Closure and Connected Component of a Planar Global Semianalytic Set Defined by Analytic Functions Definable in O-minimal Structure
aa r X i v : . [ m a t h . L O ] F e b CLOSURE AND CONNECTED COMPONENT OF A PLANARGLOBAL SEMIANALYTIC SET DEFINED BY ANALYTICFUNCTIONS DEFINABLE IN O-MINIMAL STRUCTURE
MASATO FUJITA
Abstract.
We consider a global semianalytic set defined by real analytic func-tions definable in an o-minimal structure. When the o-minimal structure ispolynomially bounded, we show that the closure of this set is a global semi-analytic set defined by definable real analytic functions. We also demonstratethat a connected component of a planar global semianalytic set defined by realanalytic functions definable in a substructure of the restricted analytic field isa global semianalytic set defined by definable real analytic functions. Introduction
A semianalytic set is defined as a subset of a real analytic manifold that is,at each point of the manifold, a finite union of sets defined by finite inequalitiesand equalities of real analytic functions defined on a neighborhood of the point. Lojasiewicz [9] proved that the closure and a connected component of a semianalyticset are also semianalytic. A subset of a real analytic manifold is called globalsemianalytic if it is a finite union of sets defined by finite inequalities and equalitiesof real analytic functions defined on the whole manifold. It is not yet known whethera connected component of a global semianalytic set is itself global semianalytic,except in some specific cases [2, 7].We want to consider this problem in the o-minimal setting. Consider an o-minimal expansion ˜ R of the real field. (See [16] for the theory of o-minimal struc-tures.) A global semianalytic set defined by real analytic functions definable in theo-minimal structure ˜ R takes the form: k [ i =1 { x ∈ R n | f i ( x ) = 0 , g i ( x ) > , . . . , g il ( x ) > } ,where f i and g ij are definable real analytic functions on R n . In this paper, we referto such sets as being globally ˜ R -definable semianalytic or globally definable semian-alytic . A global semianalytic set that is simultaneously definable in an o-minimalstructure is not necessarily a globally definable semianalytic set, an example ofwhich is shown below.Consider the restricted analytic field R an . A subset of R n is definable in R an ifand only if its closure in the n -dimensional projective space P n ( R ) is subanalytic.(See [3] for the definition of a subanalytic set.) Here, we identify R n with an opensubset of P n ( R ) under the identification given by R n ∋ ( x , . . . , x n ) (1 : x : . . . : x n ) ∈ P n ( R ). Mathematics Subject Classification.
Primary 03C64; Secondary 14P15.
Key words and phrases. o-minimal structure; global semianalytic set.
Set X = { ( t, sin( t )) | ≤ t ≤ π } . Then, X is a compact global semianalytic set;furthermore, X is definable in R an . However, its analytic closure is the sine curvethat intersects with the real line { ( t, t ∈ R } at infinitely many points. Theanalytic closure of a globally definable semianalytic set intersects with the real lineat a finite number of points. The set X is simultaneously definable in R an andglobal semianalytic, but it is not a globally definable semianalytic set.In this paper, o-minimal structure indicates an o-minimal expansion of the realfield. We show that the closure of a globally definable semianalytic set is a globallydefinable semianalytic set in the case where the o-minimal structure is polynomiallybounded. (See [12, 15] for the definition of polynomially bounded o-minimal struc-tures.) We also show that a connected component of a planar globally definablesemianalytic set is again a globally definable semianalytic set when the o-minimalstructure is a substructure of the restricted analytic field.2. Closure and Finiteness Theorem
In this section, we demonstrate that the closure of a globally definable semian-alytic set is a globally definable semianalytic set when the o-minimal structure ispolynomially bounded with a proof based on Gabrielov’s work [8, Lemma 1].
Theorem 2.1.
