Spaces of $mathbb R$ - places of rational function fields

SP A CES OF R - PLA CES OF RA TIONAL FUNCTIO N FIELDS MICHA L MAC HUR A A ND KA T ARZYNA OSIAK Abstra ct. In the pap er an answ er to a problem ”When different orders of R(X) (where R is a real clo sed field) lead to the same real pla ce ?” is giv en . W e use this result to sho w that the space of R -places of the field R ( Y ) (where R is any real clo sure of R ( X )) is not metriz able space. Thus the space M ( R ( X , Y )) is not metrizable, to o. 1. In troduction The studies on real places of formally real fi elds were initiated by Dub ois [6] and Bro w n [3]. The researc h h as b een con tinued in s ev eral pap ers: Bro wn and Marshall [4], Harman [9], Sc h ¨ ulting [16], Bec ke r and Gondard [2] and Gond ard and Marshall [11]. W e will u se notation and terminology introdu ced by Lam [13] - most of the results w e sh all recall in this section can b e found there. W e assume that the reader is somewh at familiar with the v aluation theory and the th eory of formally real (ordered ) fields. Let K b e an ord ered field. The set X ( K ) of all orders of K can b e made into a top ologica l space by usin g as a su bbase th e family of Harrison sets of the f orm H K ( a ) := { P ∈ X ( K ) : a ∈ P } , a ∈ ˙ K = K \ { 0 } . It is known that the space X ( K ) is a Bo olean sp ace (i.e. compact, Hausd orff and totally disconn ected). If P is an order of K , then the set A ( P ) := { a ∈ K : ∃ q ∈ Q + q ± a ∈ P } is a v aluation ring of K with th e maximal ideal I ( P ) := { a ∈ K : ∀ q ∈ Q + q ± a ∈ P } . 2000 Mathematics Subje ct Classific ation. Primary 12D15; Secondary 14P05. Key wor ds and phr ases. real place, spaces of real p laces. 1 2 MICHA L M ACHURA AND KA T ARZYNA OSIAK Moreo ve r, the residue fi eld k ( P ) = A ( P ) /I ( P ) is ordered b y an Archimedean order ¯ P := ( P ∩ ˙ A ( P )) + I ( P ) , where ˙ A ( P ) is the s et of units of A ( P ). The set H ( K ) = \ P ∈X ( K ) A ( P ) is called the r e al holomorp hy ring of the field K . Since k ( P ) has an Archimedean order, we consider it as a su bfield of R and the place asso ciated to A ( P ) is called an R - plac e . Moreo ve r, ev ery place of K with v alues in R is determined b y some order of K . W e denote by M ( K ) the set of all R - p laces of the fi eld K . In fact, we hav e a s urjectiv e map λ K : X ( K ) − → M ( K ) . and we can equip M ( K ) with the quotien t top ology inherited from X ( K ). Note, that in term inology of R - places w e ha v e H ( K ) = { a ∈ K : ∀ ξ ∈ M ( K ) : ξ ( a ) 6 = ∞} . F or an y a ∈ H ( K ) the m ap e a : M ( K ) − → R , e a ( ξ ) = ξ ( a ) is called the ev aluation m ap. Th e ev aluatio n maps are con tin uous in the quotien t top ology of M ( K ) and the set of ev aluation maps s eparates p oin ts of M ( K ). Therefore M ( K ) is a Hausd orff space. It is also compact as a con tin uous image of a compact sp ace. By [13, Theorem 9.11], the family of sets U K ( a ) = { ξ ∈ M ( K ) : ξ ( a ) > 0 } , for a ∈ H ( K ) . is a s ubbase for the top ology of M ( K ). T hese sets ma y not b e closed and therefore the space M ( K ) need n ot b e Bo olean. How ev er, ev ery Bo olean space is realized as a space of R - places of some formally real fi eld. F or this resu lt see [15]. On the other h and there are a lot of examples of fi elds for whic h the sp ace of R - places has a fin ite n umber of connected comp onen ts and even turns out to b e connected. If K is a real closed field, then the space M ( K ) has only one p oin t. It is w ell kn o wn (see [16], [2]) that the sp ace of M ( R ( X )) is homeomorphic to a SP A CES OF R - PLACES OF RA TIONAL FUNCT ION FIELDS 3 circle rin g. More general result states that th e space of R - places of a rational function field K ( X ) is conn ected if an d only if M ( K ) is connected (see [16], [9]). The goal of this pap er is to describ e the space