y ′ )  = max {| x − x ′ | , | y − y ′ |} generated by the nor m k ( x, y ) k = max { | x | , | y |} . In this metr ic the sq ua re K is just the open unit ball centered at 0 . Claim 3.2 . F or every f ∈ F the map f ◦ h : ε ¯ K \ ε 2 ¯ K → ¯ R is uniformly c ontinuous and henc e admi ts a c ontinu ous extension ¯ f h : ε ¯ K \ ε 2 K → ¯ R . Pr o of. Given any η > 0, w e should ﬁnd τ > 0 suc h that for an y tw o points x, x ′ ∈ ε ¯ K \ ε 2 ¯ K with ρ ( x, x ′ ) < τ we get d ( f ◦ h ( x ) , f ◦ h ( x ′ )) < η . Choo se a natural num b er m ∈ N such that 2 − m +2 < η . By the uniform cont inuit y of the function f on the co mpact set ε ¯ K \ δ m +1 K , ther e exists a real num b e r τ 1 > 0 such that for any p oints 8 T ARAS BANAKH AND OLES POTY A TYNYK ( x, x ′ ) ∈ ε ¯ K \ δ m +1 K with ρ ( x, x ′ ) < τ 1 we hav e d ( f ◦ h ( x ) , f ◦ h ( x ′ )) < η . Let τ 2 = δ m − δ m +1 be equa l to the smallest dis tance b etw een the sq uares δ m K ∂ and δ m +1 K ∂ . Now let us consider the ﬁnite set ε 2 δ m B ′ m ⊂ ε 2 K ∂ and put τ 3 = min  ρ ( a ′ , b ′ ) : a ′ , b ′ ∈ ε 2 δ m B ′ m , a ′ 6 = b ′  . W e claim that the n umber τ = min { τ 1 , τ 2 , τ 3 } has the required prop erty . The choice of τ implies that any tw o p oints x, x ′ ∈ ε ¯ K \ ε 2 ¯ K with ρ ( x, x ′ ) < τ either b oth lie in ε ¯ K \ δ m +1 K (and by the deﬁnition of τ 1 this implies that d ( f ◦ h ( x ) , f ◦ h ( x ′ )) < η ), or they b oth lie in the s ame tra p e zoid T ab , bounded by the lines δ m K ∂ , ε 2 K ∂ , [ ε 2 δ m , 1] a, [ ε 2 δ m , 1] b , where a, b are neighbor p oints in B ′ m , or , at least, in such tw o adjacent trap ezoids. The interior T ab \ ∂ T ab of the tra pez oid T ab is a connected s et whose image h ( T ab \ ∂ T ab ) do es no t intersect the algebra ic curve A m while the image f ◦ h ( T a,b \ ∂ T ab ) do es not intersect the 2 − m +1 -net C m in ¯ R . Co nsequently , diam f ◦ h ( T ab ) = diam f ◦ h ( T ab \ ∂ T ab ) < 2 − m +1 and d ( f ◦ h ( x ) , f ◦ h ( x ′ )) ≤ 2 · 2 − m +1 < η .  The functions ¯ f h , f ∈ F , comp ose a contin uous function ¯ F h = ( ¯ f h ) f ∈F : ε ¯ K \ ε 2 K → ¯ R F that extends the comp osition F ◦ h : ε ¯ K \ ε 2 ¯ K → ¯ R F . Let ¯ F ∂ = ¯ F h | ε 2 K ∂ be the restriction of ¯ F h onto the b ounda ry square ε 2 K ∂ . Also let ¯ h : ε ¯ K \ ε 2 K → ε ¯ K b e the contin uous extension of the homeomorphism h and observe that ¯ h − 1 ( 0 ) = ε 2 K ∂ . The following cla im completes the pro o f of the statement (4) of Theorem 3.1. Claim 3.3. ¯ F ( 0 ) = ¯ F ∂ ( ε 2 K ∂ ) is a Pe ano c ontinuum. Pr o of. First, we are go ing to show that ¯ F ( 0 ) = ¯ F ∂ ( ε 2 K ∂ ). Let y ∈ ¯ F ( 0 ). T his means that there exists a sequence { x n } n ∈ ω ⊂ εK ◦ such that ( 0 , y ) = lim n →∞ ( x n , F ( x n )). By the compactness of ε ¯ K \ ε 2 K , the sequence { h − 1 ( x n ) } n ∈ ω ⊂ ε ¯ K \ ε 2 ¯ K contains a s ubsequence { h − 1 ( x n k ) } k ∈ ω that conv erges to s ome p oint z ∈ ε 2 K ∂ = ¯ h − 1 ( 0 ). The contin uity of the map ¯ F h guarantees that y = lim k →∞ F ( x n k ) = lim k →∞ ¯ F h ( h − 1 ( x n k )) = ¯ F h ( z ) ∈ ¯ F ∂ ( ε 2 K ∂ ) . The conv er se inclusion ¯ F ∂ ( ε 2 K ∂ ) ⊂ ¯ F ( 0 ) is ob vious. T he equality ¯ F ( 0 ) = ¯ F ∂ ( ε 2 K ∂ ) a nd the contin uit y of ¯ F ∂ implies that ¯ F ( 0 ) is a Peano contin uum.  Now we pr ov e the statemen t (5) of Theor em 3.1. W e re call that B ′ 0 = B 0 = A 0 ∩ εK ∂ . The co nditions (13), (14) imply the following: Claim 3. 4. F or every f ∈ F and neighb or p oints a, b of the set 1 2 B 0 the map ¯ f h | [ a, b ] is monotone and its image lies in one of the se gmen t s: [0 , 1] , [ − 1 , 0] , [1 , ∞ ] , [ ∞ , − 1 ] c omp osing the pr oje ctive line ¯ R . Claim 3.4 implies: Claim 3.5. F or any neighb or p oints a, b of the set 1 2 B 0 , the map ¯ F ∂ | [ a, b ] is monotone and ¯ F ∂ ([ a, b ]) either is a singleton or a monotone ar c in ¯ R F . Claim 3 .5 implies that for any neighbor p oints a, b of the set 1 2 B 0 and any p oint y ∈ ¯ F ∂ ([ a, b ]) the preimage ( ¯ F ∂ | [ a, b ]) − 1 ( y ) is either a singleton or an arc. Let Y a,b = { y ∈ ¯ F h ([ a, b ]) : | ( ¯ F ∂ | [ a, b ]) − 1 ( y ) | > 1 } . Since [ a, b ] do es not cont ain uncoun tably man y disjoin t ar cs, the set Y a,b is at most countable and so is the se t Y = [ { Y a,b : ( a, b ) ∈ N ( 1 2 B 0 ) } . The deﬁnition o f the s et Y implies: Claim 3.6. F or e ach y ∈ ¯ R F \ Y the pr eimage ¯ F − 1 ∂ ( y ) is ﬁnite. The last statement of The o rem 3.1 is pr ov ed in the follo wing lemma, which is the most diﬃcult par t of the pro o f of Theorem 3.1. Lemma 3.7. F or any map g : ¯ F ∂ ( ε 2 K ∂ ) → ε 2 K ∂ the c omp osition g ◦ ¯ F ∂ : ε 2 K ∂ → ε 2 K ∂ is Z 2 -trivial. Pr o of. Since homoto pic maps hav e the s ame degrees, it suﬃces to show that the map g ◦ ¯ F ∂ is homotopic to some Z 2 -trivial map ˜ g : ε 2 K ∂ → ε 2 K ∂ . The construction of such a map ˜ g is rather lo ng and require some preliminary work, in pa rticular, intro ducing s o me nota tio n. W e recall that ρ s tands for the max-metr ic on the plane R 2 , d deno tes the metric of the pr o jectiv e line ¯ R . T he latter metr ic induces the max-metric d F on the F -tor us ¯ R F . By the uniform con tinuit y of the map g there is δ > 0 s uch that for any p oints x, y ∈ ¯ F ( 0 ) = ¯ F ∂ ( ε 2 K ∂ ) with d F ( x, y ) ≤ δ we get ρ ( g ( x ) , g ( y )) < ε . DIMENSION OF GRAPHOIDS OF RA TIONAL VECTOR-FUNCTIONS 9 Find m ∈ N such that 2 − m +1 < δ and consider the δ - net C m in ¯ R (which appea red in the construction of the homeomorphism h ). The de ﬁnitio n of the set C m guarantees that C 0 = { 0 , 1 , − 1 , ∞} ⊂ C m . The set C m induces the disjoint cov er C =  { c } : c ∈ C m } ∪ { ] a , b [: ( a, b ) ∈ N ( C m ) } of the pro jective line ¯ R . It follows that the closure ¯ C of e ach se t C ∈ C is a connected subset that lies in one of the interv als: [0 , 1 ], [0 , − 1], [1 , ∞ ], [ ∞ , − 1]. So, w e can e ndow each segment ¯ C with the linear order inherited fro m the extended real line [ − ∞ , ∞ ]. Now for each set C ∈ C consider the rational homeo morphism µ C : ¯ R → ¯ R deﬁned by formula: µ C ( x ) =          x if C ⊂ [0 , 1]; x + 1 if C ⊂ [ − 1 , 0); 1 − x − 1 if C ⊂ (1 , ∞ ]; − x − 1 if C ⊂ ( ∞ , − 1) . Observe tha t µ C ( ¯ C ) ⊂ [0 , 1] and the restriction µ C | ¯ C : ¯ C → [0 , 1] is strictly increa sing (with respect to the linear order o n ¯ C inherited from [ − ∞ , + ∞ ]). The cover C induces the disjoint cov e r Π C F = n Y f ∈F C f : ( C f ) f ∈F ∈ C F o of the F -to rus ¯ R F by cub es of v arious dimensions. Since C m is a δ -ne t, for each cube C ∈ Π C F its closur e ¯ C has diameter < δ (with res pec t to the metric d F ). Consequently , the image g ( ¯ C ∩ ¯ F ( 0 )) has ρ -diameter diam g ( ¯ C ∩ ¯ F ( 0 )) < ε and hence lies in so me top ologica l arc I C ⊂ ε 2 K ∂ . F or each cub e C = Q f ∈F C f ∈ Π C F consider the embedding µ C : Y f ∈F ¯ C f → [0 , 1] F , µ C : ( x f ) f ∈F 7→ ( µ C f ( x f )) f ∈F . And now the ﬁnal p or tion of deﬁnitions and notatio ns whic h should b e digested b efore the star t of the pro o f of Lemma 3.7. A pa ir ( a, b ) o f dis tinct po int s of ε 2 K ∂ is called F - admissible if [ a, b ] ⊂ ε 2 K ∂ and for every f ∈ F there is a (unique) set C f a,b ∈ C such that ¯ f h  ] a, b [  ⊂ C f a,b and the restriction ¯ f h | [ a, b ] : [ a, b ] → ¯ C f a,b is monotone. It is clear that the pro duct C a,b = Y f ∈F C f a,b ⊂ ¯ R F is an element of the cover Π C F of ¯ R F . F or each pa ir ( a, b ) of F -admissible p oints o f ε 2 K ∂ consider the sets F < a,b = { f ∈ F : ¯ f h ( a ) < ¯ f h ( b ) } , F > a,b = { f ∈ F : ¯ f h ( a ) > ¯ f h ( b ) } , F = a,b = { f ∈ F : ¯ f h ( a ) = ¯ f h ( b ) } , F 6 = a,b = F \ F = a,b = F < a,b ∪ F > a,b . Two F -admissible o rdered pair s ( a, b ), ( a ′ , b ′ ) o f neig hbor p oints o f the set ε 2 K ∂ are called F - c oher en t if F < a,b = F < a ′ ,b ′ , F = a,b = F = a ′ ,b ′ , F > a,b = F > a ′ ,b ′ , and C a,b = C a ′ ,b ′ . Two unordered pairs { a, b } , { a ′ , b ′ } of ne ig hbor points o f the set B a re ca lled F -c oher ent if the o rdered pair ( a, b ) is F - coherent either to ( a ′ , b ′ ) o r to ( b ′ , a ′ ). It is e a sy to c heck that the F -coherence rela tion is an equiv alence relation on the family of F -admissible (un)ordered pairs of p oints of ε 2 K ∂ . Claim 3.8. Ther e is a ﬁn ite subset D ⊂ ε 2 K ∂ such that: (1) 1 2 B 0 ⊂ D . (2) Any p air ( a, b ) of neighb or p oints of D is F -admissible. (3) Two u nor der e d p airs { a, b } , { a ′ , b ′ } of neighb or p oints of D ar e F -c oher ent pr ovide d ¯ F ∂  ] a, b [  ∩ ¯ F ∂  ] a ′ , b ′ [  6 = ∅ . 10 T ARAS BANAKH AND OLES POTY A TYNYK Pr o of. F o r every neighbo r p oints a, b of the set 1 2 B 0 , consider the disjoint cov er D a,b = { [ a, b ] ∩ ¯ F − 1 h ( C ) : C ∈ Π C F } of the aﬃne interv al [ a, b ] by conv ex subsets o f [ a, b ]. The conv exit y of the sets o f the cov er D a,b follows from the monotonicity of the maps ¯ f h | [ a, b ], f ∈ F . F or every set D ∈ D a,b by ∂ D we denote the bounda r y o f D in ε 2 K ∂ . Since D is conv ex, | ∂ D | ≤ 2. Let D 0 = 1 2 B 0 ∪ [  ∂ D : D ∈ D a,b , ( a, b ) ∈ N ( 1 2 B 0 )  . It is easy to see that any t wo neighbor points of the set D 0 are F -admissible. Now let us consider an increasing sequence ( D n ) n ∈ ω of ﬁnite subsets of ε 2 K ∂ deﬁned b y the following recur sive pro cedure. Assume that for some n ∈ ω a ﬁnite subset D n (containing the s et D 0 ) has b een constructed. F o r any t wo unordered pairs { a, a ′ } and { b, b ′ } of neighbor p oints of the s et D n consider the convex hull D b,b ′ a,a ′ of the set ( ¯ F ∂ | [ a, a ′ ]) − 1 ( ¯ F ∂ ([ b, b ′ ])) ⊂ [ a, a ′ ] in the aﬃne segment [ a, a ′ ] and its bounda r y ∂ D b,b ′ a,a ′ in ε 2 K ∂ , which cons is ts of a t most tw o p o in ts. Claim 3.9. ¯ F ∂ ( ∂ D b,b ′ a,a ′ ) = ¯ F ∂ ( ∂ D a,a ′ b,b ′ ) . If the interse ction ¯ F ∂ ([ a, a ′ ]) ∩ ¯ F ∂ ([ b, b ′ ]) c ontains mor e than one p oint, then doubletons ∂ D b,b ′ a,a ′ and ∂ D a,a ′ b,b ′ ar e F - c oher ent. Pr o of. The claim is trivial if the in tersection Z = ¯ F ∂ ([ a, a ′ ]) ∩ ¯ F ∂ ([ b, b ′ ]) contains at most one p oint. So, assume that this intersection co nt ains mor e than one p oint. It fo llows that Z ⊂ C a,a ′ = C b,b ′ and hence for each f ∈ F the sets C f a,a ′ and C f b,b ′ coincide and ca r ry the same linear o r der. Cho ose tw o p oints y = ( y f ) f ∈F and y ′ = ( y ′ f ) f ∈F in Z ⊂ ¯ R F for which the s et F 6 = y ,y ′ = { f ∈ F : y f 6 = y ′ f } has maximal p o ssible cardinality . It is clear that F 6 = y ,y ′ = F < y ,y ′ ∪ F > y ,y ′ , wher e F < y ,y ′ = { f ∈ F : y f < y ′ f } a nd F > y ,y ′ = { f ∈ F : y f > y ′ f } . Cho ose tw o p oints x a , x ′ a ∈ [ a, a ′ ] such that y = ¯ F ∂ ( x a ) and y ′ = ¯ F ∂ ( x ′ a ). Exchanging the p oints a, a ′ by their places, if necessa ry , w e can assume that the in terv als [ a, x a ] and [ x ′ a , a ′ ] ha ve empty intersection. Choo se unique points z a ∈ [ a, x a ] and z ′ a ∈ [ x ′ a , a ′ ] such that { z a , z ′ a } = ∂ D b,b ′ a,a ′ . Since [ x a , x ′ a ] ⊂ [ z a , z ′ a ], the monotonicity of the functions ¯ f h | [ a, a ′ ] implies that F 6 = y ,y ′ ⊂ F 6 = z a ,z ′ a and hence F 6 = y ,y ′ = F 6 = z a ,z ′ a by the maximalit y of F 6 = y ,y ′ . This fact, com bined with the monotonicity of the functions ¯ f h | [ a, a ′ ] and the choice o f the order of the p oints a , a ′ implies that F < z a ,z ′ a = F < y ,y ′ and F > z a ,z ′ a = F > y ,y ′ . Let y a = ¯ F ∂ ( z a ) a nd y ′ a = ¯ F ∂ ( z ′ a ). Now do the same for the s egment [ b, b ′ ]: choo se tw o p oints x b , x ′ b ∈ [ b , b ′ ] such that y = ¯ F ∂ ( x b ) a nd y ′ ∈ ¯ F ∂ ( x ′ b ). Replacing the p oints b, b ′ by their plac e s, if necess ary , we can assume that the interv als [ b , x b ] and [ x ′ b , b ′ ] have no common po ints. Cho ose unique points z b ∈ [ b, x b ] and z ′ b ∈ [ x ′ b , b ′ ] suc h that { z b , z ′ b } = ∂ D a,a ′ b,b ′ and let y b = ¯ F ∂ ( z b ) and y ′ b = ¯ F ∂ ( z ′ b ). It follows that F > z b ,z ′ b = F > y ,y ′ and F < z b ,z ′ b = F < y ,y ′ . W e claim that y a = y b and y ′ a = y ′ b . Assume ﬁrst that y a 6 = y b . Find a point u a ∈ [ z a , z ′ a ] such that ¯ F ∂ ( u a ) = y b and a p oints u b ∈ [ z b , z ′ b ] such that ¯ F ∂ ( u b ) = y a . Since y a 6 = y b , there is a function f ∈ F such that pr f ( y a ) 6 = pr f ( y b ) where pr f : ¯ R F → ¯ R denotes the pro jection onto the f -th coor dinate. On the set pr f ( Z ) co nsider a linear order inherited from the set C f a,a ′ = C f b,b ′ . W e lo se no generality assuming that pr f ( y a ) < pr f ( y b ). Then by the monotonicity of the function ¯ f h | [ a, a ′ ], w e g e t ¯ f h ( z a ) = pr f ( y a ) < pr f ( y b ) = ¯ f h ( u a ) ≤ ¯ f h ( z ′ a ) and hence f ∈ F < z a ,z ′ a = F < y ,y ′ . On the other hand, the monotonicity of the function ¯ f h | [ b, b ′ ] implies ¯ f h ( z b ) = pr f ( y b ) > pr f ( y a ) = ¯ f h ( u b ) ≥ ¯ f h ( z ′ b ) , and f ∈ F > z b ,z ′ b = F > y ,y ′ . This is a desired contradiction that prov es the equality y a = y b . By analogy w e can prov e the equality y ′ a = y ′ b . Now w e se e that the equality ¯ F ∂ ( ∂ D b,b ′ a,a ′ ) = { y a , y ′ a } = { y b , y ′ b } = ¯ F ∂ ( ∂ D a,a ′ b,b ′ ) implies that the doubletons ∂ D b,b ′ a,a ′ and ∂ D a,a ′ b,b ′ are F -coherent.  Deﬁne the set D n +1 as the union D n +1 = D n ∪ [ { ∂ D b,b ′ a,a ′ : ( a, a ′ ) , ( b, b ′ ) ∈ N ( D n ) } . F or every n ∈ ω consider the function p n : N ( D n +1 ) → N ( D n ) a ssigning to each ordered pair ( a, a ′ ) of neigh b or po int s of the set D n +1 a unique ordered pair ( b, b ′ ) of neighbor p oints of D n such that [ a, a ′ ] ⊂ [ b, b ′ ] and [ a, b ] ∩ [ a ′ , b ′ ] = ∅ . F or n ≤ m consider the co mpo sition p m n = p n ◦ · · · ◦ p m − 1 : N ( D m ) → N ( D n ) . DIMENSION OF GRAPHOIDS OF RA TIONAL VECTOR-FUNCTIONS 11 Claim 3.10. Ther e is n ∈ ω su ch that for any m ≥ n any p air ( a ′ , b ′ ) ∈ N ( D m ) is F -c oher ent to the p air ( a, b ) = p m − 1 ( a ′ , b ′ ) . Pr o of. The pro of of this claim relies on the K ¨ onig Lemma [12, 14.2], whic h says that a tree T is ﬁnite provided each element of T has ﬁnite degr ee and each branch of T is ﬁnite. Let us re call that a tr e e is a partially ordered set (poset) ( T , ≤ ) with the smallest element such tha t for ea ch t ∈ T , the set { s ∈ T : s ≤ t } is w ell-order ed b y the relation ≤ . F or ea ch t ∈ T , the or der type of { s ∈ T : s ≤ t } is called the height of t . The heig ht of T itself is the lea st ordinal greater than the height of each element of T . The de gr e e of an element t ∈ T is the num b er of immediate success o rs of t in T . The ro o t of a tree T is the unique element of height 0. A br anch of a tr e e T is a maximal linearly ordered subset of T . Now consider the tree T = {∅} ∪ S n ∈ ω N ( D n ). The partial order on T is deﬁned as follo ws. Given tw o vertices ( a, a ′ ) ∈ N ( D n ) and ( b, b ′ ) ∈ N ( D m ) of