2MARIOJ.EDMUNDO,MARCELLOMAMINO,ANDLUCAPRELLIarbitraryo-minimalstructu - PDF document

2MARIOJ.EDMUNDO,MARCELLOMAMINO,ANDLUCAPRELLIarbitraryo-minimalstructures.See[9].Pillay'sconjectureisanon-standardana-logueofHilbert's5oproblemforlocallycompacttopologicalgroups,roughlyitsaysthataftertakingthequotientbya\smallsubgroup"(asmallesttype-denablesub-groupofboundedindex)thequotientwhenequippedwiththethesocalledlogictopologyisacompactrealLiegroupofthesamedimension.LetDefdenotethecategoryofdenablespacesandcontinuousdenablemaps.FromthecategorytheorydenitionofmorphismsproperinDefoneobtainsasin[19,ChapterII,Proposition5.4.2andCorollary5.4.3]thelistofthemostusefulpropertiesofsuchmorphisms(Proposition3.7)whichissimilartothecorrespondinglistofpropertiesforpropermorphismsinsemi-algebraicgeometry([6,Section9])(andalsoinalgebraicgeometry[20,ChapterII,Corollary4.8]or[19,ChapterII,Proposition5.4.2andCorollary5.4.3]).InTheorem3.12weproveadenablecurvescriterionfordenablyproperex-tendingwhatwasknowintheanecaseino-minimalexpansionsoforderedgroups([7,Chapter6,Lemma(4.5)]).FromthiscriterionweobtainthecorrespondinglistofthemostusefulpropertiesofdenablypropermapsinCorollary3.13.HoweverweneedtorelatethenotionofproperinDefandthenotionofdenablyproper.ThisisachievedinTheorem3.15whereweshowthatifMhasdenableSkolemfunctions,thenforHausdorlocallydenablycompactdenablespacesproperinDefisthesameasdenablyproper.UndertheassumptionthatMhasdenableSkolemfunctions,weprovethatde-nablecompactnessofHausdordenablespacescanbecharacterizedbytheexis-tenceoflimitsofdenabletypes(Theorem2.23),extendingaremarkbyHrushovskiandLoeser([22])intheanecase,and,inTheorem3.18weproveacorrespond-ingcharacterizationofdenablypropermapsbetweenHausdorlocallydenablycompactdenablespaceswhich,whentransferredtomorphismsproperinthecat-egoryofo-minimalspectralspaces,istheanalogueofthevaluativecriterionforpropernessinalgebraicgeometry([20,ChapterII,Theorem4.7]).Asitisknown,ino-minimalstructureswithdenableSkolemfunctions,denabletypescorrespondtovaluations([24]and[26]).InTheorems4.7and4.10weshowthatdenablyproperisinvariantunderele-mentaryextensionsando-minimalexpansionsofM:InTheorem4.12weshowthatifMisano-minimalexpansionoftheorderedsetofrealnumbers,thendenablypropercorrespondstoproper.Theseinvarianceandcomparisonresultstransfertothenotionofpropermorphisminthecategoryofo-minimalspectralspaces.Behindallourmaintheoremsabovearethefollowingtwotechnicalresults:The-orem2.12whichshowthatifMhasdenableSkolemfunctions,thenHausdorde-nablycompactdenablespacesaredenablynormal;Corollary2.20whichshowsalocalalmosteverywherecurveselectionforHausdorlocallydenablycompactdenablespaces.Theorem