of R - p laces of a field R ( X ), where R is an y real closed field. The set of ord ers of R ( X ) is in one - to - one coresp ondence with Dedekind cuts of R ( X ). Therefore the set X ( R ( X )) is linearly ordered. The m ain theorem of the second section shows how the m ap λ : X ( R ( X )) − → M ( R ( X )) glues the p oin ts. The natur al app licatio n is giv en for the c ontinuous closur e of the field R . This is a field ˜ R ⊇ R suc h that R is dense in ˜ R and whic h is maximal in this r esp ect. W e shall s ho w that the spaces M ( R ( X )) and M ( ˜ R ( X )) are homeomorph ic. Using this result and mo difying a metho d of [17] we describ e in th e third section connection b et w een completeness (in the sense of u niformit y) and c ontinuity (see [1]) of an ord ered field endo w ed with v aluation top ology . Th ese results allo w us to describ e the space of R - places of the field R ( Y ), where R is a fixed real closur e of R ( X ). In the fo urth sectio n w e will sho w that it is not m etrizable. Th e space M ( R ( Y )) is kn o wn to b e sub space M ( R ( X , Y )). Therefore M ( R ( X , Y )) can not b e a metric space. 2. The rea l places of R ( X ) Let R b e a real clo sed field with its unique order ˙ R 2 . Denote b y v the v aluatio n asso ciated to A ( ˙ R 2 ). Supp ose that Γ is the v alue group of v . By [10], [17] there is one-to-one corresp ondence b etw een orders of R ( X ) and Dedekind cuts of R . If P is an ord er of R ( X ), then th e corresp ond ing cut ( A P , B P ) is giv en by A P = { a ∈ R : a < X } and B P = { b ∈ R : b > X } . On the other hand, if ( A, B ) in an y cu t in R , then a set P = { f ∈ R ( X ) : ∃ a ∈ A ∃ b ∈ B ∀ c ∈ ( a,b ) f ( c ) ∈ ˙ R 2 } is an order of R ( X ) whic h determines the cut ( A, B ). The cuts ( ∅ , R ) and ( R, ∅ ) are called the impr op er cuts . The orders determined b y these cuts are d enoted P − ∞ and P + ∞ , resp ectiv ely . A cut ( A, B ) of R is called normal if it satisfies a follo w ing condition ∀ c ∈ ˙ R 2 ∃ a ∈ A ∃ b ∈ B b − a < c. 4 MICHA L M ACHURA AND KA T ARZYNA OSIAK If A has a maximal or B h as a min imal elemen t, then ( A, B ) is a princip al cut . Ev ery a ∈ R defines tw o pr incipal cu ts w ith the corresp ond ing orders P − a and P + a . Note that if R is a real closed sub field of R , then all of prop er cuts of R are normal. Moreo ver, if R = R , then all of prop er cuts are normal and prin cipal. If A has not a maximal element and B has n ot a minimal elemen t, then w e sa y that ( A, B ) is a fr e e cut or a gap . If R is n ot conta ined in R , th en R has the abnormal gaps, i.e. gaps wh ic h are not normal. F o r example, ( A, B ) where A = { a ∈ R : a 6 0 } ∪ { a ∈ R : a > 0 ∧ a ∈ I ( ˙ R 2 ) } and B = { b ∈ R : b > 0 ∧ b / ∈ I ( ˙ R 2 ) } , is an abn ormal gap in R . In fact w e can ha v e th ree kin ds of prop er cuts: (1) principal cuts; (2) normal (bu t not principal) gaps; (3) abnormal gaps. One has to note that the corresp ondence b et w een cuts in R and orders of R ( X ) mak es the set X ( R ( X )) linearly ordered. If P and Q are differen t orders of R ( X ), then we say that P ≺ Q ⇐ ⇒ A P ⊂ A Q . Let P − a ≺ P + a b e the ord ers corresp onding to p rincipal cuts giv en b y a ∈ R Observe that the in terv al ( P − a , P + a ) is empt y . In such situations w e sh all sa y that ≺ has a step in a . The map λ : X ( R ( X )) − → M ( R ( X )) glues these steps. By [9], M ( R ( X )) is connected space. Ho w ev er it can happ en that λ glues more p oints of X ( R ( X )). Our goal is to answe r th e question: W hich p oints of X ( R ( X )) do es λ glue? More exa ctly , sup p ose th at ( A 1 , B 1 ) and ( A 2 , B 2 ) are the cuts corresp onding to the ord ers P 1 and P 2 , resp ec tiv ely . When λ ( P 1 ) = λ ( P 2 ) ? W e sh all m ak e use of S eparation Criterion [13, P rop osition 9.13], whic h allo ws to separate R - places. SP A CES OF R - PLACES