Consider a polynomially bounded o-minimal structure. Let X be aglobally definable semianalytic subset of R n . Then, the closure X is again a globallydefinable semianalytic set.Proof. We may assume that X = { x ∈ R n | f ( x ) = 0 , g ( x ) > , . . . , g k ( x ) > } without loss of generality. Here, f ( x ) and g ( x ) , . . . , g k ( x ) are definable real analyticfunctions on R n .There exists a positive integer q > Y x,c = { y ∈ X | g ( y ) > c k x − y k q , . . . , g k ( y ) > c k x − y k q } ⊂ R n contains x if and only if x ∈ X for any positive real number c . Here, k x − y k denotesthe Euclidean norm of element x − y in R n . Obviously, for any positive integer q ,the condition that x ∈ X implies that x ∈ Y x,c ; likewise, x Y x,c if x X because Y x,c ⊂ X .Finally, we consider the case in which x ∈ ∂X , where ∂X denotes the boundaryof X defined by ∂X = X \ X . Consider the positive definable function defined by δ ( x, t ) = max y ∈ X, k x − y k = t (cid:18) min ≤ i ≤ k g i ( y ) (cid:19) for any x ∈ ∂X and any sufficiently small t >
0. We show that the function δ ( x, t )is well-defined and positive. Set g ( y ) = min ≤ i ≤ k g i ( y ), which is a continuous definablefunction on R n and positive on X . Set X x,t = { y ∈ X | k x − y k = t } for x ∈ ∂X and a sufficiently small t >
0; then, g ( y ) = 0 for any y ∈ X x,t \ X x,t . Since X x,t is compact, the restriction of g to X x,t has the maximum value, which is positivebecause g is positive on X x,t . Hence, the restriction of g to X x,t has the positivemaximum value at a point in X x,t . We have shown that the function δ ( x, t ) iswell-defined and positive for any x ∈ ∂X and any sufficiently small t > LOBAL SEMIANALYTIC SETS DEFINED BY DEFINABLE FUNCTIONS 3
There exists q ∈ N with δ ( x, t ) > ct q for any positive real number c and anysufficiently small t > y t ∈ Y x,c with k x − y t k = t for any sufficiently small positive real number t . Therefore, the closureof Y x,c contains x if x ∈ ∂X .Let ˆ g i : R n × R n → R be the definable function such that ˆ g i ( x, y ) is the Taylorexpansion of g i ( y ) of order q at x . That is, we setˆ g i ( x, y ) = X µ ∈ ( N ∪{ } ) n , | µ |≤ q ∂ | µ | g i ( x ) ∂ µ X · · · ∂ µ n X n ( y − x ) µ · · · ( y n − x n ) µ n for µ = ( µ , . . . , µ n ), x = ( x , . . . , x n ) ∈ R n and y = ( y , . . . , y n ) ∈ R n . Here, | µ | = P ni =1 µ i and X , . . . , X n are the coordinate functions of R n . Set S x,c = { y ∈ R n | ˆ g i ( x, y ) > g i ( x, y ) ≥ c k y − x k q for all i = 1 , . . . , k } .We show that the closure of the set S ′ x,c := S x,c ∩ { y ∈ R n | f ( y ) = 0 } contains x if and only if x ∈ X .Assume first that x ∈ X . Therefore, x ∈ Y x, c and there exists a sequence { y ν } ⊂ Y x, c with y ν → x . Since ˆ g i ( x, y ) is the Taylor expansion of g i ( y ) ofthe order q at x , there exists C > | g i ( x ) − ˆ g i ( x, y ) | ≤ C k x − y k q +1 in aneighborhood of x . If y is sufficiently close to x , c k x − y k q > C k x − y k q +1 . Hence,we have ˆ g i ( x, y ) ≥ c k y − x k q if y ∈ Y x, c and y is sufficiently close to x , and y ν ∈ S ′ x,c for sufficiently large ν >