2.12wasonlyknowninspecialcases:itwasprovedbyBerarducciandOterofordenablemanifoldsino-minimalexpansionsofrealclosedelds([1,Lemma10.4]-theproofthereworksaswellino-minimalexpansionsoforderedgroups);itwasprovedin[15]fordenablycompactgroupsinarbitraryo-minimalstructures.Corollary2.20isanextensionofthealmosteverywherecurveselectionforclosedandboundeddenablesetsinarbitraryo-minimalstructuresprovedbyPeterzilandSteinhorn([25,Theorem2.3]). 4MARIOJ.EDMUNDO,MARCELLOMAMINO,ANDLUCAPRELLIthediagram(a;b)// _ CX[a;b] :: iscommutative.Denition2.2.LetXbeadenablespaceandCXadenablesubset.WesaythatCisdenablycompactifeverydenablecurveinCiscompletableinC(see[25]).Thefollowingiseasy:Fact2.3.SupposethatMhasdenableSkolemfunctions.Letf:X!Yacon-tinuousdenablemapbetweendenablespaces.IfKXisadenablycompactdenablesubset,thenf(K)isadenablycompactdenablesubsetofY.FordenablesubsetsXMnwiththeirinducedtopology(i.e.anedenablespaces)thenotionofdenablycompactisverywellbehaved.Indeed,wehave([25,Theorem2.1]):Fact2.4.AdenablesubsetXMnisdenablycompactifandonlyifitisclosedandboundedinMnHowever,ingeneral,unlikeinthetopologicalcase,denablycompactdenablesubsetsofadenablespacearenotHausdorandarenotevennecessarilyclosedsubsets:Example2.5(NonHausdorandnoncloseddenablycompactsubsets).Leta;b;c;d2Mbesuchthatcbad:LetXbethedenablespacewithdenablecharts(Xi;i)i=1;2givenby:X1=(fhx;yi2[c;d][c;d]:x=ygnfhb;big)[fhb;aigM2,X2=fhx;yi2[c;d][c;d]:x=ygM2andi=jXiwhere:M2!Mistheprojectionontotherstcoordinate.ThenanyopendenableneighborhoodinXofthepointhb;aiintersectsanyopendenableneighborhoodinXofthepointhb;bi:ClearlyXisdenablycompactbutnotHausdorandX2isadenablycompactsubsetwhichisnotclosed(inX).Itisdesirabletoworkinasituationwheredenablycompactsubsetsareclosed.WewillshowthatthisisthecaseinHausdordenablespaceswhenMhasden-ableSkolemfunctions.Beforeweneedtointroducesomenotations.LetXbeadenablespaceandlet(Xi;i)ikbethedenablechartsofXwithi(Xi)Mni:LetN=n1+


+nkandxapoint2M:Foreachik;leti:MN=Mn1


Mnk!Mnibethenaturalprojectionandleti:Mni!MN 6MARIOJ.EDMUNDO,MARCELLOMAMINO,ANDLUCAPRELLIIfBXisadenablesubsetand:B!M2Nisadenablemapsuchthat(x)2D(x)forallx2B,thenU(B;)=[x2BU(x;(x))isanopendenableneighborhoodofBinX:Itfollowsthat:Remark2.6.Thenotionsofopen(resp.closed)inadenablespaceXarerst-orderinthesensethatif(At)t2TisauniformlydenablefamilyofdenablesubsetsofX,thenthesetofallt2TsuchthatAtisanopen(resp.aclosed)subsetofXisadenableset.RecallthatatopologicalspaceXisregularifonethefollowingequivalentcon-ditionsholds:(1)foreverya2XandSXclosedsuchthata62S,thereareopendisjointsubsetsUandVofXsuchthata2UandSV;(2)foreverya2XandWXopensuchthata2W,thereisVopensubsetofXsuchthata2Vand