OF RA TIONAL FUNCT ION FIELDS 5 Theorem 2.1. [Separation Criterion] L et P and Q b e distinct or ders of a field K . Then λ K ( P ) 6 = λ K ( Q ) if and only if ther e exists a ∈ K such that a ∈ ˙ A ( P ) ∩ P and − a ∈ Q . Let P b e an order of R ( X ). Then A ( P ) = { f ∈ R ( X ) : ∃ q ∈ Q + ∃ a ∈ A ∃ b ∈ B ∀ c ∈ ( a,b ) q ± f ( c ) ∈ ˙ R 2 } = { f ∈ R ( X ) : ∃ a ∈ A ∃ b ∈ B ∀ c ∈ ( a,b ) f ( c ) ∈ A ( ˙ R 2 ) } , and I ( P ) = { f ∈ R ( X ) : ∀ q ∈ Q + ∃ a ∈ A ∃ b ∈ B ∀ c ∈ ( a,b ) q ± f ( c ) ∈ ˙ R 2 } . By a neighb orho o d of a cut ( A, B ) we mean an interv al ( a, b ) ⊂ R suc h that ( a, b ) ∩ A 6 = ∅ and ( a, b ) ∩ B 6 = ∅ . R emark 2.2 . Let P b e an order of R ( X ). Then A ( P ) is a set of these functions whic h on some neig hborh o o d of ( A P , B P ) ha v e v alues in A ( ˙ R 2 ) and ˙ A ( P ) con tains these fun ctions w hic h on some neighborh o o d of ( A P , B P ) hav e v alues in ˙ A ( ˙ R 2 ). According to [17, Lemma 2.2.1], ev ery cut of R determines a lo wer cut S = { v ( b − a ) : a ∈ A, b ∈ B } , in Γ. No te th at if ( A, B ) is a normal cut, then S = Γ and if ( A, B ) is an improp er cut, then S = ∅ . The sets S allo w u s to compare gaps. W e can sa y that a gap ( A 1 , B 1 ) is ” bigger” then ( A 2 , B 2 ) if S 1 ⊂ S 2 . In a similar w a y one can in ve stigate a ”distance” b etw een t wo cuts ( A 1 , B 1 ) and ( A 2 , B 2 ). Su pp ose that A 1 ⊂ A 2 and consider the set U = { v ( a ′ − a ) : a, a ′ ∈ B 1 ∩ A 2 , a < a ′ } . Supp ose that S 1 and S 2 are the lo wer cuts in Γ d etermined by ( A 1 , B 1 ) and ( A 2 , B 2 ). Lemma 2.3. U is an upp er cut in Γ . Mor e over, Γ \ ( S 1 ∩ S 2 ) ⊂ U . Pr o of. W e sh all sh o w that if γ ∈ U and γ ′ ∈ Γ and γ < γ ′ , then γ ′ ∈ U . W e h a ve γ = v ( a ′ − a ), where a and a ′ are as in the definition of U and γ ′ = v ( c ), w here c is a p ositiv e elemen t of R . Since v ( a ′ − a ) < v ( c ), w e hav e v ( c a ′ − a ) ∈ I ( ˙ R 2 ). Then c a ′ − a < 1. So a + c < a ′ . Thus a + c ∈ B 1 ∩ A 2 . W e ha v e v ( a + c − a ) = v ( c ) = γ ′ ∈ U . 6 MICHA L M ACHURA AND KA T ARZYNA OSIAK No w su pp ose that γ ∈ Γ \ ( S 1 ∩ S 2 ). Let c b e a p o sitiv e elemen t of R with v ( c ) = γ . Fix an elemen t a ∈ B 1 ∩ A 2 . Assume that γ / ∈ S 1 . Then a − c ∈ B 1 ∩ A 2 . Th us v ( a − ( a − c )) = v ( c ) = γ ∈ U . If γ / ∈ S 2 , then a + c ∈ B 1 ∩ A 2 and v ( a + c − a ) = v ( c ) = γ ∈ U .  Theorem 2.4. L et ( A 1 , B 1 ) and ( A 2 , B 2 ) b e the cuts in R c orr esp onding to the or ders P 1 and P 2 of R ( X ) , r esp e ctiv e ly. L et S 1 , S 2 and U b e the cuts in Γ define d ab ove and let λ : X ( R ( X )) → M ( R ( X )) b e the c anonic al map. (1) If S 1 6 = S 2 , then λ ( P 1 ) 6 = λ ( P 2 ) . (2) If S 1 = S 2 = S and S ∩ U 6 = ∅ , then λ ( P 1 ) 6 = λ ( P 2 ) . (3) If S 1 = S 2 = S and S ∩ U = ∅ , then λ ( P 1 ) = λ ( P 2 ) . Pr o of. Without lost of generalit y w e can assume that A 1 ⊂ A 2 . (1) S upp ose that S 1 ⊂ S 2 . Then there exist a ∈ B 1 ∩ A 2 and b ∈ B 2 suc h that v ( b − a ) / ∈ S 1 . C onsider a linear p olynomial f ( X ) = X − a b − a + 1. Th is p olynomial has a root x 0 = a − ( b − a ). If x 0 ∈ A 1 , then v ( a − x 0 ) = v ( b − a ) ∈ S 1 - a con tr adiction. Therefore x 0 ∈ B 1 . Moreo v er, f ( a ) = 1 and f ( b ) = 2. Th erefore f has p ositi v e v alues in some n eigh b ourho o d of ( A 2 , B 2 ) w hic h are u nits in A ( ˙ R 2 ) and negativ e v alues in some neigh b ourho o d of ( A 1 , B 1 ). By Remark 2.2 and Separation Criterion λ ( P 1 ) 6 = λ ( P 2 ). If S 2 ⊂ S 1 , th en f can b e defin ed as a suitable, decreasing linear p olynomial. (2) Let γ ∈ U ∩ S . T hen th ere exist a ∈ A 1 , b ∈ B 1 and c, d ∈ B 1 ∩ A 2 suc h that γ = v ( b − a ) = v ( d − c ). W e shall sho w that one can fix γ in suc h a wa y that a < b 6 c < d . If c < b , then w e tak e γ ′ = v ( c − a ). W e ha ve γ ′ > γ , so γ ′ ∈ U . If γ ′ > γ , then v ( c − a ) > v ( d − c ). Thus c − a < d − c and c + ( c − a ) < d . Therefore c + ( c − a ) ∈ B 1 ∩ A 2 . So w e can tak e γ ′ as γ , c as b and c + ( c − a ) as d . Since v ( b − a ) = v ( d − c ), th ere exists n ∈ N such that 1 n < b − a d − c < n . Th en b − a n < d − c . Consider a linear p olynomial f ( X ) such that f ( a ) = n + 1 and f ( b ) = 1 i.e. f ( X ) = n ( b − X ) b − a + 1. T his p olynomial h as a ro ot x 0 = b + b − a n < b + d − c < d . Thus f has p osit iv e v alues in a n eigh b orho o d of ( A 1 , B 1 ) whic h are units in A ( ˙ R 2 ) and negativ e v alues in a neigh b orho o d of ( A 2 , B 2 ). Using Separation Criterion we get λ ( P 1 ) 6 = λ ( P 2 ). SP A CES OF R - PLACES OF RA TIONAL FUNCT ION FIELDS 7 (3) By [13, C orollary 2,13 ] it suffices to show that ˙ A ( P 1 ) ∩ P 1 = ˙ A ( P 2 ) ∩ P 2 . (i) F rom Lemma 2.3 w e hav e U = Γ \ S . (ii) W e can assume that 0 ∈ B 1 ∩ A 2 . If a ∈ B 1 ∩ A 2 , then w e can consider the cuts: ( A 1 − a, B 1 − a ) and ( A 2 − a, B 2 − a ). Then f ( X ) is a ”separation” f unction for ( A 1 , B 1 ) and ( A 2 , B 2 ) if and on ly if f ( X + a ) is a ”separation” function for ( A 1 − a, B 1 − a ) and ( A 2 − a, B 2 − a ). Therefore the R - places d etermined by ord ers asso ciated to ( A 1 − a, B 1 − a ) and ( A 2 − a, B 2 − a ) are equal if and only if λ ( P 1 ) = λ ( P 2 ). (iii) ( A 1 , B 1 ) and ( A 2 , B 2 ) are symmetric in resp ect to 0 i.e. if a ∈ B 1 ∩ A 2 , then − a ∈ B 1 ∩ A 2 . If a ∈ B 1 ∩ A 2 and − a ∈ A 1 , then S ∋ v ( a ) = v ( − a ) ∈ U - a con tr adiction with S ∩ U = ∅ . (iv) Let A := B 1 ∩ A 2 and B = A 1 ∪ B 2 . Then v ( A ) = U ∪ {∞} and v ( B ) = S . Let v + b e a v aluation determined by order P 1 and let v − b e a v aluation determined b y ord er P 2 . The v aluations v 1 and v 2 are extensions of v . W e ha v e v ( a ) > v ( b ) , for a ∈ A, b ∈ B . Therefore v ( a ) > v ± ( X ) > v ( b ) , for a ∈ A, b ∈ B . (v) A v aluation group of v ± is bigger then Γ. In fact v ( a ) > v ± ( X ) > v ( b ) , for a ∈ A, b ∈ B . If v ± ( X ) = v ( a ) for some a ∈ A then v ± ( X a ) = 0. Th us the f unction X a has Arc h imedean v alues in some neighb ourho o d of ( A 1 , B 1 ). Therefore there exists b ∈ B suc h that v ( b a ) = 0. So v ( b ) = v ( a ) - a con tradiction. Similarly v ± ( X ) 6 = v ( b ). (vi) W e shall c hec k the v alues of v aluations v ± on linear and quadratic p olyno- mials: v ± ( X − a ) = v ± ( X ) for a ∈ A ; v ± ( X − b ) = v ( b ) for b ∈ A ; v ± (( X − c ) 2 + d 2 ) =        2 v ± ( X ) if c, d ∈ A 2 v ( d ) if c ∈ A, d ∈ B 2 v ( c ) if c ∈ B , d ∈ A 2 m in { v ( c ) , v ( d ) } if c, d ∈ B . 8 MICHA L M ACHURA AND KA T ARZYNA OSIAK (vii) S ince Γ is a divisible group, n · v ± ( X ) / ∈ Γ for n ∈ N . Th u s for ev ery p olyno- mial f ∈ R [ X ], there exist a ∈ R and n ∈ N suc h that v ± ( f ) = v ( a ) + n · v ± ( X ) and v ± ( f ) = 0 if and only if v ( a ) = 0 an d n = 0. Therefore ˙ A ( P 1 ) = ˙ A ( P 2 ) . (vii) Let F ∈ ˙ A ( P 1 ) ∩ P 1 . Then F h as a representa tion: F = f 1 f 2 = c ( X − c 1 ) · .... ( X − c k ) · [a pro d uct of sum of squares] d ( X − d 1 ) · .... ( X − d k ) · [a pro d uct of sum of squares] Since v ± ( F ) = 0, ♯ { c i ∈ A : i = 1 , ...k } − ♯ { d i ∈ A : i = 1 , ...l } ≡ 0 (mo d 2) . Th us f 1 · f 2 has an ev en n umb er of ro ots in A . Therefore F ∈ P 2 .  R emark 2.5 . The Theorem 2.4 sho ws that the map λ glues only the abnormal gaps wh ic h are close one to eac h other. F or example, the orders determined by gaps ( A 1 , B 1 ) and ( A 2 , B 2 ) where A 1 = { a ∈ − R 2 : a / ∈ I ( ˙ R 2 ) } , B 1 = { a ∈ − R 2 : a ∈ I ( ˙ R 2 ) } ∪ R 2 , A 2 = {− R 2 ∪ { a ∈ R 2 : a ∈ I ( ˙ R 2 ) } , B 2 = { a ∈ R 2 : a / ∈ I ( ˙ R 2 ) } , are alw a ys glued, since then S i = { γ ∈ Γ : γ ≤ 0 } for i = 1 , 2 and U = { γ ∈ Γ : γ > 0 } . R emark 2.6 . The orders P + a and P − a determine the same R -place, s ince U = ∅ . Also P + ∞ and P − ∞ determine the same R -place, since S = ∅ . R emark 2. 