T , we write ( a, a ′ ) ≤ ( b, b ′ ) if n ≤ m and ( a, a ′ ) = p m n ( b, b ′ ). The set ∅ is the ro ot of T and is s maller that any other non-empty element of T . It is clear that ea ch vertex of the tree T has ﬁnite degree. The monoto nicit y of the maps ¯ f h | [ a, a ′ ] for ( a, a ′ ) ∈ T implies the following fact: Claim 3.1 1. F or any two vertic es ( a, a ′ ) ≤ ( b, b ′ ) of the tr e e T we get F = a,a ′ ⊂ F = b,b ′ . Mor e over, the p airs ( a, a ′ ) and ( b, b ′ ) ar e F -c oher ent if and only if F = a,a ′ = F = b,b ′ . Now cons ide r the subtree T ′ ⊂ T cons isting of the r o ot of T and all pa irs ( a, a ′ ) ∈ N ( D n ) ⊂ T that a r e not F -coher ent to some pair ( b, b ′ ) ∈ N ( D n +1 ) ⊂ T with p n ( b, b ′ ) = ( a, a ′ ). Claim 3.11 implies tha t e a ch branch o f the tree T ′ has ﬁnite length ≤ |F | + 1 . By K¨ onig Lemma, the s ubtree T ′ is ﬁnite. Co nsequently , there is n ∈ N such that T ′ ∩ N ( D m ) = ∅ for all m ≥ n − 1. This implies that for every m ≥ n , ea ch pair ( a, a ′ ) ∈ N ( D m ) is F -coherent to the pa ir ( b, b ′ ) = p m − 1 ( a, a ′ ). This completes the pr o of of Claim 3.10.  Let D = D n +1 where the num b er n is taken from Claim 3 .10. It is clear that the set D satisﬁes the conditions (1) and (2) of Claim 3 .8. The condition (3) is veriﬁed in the following cla im. Claim 3.1 2. Two unor der e d p airs { a, a ′ } , { b, b ′ } ∈ N { D } of n eighb or p oints of the set D = D n +1 ar e F -c oher ent if ¯ F h  ] a, a ′ [  ∩ ¯ F h  ] b, b ′ [  6 = ∅ . Pr o of. W e shall consider t wo cases (and several sub cases ). 1. The intersection ¯ F h  ] a, a ′ [  ∩ ¯ F h  ] b, b ′ [  contains more than one p o int. B y Claim 3.9, the doubletons ∂ D b,b ′ a,a ′ and ∂ D a,a ′ b,b ′ are F -coherent. T ake a pair o f neigh b or p oints ( a n +2 , a ′ n +2 ) ∈ N ( D n +2 ) s uch that ] a n +2 , a ′ n +2 [ ⊂ D b,b ′ a,a ′ and p n +1 ( a n +2 , a ′ n +2 ) = ( a, a ′ ). The choice of the num be r n gua r antees that the pairs ( a n +2 , a ′ n +2 ) and ( a, a ′ ) are F -coher ent . T aking in to acco unt that [ a n +2 , a ′ n +2 ] ⊂ conv( ∂ D b,b ′ a,a ′ ) ⊂ [ a, a ′ ], we conclude that the pair { a, a ′ } is F -coher ent to the doubleton ∂ D b,b ′ a,a ′ . By ana logy w e can prov e that the pair { b, b ′ } is F -co herent to the doubleton ∂ D a,a ′ b,b ′ . Now we see that the F -coher ence o f the doubletons ∂ D b,b ′ a,a ′ and ∂ D a,a ′ b,b ′ implies the F -coherence of the pairs { a, a ′ } and { b , b ′ } . 2. The in tersection ¯ F h  ] a, a ′ [  ∩ ¯ F h  ] b, b ′ [  is a singleton cont aining a unique po int y . If both se ts ∂ D b,b ′ a,a ′ and ∂ D a,a ′ b,b ′ are doubletons, then we can use the equalit y ¯ F ∂ ( ∂ D b,b ′ a,a ′ ) = { y } = ¯ F ∂ ( ∂ D a,a ′ b,b ′ ), which implies that the doubletons ∂ D b,b ′ a,a ′ and ∂ D a,a ′ b,b ′ are F -coherent and pro ce e d as in the preceding case. 2a. Now assume tha t ∂ D b,b ′ a,a ′ is a singleton. Let ( a n , a ′ n ) = p n +1 ( a, a ′ ) a nd ( b n , b ′ n ) = p n +1 ( b, b ′ ). The choice of the nu mber n g uarantees that the pair ( a n , a ′ n ) is F -coher ent to ( a, a ′ ) a nd ( b n , b ′ n ) is F -coher ent to ( b, b ′ ). It follo ws that the intersection ¯ F h  ] a n , a ′ n [  ∩ ¯ F h  ] b n , b ′ n [  ⊃ ¯ F h  ] a, a ′ [  ∩ ¯ F h  ] b, b ′ [  = { y } is not empt y . If this in tersection is a singleton, then the con vex set D b n ,b ′ n a n ,a ′ n also is a singleton (in the o ppo site ca se, the set D b,b ′ a,a ′ = D b n ,b ′ n a n ,a ′ n ∩ ] a, a ′ [ cannot b e a singleton). In this case the singleton ∂ D b n ,b ′ n a n ,a ′ n = ∂ D b,b ′ a,a ′ belo ngs to the set D = D n +1 and is disjoint with the ope n in terv al ] a, a ′ [, which contradicts y ∈ ¯ F ∂  ] a, a ′ [  . This pr ov es that the int ersectio n ¯ F h  ] a n , a ′ n [  ∩ ¯ F h  ] b n , b ′ n [  is not a singleton. Proceeding as in the case 1, we ca n show that the pairs { a n , a ′ n } and { b n , b ′ n } are F -coherent and so are the pairs { a, a ′ } and { b, b ′ } (which are F -coherent to the pairs { a n , a ′ n } and { b n , b ′ n } , resp ectively). 2b. In case ∂ D a,a ′ b,b ′ is a singleto n, we can pro ceed b y a nalogy with the case 2a .   Now we are ready to prove that the compo sition g ◦ ¯ F ∂ is homotopic to some a map ˜ g : ε 2 K ∂ → ε 2 K ∂ of even degree. It s uﬃces to deﬁne ˜ g on each segment [ a, b ] connecting tw o ne ig hbor points of the set D . 12 T ARAS BANAKH AND OLES POTY A TYNYK W e recall that b y N { D } we deno te the family of unordered pairs of neig hbor points of the set D . The family N { D } decomp oses into pa irwise disjoint eq uiv alence classes consis ting of F -coher ent pa irs. Denote by ↔ N { D } the family of these equiv alence classes. F or each equiv ale nce clas s E ∈ ↔ N { D } let ¯ E = [  [ a, b ] : { a, b } ∈ E  and ∂ E = [  { a, b } : { a, b } ∈ E  . It is clear that ε 2 K ∂ = S { ¯ E : E ∈ ↔ N { D } } . F or each equiv alence class E ∈ ↔ N { D } we are going to construct a sp eciﬁc map ˜ g E : ¯ E → ε 2 K ∂ such that ˜ g E | ∂ E = g ◦ ¯ F ∂ | ∂ E and ˜ g E is homotopic to g ◦ ¯ F ∂ | ¯ E . This map ˜ g E will hav e a s pe c iﬁc algebra ic struc tur e which will help us to ev aluate the deg ree of the uniﬁed map ˜ g = S  ˜ g E : E ∈ ↔ N { D }  . So, ﬁx a n equiv a lence class E ∈ ↔ N { D } . Since a ny t wo unordere d pairs fro m E are F - coherent, w e ca n choo s e a function γ : E → D 2 assigning to each unordered pair { a, b } ∈ E o ne of ordered pairs ( a, b ) or ( b, a ) so that for any uno r dered pair s { a, b } , { a ′ , b ′ } ∈ ↔ N { D } the orde r ed pairs γ ( { a, b } ) and γ ( { a ′ , b ′ } ) are F - coherent. Let ~ E = γ ( E ) ⊂ N ( D ) and → N ( D ) = { ~ E : E ∈ ↔ N { D } } . The F -c o herence of a ny t wo pairs ( a, b ) , ( a ′ , b ′ ) ∈ ~ E implies that F < a,b = F < a ′ ,b ′ , F = a,b = F = a ′ ,b ′ , F > a,b = F > a ′ ,b ′ , and C a,b = C a ′ ,b ′ . So, we can put F < E = F < a,b , F = E = F = a,b , F > E = F > a,b , C f E = C f a,b for f ∈ F , C E = C a,b = Y f ∈F C f a,b , I E = I C E and µ E = µ C E : ¯ C E → [0 , 1] F where ( a, b ) ∈ ~ E is any pair. W e rec a ll that I C E is an ar c in ε 2 K ∂ that con tains the set g ( ¯ C E ∩ ¯ F ( 0 )). F or every f ∈ F consider the num ber ε f ∈ {− 1 , 0 , 1 } deﬁned by the formula ε f =      1 if f ∈ F < E 0 if f ∈ F = E − 1 if f ∈ F > E . T aking into acc o unt that the subset µ E ◦ ¯ F ∂ ( ∂ E ) ⊂ µ E ( C E ) ⊂ [0 , 1] F is ﬁnite, it is easy to ﬁnd a s equence of p o sitive real n umbers ( α f ) f ∈F such that the linear map λ E : [0 , 1] F → R , λ E : ( x f ) f ∈ F 7→ X f ∈ F ε f α f x f is injective o n the set µ E ◦ F ∂ ( ∂ E ). Claim 3.13. F or e ach p air { a, b } ∈ E the map λ E ◦ µ E is inje ct ive on the set ¯ F ∂ ([ a, b ]) . Pr o of. Assume that λ E ◦ µ E ( y ) = λ E ◦ µ E ( y ′ ) for some p oints y = ( y f ) f ∈F and y ′ = ( y ′ f ) f ∈F in ¯ F ∂ ([ a, b ]). Choose t wo p oints x, x ′ ∈ [ a, b ] such that y = ¯ F ∂ ( x ) and y ′ = ¯ F ∂ ( x ′ ). On the in terv a l [ a, b ] we consider the linear order such that a < b . W e lose no gener a lity assuming tha t x < x ′ with resp ect to this or der. The mono tonicity o f the maps ¯ f h , f ∈ F , imply that • ¯ f h ( x ) ≤ ¯ f h ( x ′ ) fo r each f ∈ F < E ; • ¯ f h ( x ) ≥ ¯ f h ( x ′ ) fo r each f ∈ F > E ; • ¯ f h ( x ) = ¯ f h ( x ′ ) fo r each f ∈ F = E . T aking in to account these inequalities, the increasing proper ty of the maps µ C and the choice of the n umbers ε f , f ∈ F , we conclude that ε f · µ C f E ( ¯ f h ( x )) ≤ ε f · µ C f E ( ¯ f h ( x ′ )) for all f ∈ F . Consequently , λ E ◦ µ E ( y ) = λ E ◦ µ E ◦ ¯ F ∂ ( x ) = X f ∈F α f ε f · µ C f E ( ¯ f h ( x )) ≤ ≤ X f ∈F α f ε f · µ C f E ( ¯ f h ( x ′ )) = λ E ◦ µ E ◦ ¯ F ∂ ( x ′ ) = λ E ◦ µ E ( y ′ ) . T aking into ac c o unt that λ E ◦ µ E ( y ) = λ E ◦ µ E ( y ′ ), we conc lude that y f = ¯ f h ( x ) = ¯ f h ( x ′ ) = y ′ f for all f ∈ F 6 = E . F or each f ∈ F = E the function ¯ f h | [ a, b ] is constant and hence y f = ¯ f h ( x ) = ¯ f h ( x ′ ) = y ′ f . Co ns equently , y = y ′ , whic h means that the map λ E ◦ µ E is injectiv e o n ¯ F ∂ | [ a, b ].  