VW.Proposition2.7.SupposethatMhasdenableSkolemfunctions.LetXbeaHausdordenablespace,a2XandKXadenablycompactsubset.Supposethata62K:Thenthereisadenablefunction:K!M2Nwith(x)2D(x)forallx2Kandthereisd2D(a)suchthat:KU(K;):U(a;d)\U(K;)=;:Inparticular,ifXisaHausdor,denablycompactdenablespace,thenXisregular.Proof.Firstweshowthefollowing:Claim2.8.TherearenitelymanydenablycompactsubsetsKi(i=1;:::;l)ofK;nitelymanycontinuousdenablefunctionsi:Ki!M2Nwithi(x)2D(x)forallx2Kiandthereisd2D(a)suchthat:K=Sli=1Ki:KSli=1U(Ki;i):U(a;d)\(Sli=1U(Ki;i))=;:Proof.WeprovetheresultbyinductionondimensionofK.IfdimK=0,thenthisfollowsbecauseXisHausdor.AssumetheresultholdsforeverydenablycompactsubsetLofXsuchthata62LanddimLdimK:SinceXisHausdor,foreachx2Kthereisd02D(x)andthereisd2D(a)suchthatU(a;d)\U(x;d0)=;:BydenableSkolemfunctionstherearedenablemapsg:K!M2Nandh:K!M2N 8MARIOJ.EDMUNDO,MARCELLOMAMINO,ANDLUCAPRELLIisadenablycompactdenableneighborhoodofK.Inparticularwehave [x2KU(x;(x))=[x2K U(x;(x)):Proof.Let:(a;b)!Sx2K U(x;(x))beadenablecurve.Wehaveshowthatthelimitlimt!b�(t)existsinSx2K U(x;(x)):BydenableSkolemfunctionsthereisadenablemap:(a;b)!Ksuchthatforeacht2(a;b)wehave(t)2 U((t);((t))):Byo-minimality,aftershrinking(a;b)ifnecessary,wemayassumethatisadenablecurveinK:SinceKisdenablycompact,letw=limt!b�(t)2K:Byo-minimality,aftershrinking(a;b)ifnecessary,wemayassumethat:(a;b)!M2Niscontinuous.Let :(a;b]!Kbethecontinuousdenablemapsuchthat j(a;b)=j(a;b):Recallthatwehave (b)=(w)2D(w)andD(w)M2Nisanopendenablesubsetby(D1).Itfollowsfromthecontinuityof :(a;b]!M2Natbthatthereisa02(a;b)suchthat (t)2D(w)forallt2[a0;b].Sinceforeachj2Iw,Xjisanopendenableneighborhoodofw,bycontinuity,aftershrinking(a0;b]ifnecessary,wemayassumethat (t)2Xjforallt2[a0;b]andallj2Iw:ThuswemusthaveIwI (t)forallt2[a0;b]:Therefore,by(D3),forallt2[a0;b]wehaveU( (t);( (t)))U(w;( (t))):Inparticular,foreacht2[a0;b)wehave(t)2 U(w;( (t))):Byhypothesisthereisd2D(w)suchthat(w)=( (b))dand U(w;d)isdenablycompact.By(D2)andcontinuityof :[a0;b]!D(w)M2N,aftershrinking(a0;b]ifnecessary,wemayfurtherassumethat( (t))dforallt2[a0;b]:Therefore,(t)2 U(w;d)forallt2[a0;b):Since U(w;d)isdenablycompact,thereexiststhelimitlimt!b�(t)2 U(w;d):Letv=limt!b�(t)2 U(w;d):Wewanttoshowthatv2 U(w;(w)):SupposenotandsetL= U(w;(w)):SinceLisdenablycompactsubsetof U(w;d),byProposition2.7,thereisadenablefunctionL:L!M2NwithL(x)2D(x)forallx2LandthereisdL2D(v)suchthat:LU(L;L):U(v;dL)\U(L;L)=;:WehaveU(w;(w))LU(L;L):IfU(w;(w))=U(L;L)thenU(w;(w))=L= U(w;(w))andsoU(w;(w))isaclosedandopendenablesubsetofX.SinceXisdenablyconnectedwewouldhaveU(w;(w))=Xandsov2 