7 . If R is a real closed sub field of R , then eve ry cut of R is n ormal. By Theorem 2.4 the sp ace M ( R ( X )) is homeomorphic to M ( R ( X )). The ab ov e remark can b e easily generalized if one r eplace R and R b y an y real closed field K and its con tin uous closur e ˜ R . Let u s recall a definition fr om [1]. Definition 2.8. The ord ered field K is called c ontinuous close d if ev ery normal cut in K is pr incipal. W e sa y that an ord ered field ˜ K is a c ontinuous closur e of K if ˜ K is con tin uous closed and K is den se in ˜ K . The con tin uous closur e ˜ K is un iquely determined for eve ry ordered field K . Moreo ve r, if K is r eal closed, then ˜ K is also real closed (see [1]). Theorem 2.9. L et R b e a r e al close d field and let ˜ R b e i ts c ontinuous closur e. Then sp ac es M ( R ( X )) and M ( ˜ R ( X )) ar e home omorphic. SP A CES OF R - PLACES OF RA TIONAL FUNCT ION FIELDS 9 Pr o of. By [14 ], the r estriction map ω : M ( ˜ R ( X )) − → M ( R ( X )), ω ( ξ ) = ξ | R ( X ) is cont inuous. It suffices to show that it is a bijection. Fix ξ ∈ M ( R ( X )). There is P ∈ X ( R ( X )) suc h that λ R ( X ) ( P ) = ξ . Let ( A P , B P ) b e a cut in R corresp onding to P . Let ˜ A = { ˜ a ∈ ˜ R : ∃ a ∈ A P ˜ a < a } and let ˜ B b e the completion of ˜ A in ˜ R . If ( ˜ A, ˜ B ) is a cut (i.e. if ( A P , B P ) is principial or abn ormal gap), then the ord er ˜ P corresp ondin g to ( ˜ A, ˜ B ) and restricted to R ( X ) is equal to P . O therwise, there exists a u nique elemen t ˜ c ∈ ˜ R such that ∀ a ∈ ˜ A ∀ b ∈ ˜ B a < ˜ c < b. Then ( ˜ A ∪ { ˜ c } , ˜ B ) is a cu t in ˜ R . Let ˜ P b e an order corresp ond ing to this cut. Observe , that ˜ P ∩ R ( X ) = P . In fact, if f ∈ R ( X ) is p ositiv e on some neigh- b orho o d of ( ˜ A ∪ { ˜ c } , ˜ B ), th en it tak es p ositiv e v alues on some n eigh b orhoo d of ( A P , B P ) (note that ˜ c is not algebraic o v er R since R is real closed). Th en in b oth cases, λ ˜ R ( X ) ( ˜ P ) | R ( X ) = ξ . T h us ω is surjectiv e. T o s ho w injectivit y of ω it suffices to observe the follo win g. Let ˜ Γ b e a v alue group of v aluation v corresp onding to the unique ord ering of ˜ R . Let ˜ A ⊂ ˜ Γ. Assume that v ( ˜ c ) ∈ ˜ A for some ˜ c ∈ ˜ R . By density of R in ˜ R in an y sufficien tly small neighborh o o d of ˜ c one can fi nd c ∈ R s uc h that v ( c ) = v ( ˜ c ). Thus { v ( c ) : c ∈ R } ∩ ˜ A = ˜ A . No w one can use Theorem 2.4 (i) and (ii).  3. Comp leteness a nd c ontinuity Notation 3.1. Let k b e a field, Γ b e a lin early ordered ab elia n group. The field of (generalized) p o w er series k ((Γ)) is the s et of formal series a = X γ ∈ Γ a γ x γ with we ll - ordered supp ort supp ( a ) = { γ : a γ 6 = 0 } . Sum and multiplicat ion are defin ed as follo w: a + b = X γ ∈ Γ ( a γ + b γ ) x γ ; ab = X γ ∈ Γ ( X δ + η = γ a δ b η ) x γ The fact that k ((Γ)) is a field was sh o wn by Hahn [8 ]. This field is ordered by lexicographic order. The natural v aluation v : k ((Γ)) − → Γ ∪ {∞} corresp ond ing 10 MICHA L M ACHURA AND KA T ARZYNA OSIAK to this order is giv en by formula v ( a ) = m in s upp ( a ) . Let R b e real closed field, v : R − → Γ ∪ {∞} b e the natural v aluation of R with the Arc himedean resid ue field k . The pro of of the follo win g theorem one can find in [17 ], compare[12 ]. Theorem 3.2. Ther e exists a field emb e dding R ֒ → k ((Γ)) pr eserving the or der and valuation. In furth er part of this c h apter we will use the notion of uniform sp aces (see [7], c h .8) and related notions: base of the uniform it y ([7], Chapter 8.1 ); completeness, Cauc h y and conv ergen t nets ([7], Chapter 8.3). One of examples of uniform spaces is a fi eld K with a v aluation v : the base of uniformity is th e family { V γ : γ ∈ Γ } where V γ = { ( a, b ) ∈ K × K : v ( a − b ) > γ } for ev ery γ ∈ Γ, and Γ is a v alue