DIMENSION OF GRAPHOIDS OF RA TIONAL VECTOR-FUNCTIONS 13 Let us r e call that by Y we denote the countable set of p o ints y ∈ ¯ R F with inﬁnite pr eimage ¯ F − 1 ∂ ( y ). The following claim plays a crucial role in the pro of o f Lemma 3.7. Claim 3.14. F or any y ∈ R \ λ E ◦ µ E ( ¯ F ∂ ( ∂ E ) ∪ Y ) the pr eimage D y = ( λ E ◦ µ E ◦ ¯ F h | ¯ E ) − 1 ( y ) is ﬁnite and c ontains even numb er of p oints. Pr o of. If D y is empt y , then there is no thing to pr ov e. So, w e assume that the set D y is not empt y . Claims 3.13 and 3.4 imply that for any pair { a, b } ∈ E the in tersection D y ∩ [ a, b ] c o ntains at most one p oint. This point belo ngs to the interior ] a, b [ as y / ∈ λ E ◦ µ E ◦ ¯ F ∂ ( ∂ E ). Since µ E ◦ ¯ F ∂ ( ¯ E ) ⊂ µ E ( ¯ C E ) ⊂ [0 , 1 ] F ⊂ R F , the preimage W = ( µ E ◦ ¯ F h ) − 1 ( R F ) is an op en neighbor ho od of the set ¯ E in ε ¯ K \ ε 2 K and η E = λ E ◦ µ E ◦ ¯ F h | W : W → R is a well-deﬁned contin uous ma p. Observe that the formula R ( u, v ) = λ E ◦ µ E ◦ F ( u , v ) = X f ∈F α f ε f · µ C f E ( f ( u, v )) determines a ra tional function on R 2 . W e claim that this rational function is not co nstant. Indeed, the set D y , b eing not empt y , contains some point x which lies in the int erv al [ a, b ] for some pair { a, b } ∈ E . Since η E ( x ) = y 6 = η E ( a ), we can ﬁnd t > 1 such that ta, tx ∈ W and η E ( ta ) 6 = η E ( tx ). Now we see that R ( h ( ta )) = η E ( ta ) 6 = η E ( tx ) = R ( h ( tx )) , which means that the rationa l function R is not constant. So , it is legal to co nsider the plane algebra ic curve A y = R − 1 ( y ). The c hoice of the p oint y guara ntees that y / ∈ η E ( ∂ E ). Since W is an op en neighborho o d of ¯ E in ε ¯ K \ ε 2 K and lim m →∞ δ m = ε 2 , there is a n umber m ∈ N so large that: • [1 , 2 δ m ε ] · ¯ E ⊂ W , • y / ∈ η E ([1 , 2 δ m ε ] · ∂ E ), and • the num b er ε m is A y -small. Let A y denote the family o f connected comp onents of the set A y ∩ ε m ¯ K ◦ . Since ε m is A y -small each s et A ∈ A y is an ε m -elementary branch of the algebra ic curve A y . By A ∗ ∈ A y we shall denote its conjugate ε m -branch. F or each ε m -elementary br anch A ∈ A y the preimag e B = h − 1 ( A ) is a curve in the “square annulus” δ m ¯ K \ ε 2 K . Let B ∗ = h − 1 ( A ∗ ) be the “c o njugate” curve to B = h − 1 ( A ). Now consider the family B y = { h − 1 ( A ) : A ∈ A y } that decomp oses in to pa irs of conjugate curves. Claim 3.15. F or any curve B ∈ B y and its closur e ¯ B in R 2 the interse ction ¯ B ∩ ε 2 K ∂ is a n on-empty c onvex subset of ε 2 K ∂ such that ¯ B ∩ ¯ E ⊂ D y . If the interse ction ¯ B ∩ ¯ E is not empty, then it is a singleton. Pr o of. W e lose no g enerality assuming that the ε n -elementary curve A = h ( B ) ∈ A y is an east ε m -elementary curve. The cons tr uction of the homeomorphism h guara n tees that the curve B coincides with the graph of some contin uous function deﬁned on the in terv al ( ε 2 , δ m ]. This implies that the in tersection ¯ B ∩ ε 2 K ∂ is a non-empt y clo sed co nv ex subset that lies in the east side { ε 2 } × [ − ε 2 , ε 2 ] of the squar e ε 2 K ∂ . T a king in to account that η E ( W ∩ B ) = { y } , we conclude that η E ( W ∩ ¯ B ) = { y } a nd hence ¯ B ∩ ¯ E ⊂ D y . If ¯ B ∩ ¯ E is not empt y , then it is a sing leton b ecause ¯ B ∩ ε 2 K ∂ is conv ex, do es not mee t the set ∂ E , and the int ersectio n ¯ B ∩ ¯ E ⊂ D y is ﬁnite.  Let B E y = { B ∈ B y : ¯ B ∩ ¯ E 6 = ∅} a nd A E y = { h ( B ) : B ∈ B E y } . F or every B ∈ B E y let π ( B ) be the unique po int of the intersection ¯ B ∩ ¯ E ⊂ D y . The following cla im completes the pro o f of Claim 3 .14 showing that the set | D y | = |B E y | has even cardina lit y .  Claim 3.16. (1) F or any p air { a, b } ∈ E and t ∈ (1 , 2 δ m ε ] the se gment [ ta, tb ] me ets at most one set B ∈ B E y . (2) The fun ction π |B E y : B E y → D y is bije ctive. (3) F or e ach curve B ∈ B E y its c onjugate curve B ∗ b elongs to B E y , so t he c ar dinality |B E y | is even. Pr o of. 1. Ass ume that for some pair { a, b } ∈ E and some t ∈ (1 , 2 δ m ε ] the segment [ ta, tb ] meets t wo distinct curves B , B ′ ∈ B E y at some p o int s u , u ′ , r esp ectively . W e lose no generality a s suming the points u, u ′ are or dered so that [ ta, u ] ∩ [ u ′ , tb ] = ∅ . Since D ⊃ 1 2 B 0 there are tw o neighbor p oints a 0 , b 0 ∈ 1 2 B 0 such that [ a, b ] ⊂ [ a 0 , b 0 ] and [ a 0 , a ] ∩ [ b, b 0 ] = ∅ . 14 T ARAS BANAKH AND OLES POTY A TYNYK Now consider the p oints h ( u ), h ( u ′ ) ∈ A y ∩ ( t − 1) ε ¯ K ◦ and observe that [ h ( u ) , h ( u ′ )] ⊂ [ h ( ta ) , h ( tb )] ⊂ [ h ( ta 0 ) , h ( tb 0 )]. The prop erty (13 ) of the set B 0 guarantees that for every function f ∈ F the r e s triction f | [ h ( ta 0 ) , h ( tb 0 )] either is constant or is injective. In particular , for each function f ∈ F 6 = a,b the restriction f | [ h ( u ) , h ( u ′ )] is injective. Then the choice of the sign ε f , guarantees that ε f f ( h ( u )) < ε f f ( h ( u ′ )) and then y = λ E ◦ µ E ◦ F ( h ( u )) < λ E ◦ µ E ◦ F ( h ( u ′ )) = y , which is the desired contradiction. 2. First w e c heck that the function π |B E y is injective. Ass ume that π ( B ) = π ( B ′ ) for t wo distinct curves B , B ′ ∈ B E y . Let x ∈ π ( B ) = π ( B ′ ) ∈ ¯ E ∩ D y and ﬁnd an o rdered pa ir ( a, b ) ∈ ~ E such that x ∈ ] a, b [. The connectednes s of the curves B and B ′ implies that for some t ∈ (1 , 2 δ m ] the segment [ ta, tb ] in tersects both cur ves B and B ′ which is forbidden by Cla im 3.1 6(1). Now we prove that the function π |B E y is sur jective. Fix any point x ∈ D y and ﬁnd an or der ed pair ( a, b ) ∈ ~ E such that x ∈ ] a, b [. Since y / ∈ η E ( { a, b } ), we conclude that ¯ F h ( a ) 6 = ¯ F h ( x ) 6 = ¯ F h ( b ). Then the choice of the s igns ε f , f ∈ F , guara ntees that η E ( a ) < η E ( x ) = y < η E ( b ). Cho o s e a num b er t ∈ (1 , 2 δ m ε ] suc h that η E ( ta ) < y < η E ( tb ) . It follows that the p oint y belong s to η E ([ ta, tb ]) and the segment [ ta, tb ] meets the preimage B = h − 1 ( A ) of some ε m -branch A of the algebraic curve A y . T aking into acco unt that A is an ε m -elementary curve and the interv als [ a, ta ] and [ b, tb ] do no t in tersect B , we conclude that the curve B has a limit p o int π ( B ) in the singleton [ a, b ] ∩ D y = { x } . 3. T ake any cur ve B ∈ B E y and consider its conjugate curve B ∗ . Cho ose an y p o int x ∗ ∈ ¯ B ∩ ε 2 K ∂ . F or the p oints x = π ( B ) and x ∗ ﬁnd pairs { a, b } , { a ∗ , b ∗ } ∈ N { D } such that x ∈ [ a, b ] and x ∗ ∈ [ a ∗ , b ∗ ]. It follows from B ∈ B E y that the pair { a, b } ∈ E . W e need to show that the pair { a ∗ , b ∗ } als o b elong s to E , whic h mea ns that { a ∗ , b ∗ } is F -coher ent to { a, b } . This will follow from Cla im 3.8(3) as so on a s we chec k that ¯ F h ( x ) = ¯ F h ( x ∗ ). Since the curves B and B ∗ are conjugated, their ima ges A = h ( B ) and A ∗ = h ( B ∗ ) are conjugated ε m -branches of the a lgebraic curve A y . Lemma 2.2 implies that lim A ∋ z → 0 F ( z ) = lim A ∗ ∋ z → 0 F ( z ) . Using the contin uit y o f the function ¯ F h at the p oints x and x ∗ , we see that ¯ F h ( x ) = lim B ∋ u → x ¯ F h ( u ) = lim B ∋ u → x F ◦ h ( u ) = lim A ∋ v → 0 F ( v ) = = lim A ∗ ∋ v → 0 F ( v ) = lim B ∗ ∋ u → x ∗ F ◦ h ( u ) = lim B ∗ ∋ u → x ∗ F h ( u ) = ¯ F h ( x ∗ ) .  