U(w;(w))whichisacontradiction.SinceU(w;(w))U(L;L)andU(L;L)isanopendenableneighborhoodofw,by(D4)thereisa002[a0;b]suchthatU(w;( (t)))U(L;L)forallt2[a00;b]:Therefore,foreacht2[a00;b]wehave(t)2 U(L;L): 10MARIOJ.EDMUNDO,MARCELLOMAMINO,ANDLUCAPRELLI(i)(x)2D(x);(ii)(x)(x);(iii) U(x;(x))isdenablycompact;(iv) U(x;(x))V:ItfollowsthatU(K;)=Sx2KU(x;(x))isanopendenableneighborhoodofKsuchthat,byLemma2.10, U(K;)= [x2KU(x;(x))=[x2K U(x;(x))isdenablycompactand U(K;)V:LetXbeadenablespace.Wesaythat:Xislocallydenablycompactifforeveryx2XandeveryopendenableneighborhoodVofxinXthereisanopendenableneighborhoodWofxinXsuchthat WVand Wisdenablycompact.XisdenablylocallycompactifforeverydenablycompactsubsetKXandeveryopendenableneighborhoodVofKinXthereisanopende-nableneighborhoodWofKinXsuchthat WVand Wisdenablycompact.Remark2.13.IfMhasdenableSkolemfunctionsandXisHausdor,thenbyTheorem2.12:Xislocallydenablycompactifandonlyifforeveryx2XthereisanopendenableneighborhoodUofxinXsuchthat Uisdenablycompact.XisdenablylocallycompactifandonlyifforeverydenablycompactsubsetKXthereisanopendenableneighborhoodUofKinXsuchthat Uisdenablycompact.Notehoweverthat,ingeneral,locallydenablycompactshouldnotbethesameasdenablylocallycompact.Denablenormalitygivestheshrinkinglemma(comparewith[7,Chapter6,(3.6)]):Fact2.14(Theshrinkinglemma).SupposethatMhasdenableSkolemfunc-tions.SupposethatXisadenablynormaldenablespace.IffUi:i=1;:::;ngisacoveringofXbyopendenablesubsets,thentherearedenableopensub-setsVianddenableclosedsubsetsCiofX(1in)withViCiUiandX=[fVi:i=1;:::;ng.2.2.Denabletypesandalmosteverywherecurveselection.Hereweex-tendthealmosteverywherecurveselection([25,Theorem2.3])toHausdorandlocallydenablycompactspaceandweshowthatdenablecompactnessofHaus-dordenablespacescanalsobecharacterizedbyexistenceoflimitsofdenabletypesextendingasimilarresultintheanecase(seetheobservationafter[22,Remark2.7.6]). 12MARIOJ.EDMUNDO,MARCELLOMAMINO,ANDLUCAPRELLI C=( C\V)[( C\W)=( CV\V)[( CW\W):Therefore, CnC=(( CV\V)nC)[(( CW\W)nC)=(( CV\V)nCV)[(( CW\W)nCW):IfCVisnotclosedinV,bythehypothesis,thereisadenablesetFV CV\VnCVsuchthatdimFVdim( CV\VnCV)andforeveryx2 CV\Vn(CV[FV)thereisadenablecurveinCVwhichhasxasalimitpoint.Similarly,ifCWisnotclosedinW,thereisadenablesetFW CW\WnCWsuchthatdimFWdim( CW\WnCW)andforeveryx2 CW\Wn(CW[FW)thereisadenablecurveinCWwhichhasxasalimitpoint.LetEVbeFVifitexistsandletitbe;otherwise.Similarly,letEWbeFWifitexistsandletitbe;otherwise.LetE=EV[EW.Since CnC=(( CV\V)nCV)[(( CW\W)nCW)wehaveE CnC:SinceC=CV[CWwealsohavethatforeveryx2 Cn(C[E)thereisadenablecurveinCwhichhasxasalimitpoint.SincedimE=maxfdimEV;dimEWganddim CnC=maxfdim( CV\VnCV);dim( CW\WnCW)gwealsohavedimEdim( CnC)asrequired.Theorem2.19(Almosteverywherecurveselection).