Abstract. The notion of relative closure (X, Y)0 of a semi-Pfaffian couple (X, Y) was introduced by Gabrielov to give a description of the o-minimal structure generated by Pfaffian functions. In this paper, an effective bound is given for the number of con
NUMBEROFCONNECTEDCOMPONENTSOFTHERELATIVECLOSURE...5Theorem12.ThestructureSiso-minimal.Moreover,ifXisade nablesetde nedbyaformulainvolvingthelimitsetsL1,...,LN,Xcanbepresentedasalimitsetwhoseformatisboundedbyane ectivefunctionoftheformatsofL1,...,LN.
2.Connectedcomponentsofalimitset
Inthissection,weestablishe ectiveboundsonthenumberofconnectedcom-ponentsoftherelativeclosureofasemi-Pfa ancouple(X,Y).Inwhatfollows,wewillalwaysassumethatYisnotempty.NotethatifYisempty,itfollowsfromDe nition8that( X)+mustbeemptytoo;then,XλiscompactandthenumberofconnectedcomponentsofX0isatmostthenumberofconnectedcomponentsofXλ.E ectiveboundsforthiscaseappearin[12,22].
Asannouncedintheintroduction,weassumeasin[6]thatXandYarerelativelycompact.Tokeeptheestimatessimple,we’llalsoassumethatthePfa anchain(f1,...,f )isde nedoverthewholeofRn×R+,sothattheboundgiveninTheorem3isapplicable.
2.1.Generalconsiderations.Weshowherehowtoreducetheproblemofcountingthenumberofconnectedcomponentsofalimitsettoaprobleminthesemi-Pfa ansetting.
LetΦbethe(squared)distancefunctiononRn×Rn:
Φ:Rn×Rn →R(7)(x,y)→|x y|2
Foranyλ>0,wecande nethedistancetoYλ,ΨλonXλby:
(8)Ψλ(x)=minΦ(x,y).y∈Yλ
ˇ:De nesimilarlyforx∈X
(9)Ψ(x)=minΦ(x,y).ˇy∈Y
Thefollowingresultappearsin[6,Theorem2.12].Theproofispresentedhereforthesakeofself-containment.
Theorem13.Let(X,Y)beasemi-Pfa ancouple.Then,thereexistsλ 1suchthatforeveryconnectedcomponentCof(X,Y)0,wecan ndaconnectedcomponentDλofthesetoflocalmaximaofΨλsuchthatDλisarbitrarilycloseto
C.
Proof.LetCbeaconnectedcomponentof(X,Y)0.Notethatbyde nitionˇSowemusthaveΦ(x,y)>0oftherelativeclosure,ifxisinC,itcannotbeinY.ˇ,andsinceYˇiscompact,wemusthaveΨ(x)>0.Also,anypointinforally∈Yˇ,henceΨ| C≡0.ˇbutnotin(X,Y)0.Sowemusthave C Y CmustbeinX,
ThismeansthattherestrictionofΨtoCtakesitsmaximuminsideofC.
Choosex0∈C,andletc=Ψ(x0)>0.Forasmallλ,thereisapointxλ∈Xλclosetox0suchthatcλ=Ψλ(xλ)isclosetoc,andisgreaterthanthemaximumofthevaluesofΨλoverpointsofXλcloseto C.Hencetheset{x∈Xλ|Ψλ(x)≥cλ}isnonempty,andtheconnectedcomponentAλofthissetthatcontainsxλisclosetoC.Thereexistsalocalmaximumx λ∈AλofΨλ.IfDλistheconnectedcomponentinthesetoflocalmaximaofΨλ,itiscontainedinZλandisclosetoC.