group of v . Let us r ecall defin itions of unif orm notions in particular case of a field K (with v aluation v and v alue group Γ). Definition 3.3. K is c omplete if ev ery cen tered f amily of closed sets, wh ic h con tains arbitrarily small sets, h as one-p oin t int ersection. A f amily F con tains arbitr arily smal l sets if ∀ γ ∈ Γ ∃ F ∈F ∀ x,y ∈ F v ( x − y ) > γ Definition 3.4. W e sa y that a n et ( a σ : σ ∈ Σ) is a Cauc h y set if ∀ γ ∈ Γ ∃ σ 0 ∈ Σ ∀ σ>σ 0 v ( a σ − a σ 0 ) > γ Similarly , w e say that a net ( a σ : σ ∈ Σ) is con verge nt to a ∈ K if ∀ γ ∈ Γ ∃ σ 0 ∈ Σ ∀ σ>σ 0 v ( a σ − a ) > γ Theorem 3.5 ([7], Theorem 8.3.20) . A uniform sp ac e X is c omplete if and only if every Cauchy net in X is c onver gent. SP A CES OF R - PLACES OF RA TIONAL FUNCT ION FIELDS 11 Let k ((Γ)) b e form al p o w er series field, where k is Arc himedean ordered. Let κ b e cardin al num b er whic h is cofinalit y of Γ. Consid er a s ubfield k κ ((Γ)) = { a ∈ k (( Γ)); | sup p ( a ) | < κ or supp ( a ) is cofinal in Γ of ord er t yp e κ } . Claim 3.6. k κ ((Γ)) is c omplete. Pr o of. Let ( a σ : σ ∈ Σ) b e a Cauc h y net, w here a σ = P γ ∈ Γ a σ γ x γ and Σ is directed set. Fix cofin al sequence ( γ δ : δ < κ ) in Γ. Since ( a σ : σ ∈ Σ) is a Cauc h y net, we ha v e ∀ δ<κ ∃ σ δ ∀ τ >σ δ v ( a σ δ − a τ ) ≥ γ δ +1 The condition v ( a σ δ − a τ ) ≥ γ δ +1 means that a σ δ q = a τ q for ev ery q < γ δ +1 . Put a ∞ q = ( a σ δ q q ∈ [ γ δ , γ δ +1 ) a σ 0 q q < γ 1 . It is easy to see that a ∞ = P γ ∈ Γ a ∞ γ x γ is a limit o f giv en net. In fact, it is sufficien t to ob serv e that a ∞ q = a σ δ q = a τ q for eve ry q < γ δ +1 .  No w w e will sho w how completeness of an ordered field imp lies its conti nuit y . Theorem 3.7. L et ( K , P ) b e an or der e d field with valuation v determine d by P . If K is c omplete, then K is c ontinuous. Pr o of. Consider a n ormal Dedekind cut ( A, B ) in K . W e will s ho w that it is a principal cut. By defin ition of a normal cut for ev er y γ ∈ Γ ther e are a γ ∈ A and b γ ∈ B su c h that v ( a γ − b γ ) > γ . The interv als [ a γ , b γ ] are closed in u niform top ology and v ( x − y ) > γ for an y x , y ∈ [ a γ , b γ ]. Moreo ver, the family F = { [ a γ , b γ ] : γ ∈ Γ } is cente red. By a definition of completeness there is a uniqu e c ∈ K su c h that \ F = \ γ ∈ Γ [ a γ , b γ ] = { c } . T o prov e that a ≤ c for ev ery a ∈ A assume the op p osite: c < a f or some a ∈ A . Let γ 0 = v ( c − a ). There is b ∈ B such that v ( a − b ) > γ 0 . Since a family F ∪ { [ a, b ] } is cen tered we ha v e also c ∈ \ ( F ∪ { [ a, b ] } ) ⊂ [ a, b ] , 12 MICHA L M ACHURA AND KA T ARZYNA OSIAK whic h is a con tr adiction. Similarly one can pro v e that c ≤ b for every b ∈ B .  Example 3.8. Let R b e a fi xed real closure of R ( X ). Then Γ = Q . Let R ( Q ) ⊂ R (( Q )) b e the set of formal p ow er series with fi nite su pp ort. W e hav e R ( Q ) ⊂ R ⊂ R ℵ 0 (( Q )) . By Theorem 3.7, R ℵ 0 (( Q )) is con tin uous. Note that R ( Q ) is den se in R ℵ 0 (( Q )). Th us R ℵ 0 (( Q )) is the contin uous closure of R . 4. The sp ace M ( R ( Y )) T o describ e the to p ology of the space M ( R ( Y )), wh ere R is a fixed rea l closure of R ( X ), w e shall mo dify some metho ds of [17]. Let R b e any real closed field and let P b e an order of th e field R ( Y ) deter- mining the v aluation v with v alue group Γ and th e Arc himedean r esidue field k . Let ( A P , B P ) b e a cut in R corresp onding to P and let S b e the lo w er cut in Γ determined by ( A P , B P ) (see section 2). A formal series ˜ p = P γ ∈ Γ ˜ p γ x γ ∈ k ((Γ)) is d efined in the follo wing w a y (see [17]): (1) If γ / ∈ S then ˜ p γ = 0. (2) Su pp ose that γ ∈ S bu t γ is not a maximal elemen t in S . Then there exists a ∈ A P and b ∈ B P suc h that v ( b − a ) > γ . Th en ˜ p γ = a γ = b γ ( ˜ p γ do es not dep end fr om the c h oise of a and b - see [17], Prop.2.2.3). (3) Su pp ose that γ is a maximal elemen t in S . Let M = { a γ ∈ A P ; ∃ b ∈ B P v ( b − a ) > γ } and N = { b γ ∈ B P ; ∃ a ∈ A P v ( b − a ) > γ } . Then there exists exactly one r ∈ R suc h th at M ≤ r ≤ N (see [17], Prop.2.2.3). Then ˜ p γ = r . R emark 4.1 . A series ˜ p do es not d etermine an order uniquely . F o r example, if R is non - Arc himedean real closed fi eld, then f ormal series ˜ p constantly equal to 0 corresp ond to orders giv en b y f ollo wing cu ts: • the prin cipal cuts in 0; • gaps b etw een infi nitely small and other Arc himedean element s; • gaps b etw een Archimedean and infin itely large elemen ts. SP A CES OF R - PLACES OF RA TIONAL FUNCT ION FIELDS 13 T o d istinguish ord ers one h as to consider the low er cuts S a nd a sign defined b elo w. Consider thr ee cases: (+) There exists a ∈ A P suc h that v ( a − ˜ p ) / ∈ S. Then tak e a sym b ol ( S, ˜ p, +) (-) Th ere exists b ∈ B P suc h that v ( b − ˜ p ) / ∈ S. T hen take a symbol ( S, ˜ p, − ) (.) F or ev er y c ∈ R , v ( c − ˜ p ) ∈ S. Then tak e a sym b ol ( S, ˜ p ) R emark 4.2 . In the case R = R ℵ 0 (( Q )) t he last case do es not hold, because ˜ p ∈ R . R emark 4.3 . ([17] p.33) T he elemen t ˜ p defined ab ov e has prop erties: (1) ∀ γ ∈ S ∃ a ∈ R v ( a − ˜ p ) ≥ γ ; (2) ∀ γ / ∈ S ˜ p γ = 0. Theorem 4.4 ([17], Theorem 2.2.6) . Ther e is one - to - one c orr esp ondenc e b etwe en or ders of the field R ( Y ) and the symb ols ( S, ˜ p, +) , ( S, ˜ p, − ) and ( S, ˜ p ) for ˜ p ∈ R ((Γ)) satisfying c onditions (1), (2) of R emark 4.3. No w we shall restrict to the case when R is a fix ed real closure of R ( X ). Let (( X )) b e a set of s eries of the form p = P p γ x γ , where γ ∈ Q ∪ {∞} , p γ ∈ R ∪ {±∞} having f ollo wing prop erties: (1) p ∞ ∈ {±∞} ; (2) the s upp ort supp ( p ) = { γ : p γ / ∈ { 0 , ±∞}} is finite or co final in Q of order t yp e ℵ 0 ; (3) if p γ = ε ∞ , then p δ = ε ∞ , for ev ery δ > γ and ε ∈ { + , −} . Note that the map ( S, ˜ p, ε ) 7− → p ∈ (( X )), wher e p γ =  ˜ p γ , for γ ∈ S ε ∞ for γ / ∈ S is a bijection. Th is fact and Th eorem 4.4 leads to follo w ing corollary . Corollary 4.5. Ther e is one - to - one c orr esp ondenc e b etwe en the or ders of the field R ℵ 0 (( Q ))( Y ) and (( X )) . W e consid er the top ology of lexicographic order on (( X )). Prop osition 4.6. (( X )) is home omorphic to R ℵ 0 (( Q ))( Y ) . 14 MICHA L M ACHURA AND KA T ARZYNA OSIAK Pr o of. T ak e the H arisson set H ( f g ) = H ( f g ) ⊂ X ( R ℵ 0 (( Q ))( Y )). The p olynomial f g tak es p ositive v alues of finite m an y interv als. An order P b elongs to H ( f g ) if and only if corresp onding cut ( A P , B P ) has neighbou rho o d on wh ic h f g take s p ositiv e v alues. So if f g tak es p ositiv e v alues on interv al ( a, b ) a v ery cut ( A P , B P ) of ( a, b ) gives an order P whic h b elongs to H ( f g ). Additionally we should chec k what happ ens at the ends a = P a γ x γ and b = P a γ x γ of the in terv al ( a, b ). Let a − ≺ a + ≺ b − ≺ b + b e a series fr om (( X )) corresp onding to th e p rincipal cuts in a and b . Note that a + and b − b elongs to H ( f g ). Th us ( a − , b + ) ⊂ H ( f g ). In other w ords H ( f g ) is a fin ite sum of in terv als ( a − , b + ) ⊂ (( X )). suc h that f g is p ositiv e on ( a, b ) ⊂ R ℵ 0 (( Q )). F rom the other side, note that an in terv al ( p, q ) ⊂ (( X )), can b e replaced by the sum of Ha rrison sets H ( f ), where f is runn ing through the all quadr atic p olynomials with ro ots a, b ∈ R ℵ 0 (( Q )) su c h that p  a + ≺ b −  q and p ositive on ( a, b ).  Corollary 4.7. The sp ac e of R - plac es of the fie ld R ℵ 0 (( Q ))( Y ) is home omor phic to the sp ac e (( M )) = ( X p γ x γ ; γ ∈ Q , p γ ∈ R ∪ {∞} ) of series satisfying fol lowing pr op erties: (1) supp ( p ) is finite or c ofinal i n Q of or der typ e ℵ 0 ; (2) if p γ = ∞ , then p δ = ∞ for every δ > γ . with the quotient top olo gy fr om (( X )) . Pr o of. By the construction of elements of (( X )) and Theorem 2.4 we ha v e that p = P p γ x γ , q = P q γ x γ ∈ (( X )) d etermine the same R - p lace if and only if p γ = q γ when one of them is a real num b er.  