(1)IfZisalocallycloseddenablesubsetofadenablemanifold,thenalmosteverywherecurveselectionholdsforZ.(2)IfZisalocallycloseddenablesubsetofadenablynormal,denablycompactdenablespace,thenalmosteverywherecurveselectionholdsforZ.Proof.(1)ByLemma2.17itisenoughtoshowthatifXisadenablemanifold,thenalmosteverywherecurveselectionholdsforX.Considerthedenablecharts(Ui;i)ki=1ofX.Sinceeachiisadenablehome-omorphism,andeachi(Ui)isanopendenablesubsetofMn,byFact2.15andLemma2.17,eachi(Ui)andsoeachUihasalmosteverywherecurveselection.NowweprovetheresultforXbyinductiononk.Thecasek=1isdone.Supposenowthattheresultholdsfordenablemanifoldswithlessorequalthanldenablechartsandk=l+1.LetV=Sli=1Ui,W=Ukand=k.Thenbytheinduc-tionhypothesisthealmosteverywherecurveselectionholdsontheopendenablesubmanifoldsVandWofX.SinceX=V[W,theresultfollowsbyLemma2.18.(2)ByLemma2.17itisenoughtoshowthatifXisaHausdor,denablycompactdenablespace,thenalmosteverywherecurveselectionholdsforX.Considerthedenablecharts(Ui;i)ki=1ofX.Bytheshrinkinglemma,thereareopendenablesubsetsVi(1il)andcloseddenablesubsetsCi(1il)suchthatViCiUiandX=[fVi:i=1;:::;lg:SinceeachCiisdenablycompactandeachiisadenablehomeomorphism,wehavethateachi(Ci)isaclosed(andbounded)denablesubsetofMniandsobyFact2.15andLemma2.17,eachi(Ci)andsoeachCihasalmosteverywherecurveselection.SobyLemma2.17,eachVihasalmosteverywherecurveselection.NowasaboveweconcludebyinductionofkthatXhasalmosteverywherecurveselection.ByTheorem2.19weseethatalmosteverywherecurveselectionholdsonlyinveryspecialdenablespaces.Despiteofthisweareluckysincelaterweonlyneedtousealmosteverywherecurveselectionlocally: 14MARIOJ.EDMUNDO,MARCELLOMAMINO,ANDLUCAPRELLI(2)EverydenabletypeonXhasalimitinX.Proof.Assume(1).ByTheorem2.12,Xisdenablynormal.Let(Xi;i)ikbethedenablechartsofXwithi(Xi)Mni:BytheshrinkinglemmatherearedenableopensubsetsVianddenableclosedsubsetsCiofX(1in)withViCiXiandX=[fCi:i=1;:::;ng.SinceXisdenablycompact,eachCiisalsodenablycompact.LetbeadenabletypeonX.ThenforsomeiwehaveCi2:Fixsuchi:Then=ei()(thetypeoni(Ci)determinedbythecollectionofdenablesubsetsfAi(Ci):�1i(A)2g)is(ratherdetermines)adenablytypeoni(Ci).Sinceij:Ci!i(Ci)Mniisadenablehomeomorphism,i(Ci)isdenablycompact.SobyFact2.22,hasalimitb2i(Ci).Letc2Cibesuchthati(c)=b:ThencisalimitofinX:Assume(2).Let:(a;b)!Xbeadenablecurve.Thenthecollectionofdenablesubsetsf([t;b)):t2(a;b)gofXdeterminesadenabletypewhichbyhypothesis,hasalimitainX.Thisa2Xisalsothelimitlimt!b�(t):3.PropermorphismsinDef3.1.Preliminaries.HerewerecallsomepreliminarynotionsforthecategoryDefwhoseobjectsaredenablespacesandwhosemorphismarecontinuousdenablemapsbetweendenablespaces.Letf:X!YbeamorphisminDef:Wesaythat:f:X!YisclosedinDef(i.e.,denablyclosed)ifforeveryobjectAofDefsuchthatAisaclosedsubsetofX,itsimagef(A)isaclosed(denable)subsetofY:f:X!Yisaclosed(resp.open)immersioniff:X!f(X)isahome-omorphismandf(X)isaclosed(resp.open)subsetofY.Proposition3.1.InthecategoryDefthecartesiansquareofanytwomorphismsf:X!Zandg:Y!ZinDefexistsandisgivenbyacommutativediagramXZYpY// pX