0. Therefore, we have x ∈ S ′ x,c . We next consider the casein which x ∈ S ′ x,c . We can show that x ∈ Y x, c in the same way as above. Hence, x ∈ X .We next demonstrate that there exists a positive integer q > Y ′ x,c,c ′ = { y ∈ S x,c | | f ( y ) | ≤ c ′ k x − y k q ′ } contains x if and only if x ∈ X for any positive real number c ′ . Note that S ′ x,c ⊂ Y ′ x,c,c ′ , so the closure of Y ′ x,c,c ′ contains x if x ∈ X . We next show that x Y ′ x,c,c ′ if x X , a claim that is obvious in the case in which f ( x ) = 0. We assume that f ( x ) = 0. If x S x,c , then we have x S ′ x,c , x X , and x Y ′ x,c,c ′ , in which casethe claim is also true. It is also obvious that the closure of Y ′ x,c,c ′ does not contain x if g i ( x ) < ≤ i ≤ k . Therefore, we have only to consider the case inwhich x ∈ S x,c and f ( x ) = min ≤ i ≤ k g i ( x ) = 0.Set V = { x ∈ R n | f ( x ) = min ≤ i ≤ k g i ( x ) = 0 , x ∈ S x,c } \ X . V is a definable set.Consider the definable function defined by δ ′ ( x, t ) = min y ∈ S x,c , k y − x k = t | f ( y ) | for any x ∈ V and sufficiently small t >
0. Set S x,c,t = { y ∈ S x,c | k y − x k = t } = { y ∈ R n | k y − x k = t, ˆ g i ( x, y ) ≥ c k y − x k q ( ∀ i ) } , which is a compact set.Because | f ( y ) | is continuous, the restriction of | f ( y ) | to S x,c,t has the minimumvalue. We have shown that function δ ′ ( x, t ) is well-defined. We next establish that M. FUJITA δ ′ ( x, t ) is positive for any x ∈ V and sufficiently small number t > x ∈ V such that δ ′ ( x, t ) = 0 for any sufficiently small t >
0. Then, there also exists a sequence { y ν } ⊂ S x,c with y ν → x and f ( y ν ) = 0,which means that y ν ∈ S ′ x,c . We now have x ∈ S ′ x,c and x ∈ X , which contradictsthe assumption that x X .We can choose a positive integer q ′ > δ ′ ( x, t ) > c ′ t q ′ for any x ∈ V ,any positive real number c ′ , and any sufficiently small t > Y ′ x,c,c ′ does not contain x if x ∈ V .Let ˆ f : R n × R n → R be the definable function such that ˆ f ( x, y ) is the Taylorexpansion of f ( y ) of order q ′ at x . We can show that the closure of set T x,c,c ′ = { y ∈ S x,c | | ˆ f ( x, y ) | ≤ c ′ k y − x k q ′ } contains x if and only if x ∈ X , as in the case of S ′ x,c . (We omit this proof.)For a fixed x , T x,c,c ′ is a semialgebraic set. According to the quantifier elimi-nation theorem [4, Proposition 5.2.2], [13], the condition that the closure of T x,c,c ′ contains x is equivalent to a semialgebraic condition on coefficients of polynomialsin the variable y defining T x,c,c ′ . The coefficients are the partial derivatives of f and g i , which are definable and real analytic. Hence, X is a globally definablesemianalytic set. (cid:3) In semialgebraic geometry, a closed (resp. open) set is a finite union of basicclosed (resp. open) sets [4, Theorem 2.7.2]. A similar finiteness theorem follows:
Proposition 2.2 (Finiteness Theorem) . Consider a polynomially bounded o-minimalstructure. An open ( resp. closed ) globally definable semianalytic set S takes theform S = k [ i =1 l \ j =1 { x ∈ R n | f ij ( x ) > ( resp. ≥ ) 0 } Here, f ij ( x ) are definable real analytic functions on R n .Proof. We only prove the case in which S is open because by taking the complement,the closed case is clear.Set S is a finite union of global semianalytic sets of the form S ′ = { x ∈ R n | f ( x ) = 0 , g ( x ) > , . . . , g m ( x ) > } .Here, f ( x ) and g ( x ) , . . . , g m ( x ) are definable real analytic functions on R n . Let k x k denote the Euclidean norm of x ∈ R n . We may assume that | f ( x ) | ≤ √ k x k for any x ∈ R n by replacing f ( x ) with the bounded function f ( x ) √ (1+ f ( x ) )(1+ k x k ) ifnecessary. We may also assume that | g i ( x ) | ≤ √ k x k for all 1 ≤ i ≤ m .Let B be the open unit ball in R n centered at the origin, i.e., B = { x ∈ R n | k x k < } . Let D be the closed unit ball in R n centered at the origin, i.e., D = { x ∈ R n | k x k ≤ } . Consider the Nash diffeomorphism ϕ : R n → B given by ϕ ( x ) = (cid:18) x √ k x k , . . . , x n √ k x k (cid:19) for x = ( x , . . . , x n ). Note that ϕ ( S ) is open in R n . The definable functions F : D → R and G i : D → R are defined as follows: LOBAL SEMIANALYTIC SETS DEFINED BY DEFINABLE FUNCTIONS 5 F ( u ) = (cid:26) f ◦ ϕ − ( u ) if u ∈ B, G i ( u ) = (cid:26) g i ◦ ϕ − ( u ) if u ∈ B, | F ( u ) | ≤ p − k u k and | G i ( u ) | ≤ p − k u k for any u ∈ B . Hence, functions F and G i are continuous on D .Set g ( x ) = Q mi =1 ( | g i ( x ) | + g i ( x )); in this case, g ( x ) = 0 if f ( x ) = 0 and x S .Set G ( u ) = Q mi =1 ( | G i ( u ) | + G i ( u )). For the compact set D \ ϕ ( S ), we have G ( u ) = 0if F ( u ) = 0. For some positive integer N and e c >
0, by the Lojasiewicz inequality[12, 5.4], G N ≤ e c | F | on D \ ϕ ( S ). Hence, g N ≤ e c | f | on R n \ S . Then, set S ′′ = { e c f < (2 m Q mi =1 g i ) N , g > , . . . , g m > } is contained in S and contains S ′ . By replacing S ′ with S ′′ , we can show the proposition inductively. (cid:3) Separation
In this section, we show that a connected component of a planar globally defin-able semianalytic set is again a globally definable semianalytic set in the case inwhich the o-minimal structure is a substructure of the restricted analytic field. Webegin by illustrating several lemmas.A global analytic subset of a paracompact Hausdorff real analytic manifold is thecommon zero set of finite real analytic functions defined on the manifold. A globalanalytic set X is irreducible if it is not the union of two proper global analyticsubsets of X . (See [1, Chapter VIII] for global analytic sets.) Lemma 3.1.
Consider an o-minimal structure ˜ R that admits analytic decompo-sition. (See [14] for the definition of analytic decomposition.) Let X be a properglobal analytic subset of R that is definable in ˜ R . Then, one-dimensional irre-ducible global analytic subsets of X are definable, and at most a finite number ofthem exist.Proof. Because ˜ R admits analytic decomposition, set X is the union of finite cellsdefinable in ˜ R that are analytically diffeomorphic to Euclidean spaces. These cellssatisfy the following conditions: First, the intersection of two cells is empty. Second,the boundary of a cell is a union of cells of smaller dimension. Furthermore, theyare real analytic submanifolds of R because they are either points, or sets of theforms { ( x, y ) ∈ R | x = a, y ∈ I } and { ( x, y ) ∈ R | y = ξ ( x ) , x ∈ I } ,where a is a constant, I is an open interval in R , and ξ ( x ) is a definable realanalytic function defined on I . Hence, a one-dimensional cell is contained in theset of regular points of X .If two one-dimensional irreducible global analytic subsets of X intersect at point z , the ring of analytic function germs on X at z is not an integral domain. By [11,Theorem 20.3], it is also not a regular local ring. Hence, z is a singular point of X . A cell is contained in a single irreducible global analytic subset of X . Hence,a one-dimensional irreducible global analytic subset of X is a union of cells and isdefinable. Furthermore, the number of one-dimensional irreducible global analyticsubsets of X is finite. (cid:3) Lemma 3.2.