Lemma 4.8. The c e l lularity of (( M )) i s not smal ler then c ontinuum c . Pr o of. W e will define a family of parwise disjoint op en sets of card inalit y c . Let t ∈ R and let U t b e a set whic h con tains all series c t with prop ertie s: · c t γ = 0 for γ < 1 , γ 6 = 0 · c t 0 = t · c t 1 ∈ ( − 1 , 1) SP A CES OF R - PLACES OF RA TIONAL FUNCT ION FIELDS 15 Observe that: (1) U t is nonempty . (2) U t is op en, b ecause its inv erse image in (( X )) is an in terv al ( a t , b t ), where a t γ =    c t γ for γ < 1 − 1 for γ = 1 + ∞ f or γ > 1 b t γ =    c t γ for γ < 1 1 for γ = 1 −∞ for γ > 1 . (3) F o r t < s , ( a t , b t ) < ( a s , b s ) and by using Theorem 2.4 w e ha ve that U t ∩ U s = ∅ .  Corollary 4.9. (( M )) i s not metrizable. Pr o of. Since cellularit y of any space is not greater than density , (( M )) can not b e separable. Since every compact, metric space is separable, (( M )) can n ot b e metrizable.  Using Theorem 2.9, Examp le 3.8, Corollary 4.7 an d Corollary 4.9 we get: Corollary 4.10. L et R b e a fixe d r e al closur e of R ( X ) . The sp ac e M ( R ( Y )) is not metrizable. Corollary 4.11. The sp ac e M ( R ( X, Y )) is not metrizable. Pr o of. Let R b e a fixed real closure of R ( X ). M ( R ( Y )) is a s ubspace of M ( R ( X , Y )) (see [5, Lemm a 8]).  Ac knowledgmen t . W e th ank Prof. A. S ladek and Prof. M. Kula who sp en t time on r eading th is pap er and ga v e us v aluable commen ts. Referen ces [1] R. Baer, Ditche, Ar chime dizit¨ at und Starrheit ge or dneter K¨ orp er , Math. Ann. 188 (1970), 165 - 205. [2] E. Becke r, D. Gondard, Notes on the sp ac e of r e al plac es of a formal ly r e al field , R eal Analytic and Algebraic Geometry , W. de Gruyter (1995), 21-46. [3] R. Bro wn, R e al plac es and or der e d fields , Rocky Mount. J.Math. 1 (1971), 633-636. [4] R. B row n, M. Marshall , The r e duc e d the ory of quadr atic f orms , Ro c ky Moun t. J.M ath. 11 (1981), 161-175 . [5] T.C. Crav en, The top olo gic al sp ac e of or derings of r ational function field , Du ke Math. J. 41 , (1974), 339 - 347. [6] D.W. Dub ois, Infinite primes and or der e d field , D issertationes Math. 69 , (1970), 1-43. [7] R. Engelking, Gener al T op olo gy , PWN, W arsa w, (1977) [8] H.Hahn , ¨ Ub er die nichtar chim e dischen Gr¨ ossensyst eme , Sitzun gsberich te d er Kaiser- lic hen Ak ademie d er Wissenschaften, Vienna, Section I Ia, 116 (1907), 601 - 653 16 MICHA L M ACHURA AND KA T ARZYNA OSIAK [9] J. H arman, Chains of higher level or derings , Con temp. Math. 8 , ( 1982), 141-174. [10] R . Gilmer, Extension of an or der to a simple tr ansc endental extension , Contemp. Math. vol.8 , (1982), 113 - 118. [11] D.Gondard, M. Marshall T owar ds an abstr act description of the sp ac e of r e al plac es , Con temp. Math. 253 , (2000), 77 - 113. [12] I. Kaplansky , Maximal fields with valuations , Duke Math. J. 9 , (1942), 303 - 321. [13] T.Y. Lam, Or derings, valuations and quadr atic forms , CBMS Regional Conf. Ser. Math., 52. Published for the Conf. Board of th e Math. Sciences, W ashington, D.C., by AMS, 1983 [14] K. Osiak, The Bo ole an sp ac e of higher level or derings , F und . Math. 196 (2007), 101 - 117. [15] K. Osiak, The Bo ole an sp ac e of r e al plac es , submitted. [16] H . W. Sch¨ ulting, On r e al plac es of a field and their holomorphy ring , Comm. Alg. 10 , (1982), 1239-1284. [17] M. Zeka v at, Or derings, cuts and formal p ower series , PhD. Thesis, Univ. of Sask atchew an, 2001. Institute of Ma thema tics, Silesian Unive rsity, Ba nk o w a 14, 40-007 Ka towice, Poland E-mail addr ess : kosiak@ux2.ma th.us.edu.pl, machura@ux2.math.u s.edu.pl