Now w e ca n contin ue the pro of of Lemma 3.7. Cho ose a ﬁnite subset N E ⊂ R s uch that • the conv ex hull conv( N E ) o f N E contains the compac t subset η E ( ¯ E ) of R ; • η E ( ∂ E ) ⊂ N E ; • for any neighbor p oints a, b of η E ( ∂ E ) the int erv al ] a, b [ has no n-empty in tersection with the set N E . Fix a contin uous map ϕ E : R → I E ⊂ ε 2 K ∂ such that • ϕ E ◦ η E | ∂ E = g ◦ ¯ F ∂ | ∂ E ; • for any neighbor p oints a, b of the set N E the r estriction ϕ | [ a, b ] : [ a, b ] → I E is injective. Finally , put ˜ g E = ϕ E ◦ η E | ¯ E : ¯ E → I E ⊂ ε 2 K ∂ . T aking into account that ˜ g E ( ¯ E ) ∪ g ◦ ¯ F ∂ ( ¯ E ) ⊂ I E and ˜ g E | ∂ E = g ◦ ¯ F ∂ | ∂ E , we see that the maps ˜ g E , g ◦ ¯ F ∂ | ¯ E : ¯ E → I E are homotopic by a homotopy h E : ¯ E × [0 , 1] → I E such that • h E ( x, 0) = ˜ g E ( x ) , h E ( x, 1) = g ◦ ¯ F ∂ ( x ) for all x ∈ ¯ E and • h E ( x, t ) = ˜ g E ( x ) = g ◦ ¯ F ∂ ( x ) for all x ∈ ∂ E and t ∈ [0 , 1]. The maps ˜ g E , E ∈ ↔ N { D } , comp os e a ma p ˜ g : ε 2 K ∂ → ε 2 K ∂ deﬁned b y ˜ g | ¯ E = ˜ g E for E ∈ ↔ N { D } . Also, the ho motopies h E : ¯ E × [0 , 1 ] → I E ⊂ ε 2 K ∂ , E ∈ ↔ N { D } , c o mp o se a homotopy h : ε 2 K ∂ × [0 , 1] → ε 2 K ∂ betw een the maps ˜ g and g ◦ ¯ F ∂ . The pro o f of Lemma 3 .7 is ﬁnished b y the following cla im. Claim 3.17. The map ˜ g is Z 2 -trivial. DIMENSION OF GRAPHOIDS OF RA TIONAL VECTOR-FUNCTIONS 15 Pr o of. T o sho w that the map ˜ g is Z 2 -trivial, we shall apply Lemma 2.4. Pick any point y 0 ∈ ε 2 K ∂ which do es no t belo ng to the countable set [  ˜ g E ( ∂ E ) ∪ ϕ ( N E ) ∪ ϕ E ◦ λ E ◦ µ E ( Y ∩ ¯ C E ) : E ∈ ↔ N { D }  . F or every equiv a lence class E ∈ ↔ N { D } cons ider the set Y E = ϕ − 1 E ( y 0 ) which is ﬁnite b y the c hoice of the function ϕ E . By Claim 3.14, for every point y ∈ Y E the preimag e D y = ( λ E ◦ µ E ◦ ¯ F h | ¯ E ) − 1 ( y ) is ﬁnite and co n tains even nu mber of p oints. Since y 0 / ∈ ˜ g ( ∂ E ), w e get D y ⊂ ¯ E \ ∂ E . Then the preimag e ˜ g − 1 E ( y 0 ) = S y ∈ Y E D y lies in ¯ E \ ∂ E and co ntains even num ber of po ints. Unifying these preimages , we conclude that the preimag e ˜ g − 1 ( y 0 ) = [ { ˜ g − 1 E ( y 0 ) : E ∈ ↔ N { D } } has even car dinality a nd lies in the s et ε 2 K ∂ \ D . It rema ins to chec k that each p oint x ∈ ˜ g − 1 ( y 0 ) has a ne ig hborho o d U x ⊂ ε 2 K ∂ such that the map ˜ g | U x is mono tone. Find tw o neighbor points a, b of the set D such that x ∈ ] a , b [. Let E b e the F -coherence cla ss of the pair { a, b } . By Claims 3.5 and 3.1 3, the ma p λ E ◦ µ E ◦ F h | [ a, b ] is monotone. Now consider the po int y = λ E ◦ µ E ◦ ¯ F h ( x ) and observe that y / ∈ N E (as ϕ E ( y ) = y 0 / ∈ ϕ E ( N E )). The c hoice of the function ϕ E guarantees that the p oint y has a neighborho o d V y ⊂ R \ N E such that the restr iction ϕ E | V y is injective (and hence monotone). Then the neighbor ho o d U x = ( λ E ◦ µ E ◦ ¯ F h | [ a, b ]) − 1 ( V y ) ha s the desired prop erty: the restriction ˜ g | U x = ˜ g E | U x = ϕ E ◦ λ E ◦ µ E ◦ ¯ F h | U x is monotone.    4. Inverse spectra In the pro of of Theore ms 1.2 and 1.3 we s hall widely use the technique of inv erse s pe c tr a, se e [9], [5]. F ormally sp eaking an inverse sp e ctrum in a category C is a cont rav aria nt functor S : Σ → C from a directed partia lly ordered set Σ to the categor y C . A partially ordere d set (brieﬂy a p os et) Σ is called dir e cte d if for a ny elemen ts α, β ∈ Σ there is an element γ ∈ Σ suc h that γ ≥ α and γ ≥ β . Each po s et Σ can b e identiﬁed with a catego ry whose ob jects are elements of Σ and t wo ob jects α, β ∈ Σ are linked by a single mor phism α → β if a nd only if α ≤ β . An inv erse sp ectrum S : Σ → C can b e written directly as the family { X α , p β α , Σ } consisting of ob jects X α of the category C , indexed by elemen ts α of the p o set Σ, and bo nding mor phisms p β α : X β → X α deﬁned fo r an y indices α ≤ β in Σ, so that for any indices α ≤ β ≤ γ in Σ the following tw o conditions are satisﬁed: • p γ α = p β α ◦ p γ β and • p α α is the identit y morphism o f X α . Inv er se sp e ctra ov er a pose t Σ in a c a tegory C form a categor y C Σ whose morphisms are na tur al transfor mations of functors. In other w ords, for tw o in verse spe c tra S = { X α , p β α , Σ } a nd S ′ = { X ′ α , π β α , Σ } a morphism f : S → S ′ in C Σ is a family of mor phis ms f = { f α : X α → X ′ α } α ∈ Σ of the categor y C such tha t for any indices α ≤ β in Σ the following square is commutativ e: X β p β α   f α / / X ′ β π β α   X α f α / / X ′ α There is a functor ( · ) Σ : C → C Σ assigning to each ob ject X o f C the inv e rse spectrum X Σ = { X α , p β α , Σ } where X α = X and p β α is the identit y map of X for all α ≤ β in Σ. T o ea ch morphism f : X → Y of the category C the functor ( · ) Σ assigns the morphism f Σ = { f α } α ∈ Σ where f α = f for all α ∈ Σ. F or a n inv erse spe c trum S : Σ → C its limit is a pair ( X , p ) consisting of an ob ject X of C and a morphis m p = { p α } α ∈ Σ : X Σ → S in the category C Σ such that for any other pair ( Z , π ) consisting of an ob ject Z of C and a morphism π = { π α } α ∈ Σ : Z Σ → S there is a unique morphism f : Z → X such that π = p ◦ f Σ . This deﬁnition implies that a limit ( X , p ) of S if exists, is unique up to the iso mo rphism. Because of tha t the space X is denoted by lim S a nd called the limit of the inv erse s pec trum S . In this pap er we shall be mainly in terested in inv e r se sp ectra in the category Co mpEpi of compact Hausdo rﬀ spaces and their con tinuous surjective maps . In this case , each in verse sp ectrum S = { X α , p β α , Σ } has a limit ( X , p ) consisting of the closed subspace X =  ( x α ) α ∈ Σ ∈ Y α ∈ Σ X α : p β α ( x β ) = x α for a ll α ≤ β in Σ  16 T ARAS BANAKH AND OLES POTY A TYNYK of the T ychonoﬀ pro duct Q α ∈ Σ X α and the morphism p = ( p α ) α ∈ Σ : X Σ → S where p α : X → X α , p α : ( x α ) α ∈ Σ 7→ x α , is the α -th co or dinate pro jection. Using the tec hnique of in verse sp ectra, we sha ll reduce the problem of inv estigation of the graphoid ¯ Γ( F ) of an arbitrar y family F ⊂ R ( x 1 , . . . , x k ) to studying the graphoids ¯ Γ( α ) of ﬁnite subfamilies α of F . Namely , g iven any family F ⊂ R ( x 1 , . . . , x k ) of rational functions of k -v ariables, co nsider the set Σ = [ F ] <ω of ﬁnite subsets of F , partially o rdered by the inclusion relation ⊂ . Endow ed with this relation, Σ = [ F ] <ω bec omes a directed p os e t. F or any elements α ≤ β of Σ (whic h ar e ﬁnite subsets α ⊂ β of F ) we ca n consider the c o ordinate pr o jection p β α : ¯ Γ( β ) → ¯ Γ( α ). In such a w ay we obtain the inv erse spectrum S F = { ¯ Γ( α ) , p β α , Σ } consisting of gr aphoids of ﬁnite s ubfamilies o f F . F or each ﬁnite subset α ∈ Σ of F the limit pro jection p α : ¯ Γ( F ) → ¯ Γ( α ) co incides with the corres p o nding co ordina te pro jection (we reca ll that ¯ Γ( F ) = ¯ R 2 × ¯ R F while ¯ Γ( α ) ⊂ ¯ R 2 × ¯ R α ). The crucia l fact that follows from the deﬁnition of ¯ Γ( F ) is the following lemma: Lemma 4.1. Th e gr aphoi d ¯ Γ( F ) to gether with the limit pr oje ctions p α : ¯ Γ( F ) → ¯ Γ( α ) , α ∈ Σ , is the limit of the inverse sp e ctrum S F = { ¯ Γ( α ) , p β α , Σ } c onsisting of gr aphoid s ¯ Γ( α ) of ﬁnite subfamilies α ⊂ F . 