Consider closed semianalytic set germs A and B of R n at the origin O such that the intersection A ∩ B is the origin. There exists an open semialgebraic M. FUJITA subset C of R n satisfying the conditions that A ⊂ C ∪ { O } and B ∩ C = { O } as setgerms at the origin.Proof. There exists an open semianalytic set germ C ′ at the origin with A ⊂ C ′ ∪{ O } and B ∩ C ′ = { O } by [7, Lemma 2.4]. Let C ′ = S li =1 T mj =1 { f ij > } , where f ij ( x ) are real analytic functions defined on a neighborhood of the origin. By [7,Lemma 2.5], there exists a positive integer µ such that any semianalytic set germ C = S li =1 T mj =1 { g ij > } satisfies the conditions that A ⊂ C ∪{ O } and B ∩ C = { O } as set germs at the origin if f ij − g ij ∈ m µ , where m is the maximal ideal of the ringof real analytic function germs at the origin. We can choose g ij as polynomials andhence, we can choose C as a semialgebraic set. (cid:3) Lemma 3.3.
Consider an o-minimal substructure ˜ R of the restricted analytic field R an . Let C be a globally ˜ R -definable semianalytic subset of R . Let X and Y be ˜ R -definable subsets of R with X ∪ Y = C and X ∩ Y = ∅ . Assume further that thereexist a non-zero ˜ R -definable real analytic function g on R , a finite subset P of R ,and finitely many one-dimensional R an -definable irreducible global analytic subsets Z , . . . , Z m of R such that X ∩ Y ⊂ P ∪ ( S mi =1 Z i ) and the function g vanishes on Z i for all i = 1 , . . . , m . Then X and Y are both globally ˜ R -definable semianalytic.Proof. We simply call a ˜ R -definable set definable in this proof-by-induction on m .Set W = X ∩ Y .We first consider the case in which m = 0. In this case, W is either empty orconsists of a finite number of points. Let q be the cardinality of W . We show thelemma by induction on q . For the case in which q = 0, W is empty. Consider theNash map ϕ : R → S , defined by ϕ ( x ) = (cid:16) x k x k , x k x k , −k x k k x k (cid:17) . Here, S isthe sphere { ( x , x , x ) ∈ R | x + x + x = 1 } in R and x = ( x , x ). It is aNash diffeomorphism to the image S \ { S } , where S = (0 , , − X = ϕ ( X )and Y = ϕ ( Y ). Thus, W ′ = X ∩ Y is either empty or consists of a single point S . In the case in which W ′ is empty, let d X , d Y : R → R be the distance functionsto the closures X and Y , respectively. Define a continuous definable function d : R → R by d ( x ) = d Y ( x ) − d X ( x ) d X ( x ) + d Y ( x ) ; then, the restriction of d to X is 1 and therestriction of d to Y is −
1. The restriction of d to S is a continuous function definedon the compact set S . There exists a polynomial function P : R → R satisfyingthe condition that | P ( x ) − d ( x ) | < on S by the Stone-Weierstrass theorem. Wenow have C ∩ { x ∈ R | P ( ϕ ( x )) > } = X . Hence, X is a globally definablesemianalytic set. We can also show that Y is a globally definable semianalytic set.We next consider the case in which W ′ consists of a single point. Since the o-minimal structure is a substructure of the restricted analytic field R an , sets X and Y are both definable in R an . It is subanalytic by the definition of the restrictedanalytic field R an . Hence it is semianalytic by [3, Theorem 6.1]. Therefore, thereexists an open semialgebraic subset V of R such that X ⊂ V and Y ∩ V = ∅ in a neighborhood of the point S by Lemma 3.2. Set V = ϕ − ( V ), then it isa semialgebraic set. There exists a positive real number R such that C ∩ { x ∈ R | k x k > R } ∩ V = X ∩ { x ∈ R | k x k > R } . Hence X ∩ { x ∈ R | k x k > R } is aglobally definable semianalytic set. Applying the lemma for the case where W ′ isempty to the set C ∩ { x ∈ R | k x k ≤ R } , we can show that X ∩ { x ∈ R | k x k ≤ R } LOBAL SEMIANALYTIC SETS DEFINED BY DEFINABLE FUNCTIONS 7 is also globally definable semianalytic. We have shown that X = ( X ∩ {k x k ≤ R } ) ∪ ( X ∩ {k x k > R } ) is globally definable semianalytic. We have finished theproof of the lemma for m = 0 and q = 0.We next consider the case in which m = 0 and q >
0. Let p be a point in W . Remark that X and Y are semianalytic sets because a connected componentof a seminalaytic set is semianalytic by [3, Corollary 2.7, Corollary 2.8]. We canconstruct an open semialgebraic set U with X ⊂ U and Y ∩ U = ∅ in a neighborhoodof p by Lemma 3.2. The cardinality of the set X ∩ U ∩ Y ∩ U is smaller than that of W . The sets X ∩ U and Y ∩ U are globally definable semianalytic set by induction,as are X \ U and Y \ U . Hence, X = ( X ∩ U ) ∪ ( X \ U ) and set Y are globallydefinable semianalytic sets, finishing the proof of the lemma for m = 0.We finally consider the case in which m >
0. Let z be an arbitrary regularpoint of Z . Let O z be the ring of real analytic function germs on R at z . Let I Z ,z be the prime ideal of O z of the germs of real analytic functions vanishingon Z at z and is a principal ideal by [11, Theorem 20.1, Theorem 20.3]. Assumethat there exists an ˜ R -definable (resp. R an -definable) real analytic function ψ on R with ψ ∈ I kZ ,z for some positive integer k . Then, we can easily demonstratethat there exists an ˜ R -definable (resp. R an -definable) real analytic function ψ ′ on R with ψ ′ ∈ I k − Z ,z . In fact, let x , x be the coordinate functions of R . Thereexist real numbers a and b such that vector ( a, b ) is neither zero nor parallel to thetangent line T z Z of Z at z . Set ψ ′ = a ∂ψ∂x + b ∂ψ∂x . Thus, ψ ′ ∈ I k − Z ,z .Fix a regular point z of Z . Let A be the ring of real analytic functions on R definable in R an . Let m be the maximal ideal of A given by m = { f ∈ A | f ( z ) = 0 } .The ring A m is a two-dimensional regular local ring by [6, Remark 2, Proposition5] and its proof. Let p be the prime ideal of A given by p = { f ∈ A | f ( x ) =0 for all x ∈ Z } . Then, I = p A m is a principal ideal by [11, Theorem 20.1,Theorem 20.3]. Let h ∈ p be the generator of I . As a corollary of the above claim,we have h ∈ I Z ,z \ I Z ,z . There also exists an ˜ R -definable real analytic function f ∈ p with f ∈ I Z ,z \ I Z ,z by the same claim, because the ˜ R -definable function g is in I Z ,z . Since h is a generator of I , there exist v ∈ A and w ∈ A \ m with wf = vh . We have v p and h , f ∈ p \ p because f , h ∈ I Z ,z \ I Z ,z .Set W = { f > } ∩ X ∩ { f > } ∩ Y . Consider an arbitrary regular point z ∈ Z with w ( z ) = 0. Let h z be the generator of I Z ,z ; then, f = v z h z forsome v z ∈ O z \ I Z ,z because v p , h ∈ p \ p and Z is an irreducible analyticset. If v z ( z ) = 0, then W is empty