5. Extension dimension of limit sp aces o f inverse spectra In this sectio n we shall ev alua te the extensio n dimension of limit spaces of inverse sp ectra in the ca tegory CompEpi . This information will b e then used in the pro o fs of Theorems 1.2 and 1.3. W e shall say that a topolo gical space Y is a n absolute n eighb orho o d exten sor for c omp act Hausdorﬀ sp ac es and write Y ∈ ANE ( Comp ) if each map f : A → Y deﬁned on a closed subspa ce A of a co mpa ct Hausdor ﬀ space X has a contin uo us extensio n ¯ f : N ( A ) → Y deﬁned on a neigh b orho o d N ( A ) of A in X . Let us recall that a space X has ex tens ion dimensio n e-dim( X ) ≤ Y if each map f : A → Y deﬁned on a closed subspace A of X ha s a contin uous extensio n ¯ f : X → Y . The following lemma should be known in E xtension Dimension Theo ry but we could not ﬁnd a precise reference . So, we have decide d to give a pro of for conv enience of the reader . Lemma 5.1. L et  X , ( p α )  b e a limit of an inverse sp e ctrum S = { X α , p β α , Σ } in the c ate gory CompE pi . The limit sp ac e X has extension di mension e-dim( X ) ≤ Y for some sp ac e Y ∈ ANE ( Comp ) if and only if fo r any α ∈ Σ and a map f α : A α → Y deﬁne d on a close d subsp ac e A α of the sp ac e X α ther e ar e an index β ≥ α in Σ and a map ¯ f β : X β → Y that extends the map f α ◦ p β α | A β : A β → Y deﬁne d on the close d subset A β = ( p β α ) − 1 ( A α ) of X β . Pr o of. First we prov e the “if ” part of the lemma. T o prov e that X has extension dimension e-dim( X ) ≤ Y , ﬁx a map f : A → Y deﬁned on a closed subset A of the s pace X . Embed the space X into a Tyc ho noﬀ cube [0 , 1] κ . Since Y ∈ ANE ( Comp ), the map f admits a contin uo us extension ˜ f : O ( A ) → Y deﬁned on a n op en neighborho o d O ( A ) of A in [0 , 1] κ . Next, ﬁnd a clos ed neighbor ho o d ˜ A ⊂ O ( A ) of A in [0 , 1] κ . Let U b e a cov e r of [0 , 1] κ by op en co nv ex subsets suc h that S t ( ˜ A, U ) := [  U ∈ U : ˜ A ∩ U 6 = ∅  ⊂ O ( A ) . Claim 5.2. Ther e is an index α ∈ Σ and a c ontinu ou s map r α : ˜ A α → O ( A ) deﬁne d on the close d subset ˜ A α = p α ( ˜ A ) of X α such that the map r α ◦ p α | ˜ A is U -ne ar to the identity emb e dding ˜ A → O ( A ) in t he sense that for e ach x ∈ ˜ A the doubleton { x, r α ◦ p α ( x ) } lies in some set U ∈ U . Pr o of. Let V be an op en cover of [0 , 1] κ that star -reﬁnes the cov er U (the latter means that for every V ∈ V its V -star S t ( V , V ) lies in some set U ∈ U ). It is well-kno wn that the topo logy o f the limit spa ce X of the s pe c trum S is g enerated b y the base consisting o f the s ets p − 1 α ( U α ) wher e α ∈ Σ a nd U α is an op en set in X α . Here p α : X → X α stands for the limit pr o jection. This fact allows us to ﬁnd for every z ∈ ˜ A an index α z ∈ Σ and an op en neig hborho o d W z ⊂ X α z of p α z ( z ) s uch that the neighborho o d p − 1 α z ( W z ) o f z lies in some set V z ∈ V . The op en co ver { p − 1 α z ( W z ) : z ∈ ˜ A } of the compact subset ˜ A admits a ﬁnite sub cov er { p − 1 α z ( W z ) : z ∈ F } . Here F is a suitable ﬁnite subset of ˜ A . Since the index set Σ is dire c ted, there is an index α ∈ Σ such that α ≥ α z for a ll z ∈ F . Changing the s ets W z by ( p α α z ) − 1 ( W z ), we ca n assume that α z = α for all z ∈ F . Then W = { W z : z ∈ F } is an op en cover of the clo sed subset ˜ A α = p α ( ˜ A ) of the compact s pace X α . Let { λ z : ˜ A α → [0 , 1 ] } z ∈ F be a partition of unity , sub ordina ted to the cov er W in the sense that λ − 1 z  ]0 , 1]  ⊂ W z for a ll z ∈ F . Consider the map r α : ˜ A α → [0 , 1] κ deﬁned b y r α ( x ) = X z ∈ F λ z ( x ) · z . W e claim that this map has the required pr op erty: r α ◦ p α | ˜ A is U - near to the iden tity embedding ˜ A → O ( A ). DIMENSION OF GRAPHOIDS OF RA TIONAL VECTOR-FUNCTIONS 17 Given an y point x ∈ ˜ A , co nsider the ﬁnite set E = { z ∈ F : λ z ( p α ( x )) > 0 } . It follows that r α ( p α ( x )) = X z ∈ E λ z ( p α ( x )) · z . Observe that fo r every z ∈ E we get x ∈ p − 1 α ( p α ( x )) ⊂ p − 1 α  λ − 1 z  ]0 , 1]  ⊂ p − 1 α ( W z ) ⊂ V z and hence E ∪ { x } ⊂ [ z ∈ E V z ⊂ S t ( x, V ) ⊂ U for s ome op en con vex set U ∈ U . The conv exity of the set U guara nt ees that this set con tains the following conv e x combination: r α ( p α ( x )) = X z ∈ E λ z ( p α ( x )) · z .  Claim 5.2 implies that r α ( ˜ A α ) = r α ◦ p α ( ˜ A ) ⊂ S t ( ˜ A, U ) ⊂ O ( A ) , so the comp osition f α = ˜ f ◦ r α : ˜ A α → Y is a well-deﬁned con tinuous map. By o ur assumption, ther e is an index β ≥ α and a contin uous map ¯ f β : X β → Y that extends the map f β = f α ◦ p β α | ˜ A β , where ˜ A β = ( p β α ) − 1 ( ˜ A α ) ⊃ p β ( ˜ A ). Observe that ¯ f β ◦ p β | ˜ A = f β ◦ p β | ˜ A = f α ◦ p β α ◦ p β | ˜ A = f α ◦ p α | ˜ A = ˜ f ◦ r α ◦ p α | ˜ A. Using the Urysohn Lemma, ch o ose a contin uous function ξ : X → [0 , 1] s uch that ξ ( A ) ⊂ { 1 } and X \ ˜ A ⊂ ξ − 1 (0). Claim 5.2 implies that for ev ery x ∈ ˜ A the conv ex combination ξ ( x ) x + (1 − ξ ( x )) r α ( p α ( x )) lies in S t ( ˜ A, U ) ⊂ O ( A ) so, the function ¯ f : X → Y , ¯ f : x 7→      ˜ f  ξ ( x ) x + (1 − ξ ( x )) r α ◦ p α ( x )  if x ∈ ˜ A ¯ f β ◦ p β ( x ) if x ∈ ξ − 1 (0), ¯ f β ◦ p β ( x ) = ˜ f ◦ r α ◦ p α ( x ) if x ∈ ξ − 1 (0) ∩ ˜ A , is a well-deﬁned contin uous ex tension o f the map f = ˜ f | A , witnessing that e - dim( X ) ≤ Y . Now we pr ov e the “only if ” par t of the lemma. Assume that e-dim( X ) ≤ Y . Fix an index α ∈ Σ and a contin uous map f α : A α → Y deﬁned on a close d subset A α of X α . W e need to ﬁnd an index β ≥ α in Σ and a contin uous map ¯ f β : X β → Y that ex tends the map f α ◦ p β α | A β deﬁned on the subset A β = ( p β α ) − 1 ( A α ) o f X β . Since Y ∈ ANE ( Comp ), the map f α admits a contin uous extensio n ˜ f α : ˜ A α → Y deﬁned on a closed neigh b orho o d ˜ A α of A α in X α . The n ˜ A = p − 1 α ( ˜ A α ) is a clos ed neig hborho o d of the clos ed s et A = p − 1 α ( A α ) in X . Since e-dim( X ) ≤ Y , the ma p ˜ f α ◦ p α | ˜ A ha s a contin uo us extens io n ¯ f : X → Y . Embed the co mpact Hausdorﬀ spa ce K = ¯ f ( X ) ⊂ Y in a Tyc honoﬀ cub e [0 , 1] κ of a suitable w eight κ . Since Y ∈ ANE ( Comp ), the identit y em bedding K → Y admits a cont inuous extens io n ψ : O ( K ) → Y deﬁned on an o p en neighborho o d O ( K ) of K in [0 , 1] κ . Let U be a cov er of [0 , 1] κ by open con vex subsets suc h that S t ( K, U ) ⊂ O ( K ). Repe a ting the arg umen t of Claim 5 .2, we can ﬁnd a n index β ≥ α in Σ a nd a contin uo us map f β : X β → [0 , 1] κ such that the comp os ition f β ◦ p β is U - near to the map ¯ f : X → K ⊂ [0 , 1] κ . Consider the closed neighborho o d ˜ A β = ( p β α ) − 1 ( ˜ A α ) ⊃ p β ( ˜ A ) of the set A β = ( p β α ) − 1 ( A α ) in the space X β . Using the Ur ysohn L emma, choo se a contin uous function ξ : X β → [0 , 1] suc h that A β ⊂ ξ − 1 (1) and X β \ ˜ A β ⊂ ξ − 1 (0). Given an y point y ∈ ˜ A β , choose a p oint x ∈ p − 1 β ( y ) ⊂ ˜ A (which exists by the surjectivity of the limit pro jection p β ), a nd obser ve tha t { ˜ f α ◦ p β α ( y ) , f β ( y ) } = { ˜ f α ◦ p α ( x ) , f β ◦ p β ( x ) } = { ¯ f ( x ) , f β ◦ p β ( x ) } ⊂ U ⊂ S t ( K, U ) ⊂ O ( K ) for some set U ∈ U accor ding to the c hoice of the map f β . The n the conv ex combination ξ ( x ) ˜ f α ( p β α ( x ))+ (1 − ξ ( x )) f β ( x ) also b elong s to U ⊂ O ( K ), which implies that the map ¯ f β : X β → Y , ¯ f β ( x ) =      ψ  ξ ( x ) ˜ f α ◦ p β α ( x ) + (1 − ξ ( x )) f β ( x )  if x ∈ ˜ A β ψ ( f β ( x )) if x ∈ ξ − 1 (0) ψ ( f β ( x )) = ψ ◦ ˜ f α ◦ p β α ( x )) if x ∈ ˜ A β ∩ ξ − 1 (0) is a well-deﬁned contin uous ex tension o f the map ψ ◦ ˜ f α ◦ p β α | A β = f α ◦ p β α | A β .  