in a neighborhood of z . Set T = Z ∩ (cid:16) v − (0) ∪ h − (0) \ Z ∪ w − (0) (cid:17) ∪ Sing( Z ), where Sing( Z ) denotes the singularlocus of Z . We can easily show that Sing( Z ) is a finite set as in Lemma 3.1. Thus,we have W ∩ Z ⊂ T . The set T is an R an -definable set of dimension smaller thanone. Hence, there exists a finite subset P ′ ⊂ R with W ⊂ P ′ ∪ ( S mi =2 Z i ). Both { f > }∩ X and { f > }∩ Y are globally definable semianalytic sets by induction.Using the same reasoning, { f < } ∩ X and { f < } ∩ Y are globally definablesemianalytic sets. By the lemma for m = 0, { f = 0 } ∩ X and { f = 0 } ∩ Y arealso globally definable semianalytic sets. We have therefore shown that X and Y are globally definable semianalytic sets. (cid:3) The following theorem is the main theorem in this section.
M. FUJITA
Theorem 3.4.
Consider an o-minimal substructure ˜ R of the restricted analyticfield. A connected component of a globally definable semianalytic subset of R isalso a globally definable semianalytic set.Proof. Let C be a globally definable semianalytic subset of R . We may assumethat C is of the form { x ∈ R n | f ( x ) = 0 , g ( x ) > , . . . , g l ( x ) > } ,where f and g j are definable real analytic functions on R n for any 1 ≤ j ≤ l . Let X be a connected component of C . Set Y = C \ X . Both X and Y are definablebecause a connected component is definable by [16, Proposition 2.18]. We show that X is a globally definable semianalytic set. Set W = X ∩ Y . The theorem followsfrom Lemma 3.3 if W is either empty or of dimension zero. We may assume that W is one-dimensional. The boundary of a definable set is of a dimension smaller thanthat of the original definable set by [16, Theorem 4.1.8]. Hence, X and Y are bothtwo-dimensional. We may assume that the real analytic function f is identically 0.We show that X and Y satisfy the conditions of Lemma 3.3. Set g ( x ) = Q li =1 g i ( x ); thus, it is a definable real analytic function. Consider the analyticset Z = { x ∈ R | g ( x ) = 0 } . The restricted analytic field admits analytic decom-position by [14, Theorem 8.8], [5, Theorem 4.6] and [10, Proposition 3.1.14]. Hencethere exist a finite subset P of R and finitely many one-dimensional R an -definableirreducible global analytic subsets Z , . . . , Z m of R such that Z = P ∪ ( S mi =1 Z i ),by Lemma 3.1.We show that X ∩ Y ⊂ Z . Taking a point z ∈ X ∩ Y , there exist a smallpositive real number ε > γ X : [0 , ε ) → R and γ Y : [0 , ε ) → R such that z = γ X (0) = γ Y (0), γ X ( t ) ∈ X and γ Y ( t ) ∈ Y for all0 < t < ε by the curve selection lemma [16, Corollary 6.1.5]. The functions g i ◦ γ X and g i ◦ γ Y are continuous and positive on the open interval (0 , ε ). Hence, we have g i ( z ) ≥
0. If g i ( z ) is positive for all i = 1 , . . . , m , then z is contained in C . In thiscase, there is a continuous path in C connecting X with Y , which contradicts theassumption that X is a connected component of C . We have shown that g i ( z ) = 0for some 1 ≤ i ≤ l . In particular, we have X ∩ Y ⊂ Z . Since X and Y satisfy theconditions of Lemma 3.3, X is a globally definable semianalytic set. (cid:3) References [1] C. Andradas, L. Br¨ocker and J.-M. Ruiz,
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