Lemma 5.1 implies the following known fact on preserv a tion of extension dimensio n by inv erse limits. 18 T ARAS BANAKH AND OLES POTY A TYNYK Corollary 5.3. L et  X , ( p α )  b e a limit of an inverse sp e ctrum S = { X α , p β α , Σ } in the c ate gory CompEpi . The limit sp ac e X has ex tension dimension e-dim( X ) ≤ Y for s ome sp ac e Y ∈ ANE ( Comp ) pr ovide d that e-dim( X α ) ≤ Y fo r al l α ∈ Σ . By [10, 3 .2.9], a compact Ha us dorﬀ space X has cov er ing dimens ion dim X ≤ n if and only if e-dim( X ) ≤ S n where S n denotes the n -dimensional sphere. This fact co m bined with Corollar y 5.3 yie lds the following w ell-known fact [10, 3.4.11]: Corollary 5.4. L et  X , ( p α )  b e a limit of an inverse sp e ctrum S = { X α , p β α , Σ } in the c ate gory CompEpi . The limit sp ac e X has dimension dim( X ) ≤ n for some n ∈ ω pr ovide d that dim ( X α ) ≤ n for al l α ∈ Σ . 6. Proof o f Theorem 1. 2 In this section we present a pro of of Theo rem 1.2. Given any non-empty fa mily of rational functions F ⊂ R ( x, y ) we need to prov e that the g raphoid ¯ Γ( F ) has dimension dim( ¯ Γ( F )) = 2 . Lemma 6.1. The gr apho id ¯ Γ( F ) has dimension dim( ¯ Γ( F )) ≤ 2 . Pr o of. By Lemma 4.1, the gr aphoid ¯ Γ( F ) is homeo morphic to the limit space o f the inv erse sp ectrum S F = { ¯ Γ( α ) , p β α , [ F ] <ω } that consists of gra pho ids ¯ Γ( α ) of ﬁnite subfamilies α ⊂ F . Now Co rollary 5.4 will imply that dim ¯ Γ( F ) ≤ 2 as s o on as w e c heck that dim ¯ Γ( G ) ≤ 2 for any ﬁnite subfamily G ⊂ F . Since G is ﬁnite, the set dom( G ) = T f ∈G dom( f ) is coﬁnite in ¯ R 2 . Iden tify the family G with the partial c ontin uous function G : do m( G ) → ¯ R G , G : x 7→ ( f ( x )) f ∈G and let ¯ G b e the g raphoid extensio n of G . Then ¯ Γ( G ) = Γ( ¯ G ) and hence ¯ Γ( G ) = Γ( ¯ G ) = Γ( G ) ∪ [  { z } × ¯ G ( z ) : z ∈ ¯ R 2 \ dom( G )  . Theorem 3.1(6) implies that for every p oint z ∈ dom( G ) the se t ¯ G ( z ) has dimension dim( ¯ G ( z )) ≤ 1 . Since the graph Γ( G ) is homeomor phic to the c o ﬁnite set ¯ R 2 \ dom( G ), it has dimensio n dim(Γ( G )) ≤ dim( ¯ R 2 ) = 2. Now Theor e m of Sum [1 0, 1 .5.3] implies that dim ¯ Γ( G ) ≤ sup { dim(Γ( G )) , dim ¯ G ( z ) : z ∈ ¯ R 2 \ dom( G ) } ≤ 2 .  Lemma 6.2. dim ¯ Γ( F ) ≥ 2 . Pr o of. Since dim ¯ Γ( F ) ≤ 1 if a nd only if e-dim( ¯ Γ( F )) ≤ K ∂ , it suﬃces to chec k that e-dim( ¯ Γ( F )) 6≤ K ∂ . T o prove this fact, we shall apply Lemma 5.1. By Lemma 4.1, the graphoid ¯ Γ( F ) is the limit of the spec trum S F = { ¯ Γ( α ) , p β α , [ F ] <ω } . The smallest element of the p o s et [ F ] <ω is the empty set. Its g raphoid ¯ Γ( ∅ ) can b e identiﬁed with the torus ¯ R 2 . Let A ∅ = 2 ¯ K \ K where K = ( − 1 , 1 ) 2 is the op en square in the plane R 2 endow ed with the max-no r m k ( x, y ) k = max {| x | , | y |} . Consider the map f ∅ : A ∅ → K ∂ , f : ( x, y ) 7→ ( x,y ) k ( x,y ) k . Ass uming that e-dim ¯ Γ( F ) ≤ K ∂ , and applying Lemma 5 .1, we can ﬁnd a ﬁnite subset β ⊂ F and a contin uous map f β : ¯ Γ( β ) → K ∂ that extends the map f ∅ ◦ p β ∅ | A β : A β → K ∂ where A β = ( p β ∅ ) − 1 ( A ∅ ). The ﬁnite family β ⊂ F thoug ht as a partial function β : dom( β ) → ¯ R β is deﬁned on a coﬁnite subs e t dom( β ) of ¯ R 2 . So, we can ﬁnd a rea l num b er t ∈ [1 , 2] such that tK ∂ ⊂ dom( β ). Consider the ﬁnite set Z = t ¯ K \ dom( β ) in tK . Using Theo rem 3 .1, we can ﬁnd ε > 0 so small that (1) tK ∂ ∩ ¯ B ( Z , ε ) = ∅ ; (2) ¯ B ( z , ε ) ∩ ¯ B ( z ′ , ε ) = ∅ for any distinct p oints z , z ′ ∈ Z ; (3) there is a homeo morphism h : t ¯ K \ ¯ B ( Z , ε 2 ) → t ¯ K \ Z s uch that (a) h is iden tity on the set t ¯ K \ B ( Z , ε ), (b) h has cont inuous extension ¯ h : t ¯ K \ B ( Z, ε 2 ) → t ¯ K such that ¯ h − 1 ( z ) = S ( z , ε 2 ) fo r every z ∈ Z ; (c) the comp osition β ◦ h : t ¯ K \ ¯ B ( Z , ε 2 ) → ¯ R β has a contin uous extens io n ¯ β h : t ¯ K \ B ( Z, ε 2 ) → ¯ R β ; (d) for every z ∈ Z and any map ϕ z : ¯ β ( z ) → K ∂ the compos ition ϕ z ◦ ¯ β h | S ( z , ε 2 ) : S ( z , ε 2 ) → K ∂ is Z 2 -trivial. It fo llows that the map ψ = ( ¯ h, ¯ β h ) : t ¯ K \ B ( Z, ε 2 ) → ¯ Γ( β ) , ψ : z 7→ ( ¯ h ( z ) , ¯ β h ( z )) , is con tinu ous and for every z ∈ Z the map f β ◦ ψ | S ( z , ε 2 ) : S ( z , ε 2 ) → K ∂ is Z 2 -trivial. Then by Lemma 2.5, the map ψ | tK ∂ : tK ∂ → K ∂ also is Z 2 -trivial, which is impos sible as this map is a homeo mo rphism, which induces an isomor phism of the homo logy gro ups H 1 ( tK ∂ ; Z 2 ) and H 1 ( K ∂ ; Z 2 ). This contradiction completes the pro of o f Lemma 6.2.  DIMENSION OF GRAPHOIDS OF RA TIONAL VECTOR-FUNCTIONS 19 7. Proof o f Theorem 1. 3 Assume that F ⊂ R ( x, y ) is a family of rational functions, containing a family of linear fractional tra nsformations  x − a y − b : ( a, b ) ∈ D  for s o me dense subset D of R 2 . By Theorem 1.2, dim( ¯ Γ( F )) = 2 . By Alexandroﬀ Theorem [6, 1.4], dim Z ( X ) = dim( X ) for each ﬁnite-dimensio nal compact Ha usdorﬀ s pa ce X . Co nsequently , dim Z ( ¯ Γ( F )) = dim( ¯ Γ( F )) = 2 . Now let G b e a non-triv ia l 2- div isible ab elia n gr oup. W e need to show that dim G ( ¯ Γ( F )) = 1. T o see that dim G ( ¯ Γ( F )) > 0, take any Eilenberg- MacLane complex K ( G, 0), for example, take the gr o up G endow ed with the discrete topolo gy . Since the space ¯ Γ( F ) is connected, any injectiv e map f : A → G deﬁned on a do ubleton A = { a, b } ⊂ ¯ Γ( F ) has no contin uous extensio n ¯ f : ¯ Γ( F ) → G , which means that e-dim( ¯ Γ( F )) 6≤ G and dim G ( ¯ Γ( F )) 6≤ 0 . The inequalit y dim G ( ¯ Γ( F )) ≤ 1, whic h is equiv alent to e-dim( ¯ Γ( F )) ≤ K ( G, 1), follows from the subsequent a bit more general res ult: Lemma 7. 1. e- dim( ¯ Γ( F )) ≤ Y for any p ath-c onne cte d sp ac e Y ∈ ANE ( Comp ) with 2-divisible fundamental gr oup π 1 ( Y ) . Pr o of. T o show that e-dim( ¯ Γ( F )) ≤ Y we shall apply Lemma 5.1. By Lemma 4.1, the gra phoid ¯ Γ( F ) is homeomorphic to the limit spa ce of the inv erse sp ectrum S F = { ¯ Γ( α ) , p β α , [ F ] <ω } . Giv en a ﬁnite subset α ∈ [ F ] <ω and a map f α : A α → Y deﬁned on a clos ed subset A α of the graphoid ¯ Γ( α ), we need to ﬁnd a ﬁnite subse t β ⊃ α of F a nd a contin uo us function ¯ f β : ¯ Γ( β ) → Y that extends the map f α ◦ p β α | A β deﬁned on the set A β = ( p β α ) − 1 ( A α ). W e can think of the family α ⊂ F as a partial function α : dom( α ) → ¯ R α deﬁned on the coﬁnite set dom( α ) in ¯ R α . Let ¯ α : ¯ R 2 ⊸ ¯ R α be the grapho id extension of α . Its graph Γ( ¯ α ) coincides w ith the graphoid ¯ Γ( α ) of α . By Theorem 3.1(6), for every point z of the ﬁnite set Z = { ( ∞ , ∞ ) } ∪  ¯ R 2 \ dom( α )  the image ¯ α ( z ) is a singleton or a ﬁnite union of a rcs. C o nsequently , the set Γ( ¯ α | Z ) = S z ∈ Z { z } × ¯ α ( z ) is a ﬁnite union of singletons or arcs. Using the path-connectedness of the space Y ∈ ANE ( Comp ), w e ca n extend the map f α to a con tinuous map f ′ α : A α ∪ Γ( ¯ α | Z ) → Y . Since Y ∈ ANE ( Comp ), the map f ′ α : A α ∪ Γ( ¯ α | Z ) → Y has a contin uous extension ˜ f α : ˜ A α → Y deﬁned on a closed neighborho o d ˜ A α of the set A α ∪ Γ( ¯ α | Z ) in ¯ Γ( α ). The bo undary ∂ ˜ A α of ˜ A α in ¯ Γ( α ) is a compact subset o f Γ( ¯ α ) \ Γ( ¯ α | Z ) ⊂ Γ( α ). The pro jection p α ∅ : Γ( ¯ α ) → ¯ R 2 maps homeomorphica lly the graph Γ( α ) o nto the coﬁnite subset dom( α ) of the torus ¯ R 2 . Replacing ˜ A α by a smaller (more regula r) neighbo rho o d, if necessary , we ca n ass ume that the b oundar y ∂ ˜ A α is a top ological graph, tha t is, a ﬁnite union of ar cs that are disjoint or meet b y their end-p oints. Adding to ˜ A α a ﬁnite union of a rcs, we ca n enlar g e the set ˜ A α to a clos e d set ¯ A α ⊂ ¯ Γ( α ) whose b o unda ry ∂ ¯ A α is a top olo g ical graph such that • the family C of connected comp onents of ¯ Γ( α ) \ ¯ A α is ﬁnite and • for each connected co mpo nent C ∈ C the clo sure ¯ C is homeomorphic to the closed square ¯ K = [ − 1 , 1] 2 . Using the path-connectednes s of the spa ce Y ∈ ANE ( Comp ), we can e xtend the map ˜ f α to a contin uous map ¯ f α : ¯ A α → Y . F or every connected comp onent C ∈ C use the density of the set D in R 2 and ﬁnd a point ( a C , b C ) ∈ D ∩ C . Now consider the ﬁnite subfamily β = α ∪ n x − a C y − b C : C ∈ C o ⊂ F , which determines a pa rtial c ontin uous function β : dom( β ) → ¯ R β deﬁned on the co ﬁnite set dom( α ) \ ( { ( ∞ , ∞ ) } ∪ { ( a C , b C ) : C ∈ C } ) of ¯ R 2 . W e cla im that there is a contin uo us function ¯ f β : ¯ Γ( β ) → Y that extends the map f α ◦ p β α | A β deﬁned on the se t A β = ( p β α ) − 1 ( A α ). Put ¯ A β = ( p β α ) − 1 ( ¯ A α ) and observe that the c omplement ¯ Γ( β ) \ ¯ A β is the union of connected comp onents C β = ( p β α ) − 1 ( C ), C ∈ C , whic h ar e grapho ids of the rationa l functions x − a C y − b C restricted to the op en 2-disks p α ∅ ( C ). Such graphoids are homeomor phic to the o pe n M¨ obius band. F or every C ∈ C the clos ure ¯ C β of C β in ¯ Γ( β ), b eing homeo morphic to the closed M¨ o bius band, is homeomorphic to the quotient s pace of the “sq ua re ann ulus” ¯ K \ 1 2 K by the eq uiv alence relatio n that identiﬁ es the pair s of opp osite po int s on the inner bo undary square 1 2 K ∂ . Let q C : ¯ K \ 1 2 K → ¯ C β be the corr e sp onding quotient map. Fix a contin uous map σ : [0 , 1] → K ∂ such that • σ (0) = σ (1 ), • σ | [0 , 1) : [0 , 1) → K ∂ is bijective, • for any t ∈ [0 , 1 2 ] the p oints σ ( t + 1 2 ) = − σ ( t ). 20 T ARAS BANAKH AND OLES POTY A TYNYK The map γ C = ¯ f α ◦ p β α ◦ q C ◦ σ : [0 , 1] → Y determines a lo op in Y , whose equiv a lence clas s is an element of the fundamen tal group π 1 ( Y ) of Y . Since the group π 1 ( Y ) is 2-divisible, there is a lo op δ C : [0 , 1 ] → Y such whose square δ 2 C : [0 , 1] → Y , δ 2 C : t 7→ ( δ C (2 t ) if 0 ≤ t ≤ 1 2 , δ C (2 t − 1) if 1 2 ≤ t ≤ 1, is homo topic to the lo op γ C by a ho motopy that do es not mo ve the points 0 and 1. Now consider the lo op ˜ γ C = ¯ f α ◦ p β α ◦ q C | K ∂ : K ∂ → Y and observe that γ C = ˜ γ C ◦ σ . Le t ˜ δ 2 C : 1 2 K ∂ → Y be a unique map such that ˜ δ 2 C ◦ 1 2 σ = δ 2 C . Here 1 2 σ : [0 , 1] → 1 2 K ∂ is the loo p assigning to each t ∈ [0 , 1] the p o in t 1 2 σ ( t ) of the s quare 1 2 K ∂ . The homotopy b etw ee n the lo ops γ C and δ 2 C allows us to ﬁnd a contin uous map ˜ h C : ¯ K \ 1 2 K → Y such that ˜ h C | K ∂ = ˜ γ C and ˜ h C | 1 2 K ∂ = ˜ δ 2 C . The deﬁnition of σ and δ 2 C guarantees that ˜ δ 2 C ( x ) = ˜ δ 2 C ( − x ) for any point x ∈ 1 2 K ∂ . Hence there is a unique contin uo us map h C : ¯ C β → Y suc h that ˜ h C = h C ◦ q C . It follows from ˜ h C | K ∂ = ˜ γ C that h C | ∂ ¯ C β = ¯ f α ◦ p β α | ∂ ¯ C β . This implies that the map f β : ¯ Γ( β ) → Y , f β ( x ) = ( ¯ f α ◦ p β α ( x ) if x ∈ ¯ A β , h C ( x ) if x ∈ ¯ C β for some C ∈ C , is a well-deﬁned contin uous ex tension o f the map f α ◦ p β α | A β .  8. Some Open Problems In light of Theorem 1.3 the following problem a rises naturally: Problem 8.1. Has the gr aphoid ¯ Γ( F ) of any family F ⊂ R ( x, y ) the c ohomolo gic al di mension dim G ( ¯ Γ( F )) = 2 for any ab elian gr oup G that is not 2-divisib le? The answer to this problem is aﬃrmative if the following problem has a n aﬃrmative a nswer. Problem 8.2. L et F ⊂ R ( x, y ) b e a ﬁnite family , ¯ F : ¯ R 2 ⊸ ¯ R F b e its gr apho id extension, and z ∈ ¯ R 2 b e an arbitr ary p oint. Is ¯ F ( z ) a singleton or a ﬁnite u nion of analytic ar cs in ¯ R F . An arc A in ¯ R n is called analytic if A = ~ α ([0 , 1]) for some vector function ~ α : [0 , 1] → ¯ R n that has a nalytic co ordinate functions α 1 , . . . , α n : [0 , 1] → ¯ R . Here w e identify the pro jective line ¯ R with the unit circle on pla ne via the s tereogr aphic pro jection. In case o f p ositive answer to Problem 8.2 the pro o f of the inequality dim( ¯ Γ( F )) ≥ 2 can b e m uch simpliﬁed (Lemma 3 with its extremely long pro of will be not required). References [1] P . S. Aleksandro v, Intr o duction to homolo gic al dimension the ory and gener al c ombinatorial top olo gy , Nauk a, Mosco w, 1975 (in Russ ian). [2] T. Banakh, Y a. Kholy avk a, M. Mach ura, O. P oty atyn yk, K. Osiak, The dimension of t he sp ac e of r ea l plac es of a function ﬁeld , preprint. [3] E. Beck er, D. Gondard, Notes on the sp ac e of r ea l plac es of a formal ly r e al ﬁeld , in: Real analytic and algebraic geometry (T rento, 1992), 21–46, de Gruyter, Berlin, 1995. [4] E. Brieskorn, H. Egbert; Kn¨ orrer, Plane algebr aic curves , Bir khuser V erl ag, Basel, 1986. [5] A. Chigogidze, Inverse Sp e ctr a , North-Holland Publ., Ams terdam, 1996. [6] A. Dranishniko v Cohomolo gica l dimension the ory of c omp act metric sp ac es , T op ology A tlas Invited Cont ributions, 6 :3 (2001), 61 pp. (arXiv:math/0501523). [7] A. Dranishniko v, J. Dydak, Extension the ory of sep ar able met rizable sp ac es wit h applic ations to dimension the ory , T rans. Amer. Math. Soc. 353 :1 (2001), 133–156. [8] I. Efrat, K. Os iak, T op olo gic al sp ac es as sp ac es of R-plac es , J. Pure Appl. Algebra 215 :5 (2011), 839–846. [9] R. Engelking, Genera l T op olo gy , Heldermann V erlag, Berli n, 1989. [10] R. Engelking, The ory of dimensions, ﬁnite and inﬁnite , Heldermann V erlag, Lemgo, 1995. [11] A. Hatc her, A lgebr aic T op olo g y , Cam bridge Univ. Pr ess, Cambridge, 2002. [12] W. Just, M. W eese, Disco vering mo dern set the ory. II. Set -the or etic to ols for every mathematician , Amer. Math. So c., Providence , RI, 1997. [13] E. Kunz, Intr o duction to plane algeb ra ic curves , Birkh¨ auser Boston, Inc., Boston, MA, 2005. [14] M. Mach ur a, M. Mar s hall, K. Osiak, Met rizability of t he sp ac e of R-plac es of a r e al function ﬁeld , Math. Z. 266 :1 (2010), 237–242. [15] W. Rudin, R e al and Complex Ana lysis , McGraw-Hill Bo ok Co., New Y ork, 1987. Instytut M a tem a tyki, Jan Kochanowski University, Kilece, Poland and Dep ar tment of Mathema tics, Iv an Franko Na tional Un iversity of L viv, Ukraine E-mail addr ess : tbanakh@yahoo .com Dep ar tment of Ma thema tics, Iv an Franko Na tional University of L viv, Ukraine E-mail addr ess : oles2008@gmai l.com

Dimension of graphoids of rational vector-functions

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