Yg Xf// ZwherethemorphismspXandpYareknownasprojections.TheCartesiansquaresatisesthefollowinguniversalproperty:foranyotherobjectQofDefandmor-phismsqX:Q!XandqY:Q!YofDefforwhichthefollowingdiagramcommutes,QqX qY%% u## XZYpX pY// Yg Xf// Z 16MARIOJ.EDMUNDO,MARCELLOMAMINO,ANDLUCAPRELLILetf:X!YbeamorphisminDef:Wesaythat:f:X!YisuniversallyclosedinDefifforanymorphismg:Y0!YinDefthemorphismf0:X0!Y0inDefobtainedfromthecartesiansquareX0f0// g0 Y0g Xf// YinDefisclosedinDef.Denition3.3.Wesaythatamorphismf:X!YinDefisproperinDefiff:X!YisseparatedanduniversallyclosedinDef.Denition3.4.WesaythatanobjectZofDefiscompleteinDefifthemorphismZ!ptisproperinDef:BelowwewillrelatethenotionofproperinDefandcompleteinDefwiththeusualnotionsofdenablyproperanddenablycompact.3.2.SeparatedandproperinDef.Herewelistthemainpropertiesofmor-phismsseparatedorproperinDef.FromRemark3.2andthewaycartesiansquaresaredenedinDefweeasilyobtainthefollowing:Remark3.5.Letf:X!YbeamorphisminDef.Thenthefollowingareequivalent:(1)f:X!YisseparatedinDef.(2)Thebersf�1(y)offareHausdor(withtheinducedtopology).Directlyfromthedenitions(asin[18,ChapterI,Propositions5.5.1and5.5.5])ormoreeasilyfromRemark3.5thefollowingisimmediate:Proposition3.6.InthecategoryDefthefollowinghold:(1)OpenandclosedimmersionsareseparatedinDef.(2)AcompositionoftwomorphismsseparatedinDefisseparatedinDef.(3)Iff:X!YisamorphismoverZseparatedinDefandZ0!Zisabaseextension,thenthecorrespondingbaseextensionmorphismf0:XZZ0!YZZ0isseparatedinDef.(4)Iff:X!Yandf0:X0!Y0aremorphismsoverZseparatedinDef,thentheproductmorphismff0:XZX0!YZY0isseparatedinDef.(5)Iff:X!Yandg:Y!ZaremorphismssuchthatgfisseparatedinDef,thenfisseparatedinDef. 18MARIOJ.EDMUNDO,MARCELLOMAMINO,ANDLUCAPRELLILetZ0!ZbeamorphisminDef.ThenXZZ0fidZ0// p&& YZZ0p0 Z0isacommutativediagram,withfidZ0surjectiveandpclosedinDefbyhypothesis.Itfollowsthatp0isclosedinDefasrequired.(6)Supposethatf:X!YisamorphisminDefandletfVigikbeanitecoverofYbyopendenablesubsets.Ifg:Y0!YisamorphisminDef,thenff�1(Vi)gik(resp.fg�1(Vi)gik)isanitecoverofX(resp.Y0)byopendenablesubsetsandff�1(Vi)Yg�1(Vi)gikisanitecoverofXYY0byopendenablesubsets.Onetheotherhand,f�1(Vi)Yg�1(Vi)=f�1(Vi)Vig�1(Vi)andf�1(Vi)Vig�1(Vi)i// p0i XYY0p0 g�1(Vi)j// Y0isacommutativediagramwithiandjtheinclusions,p0theprojectionandp0itherestrictionofp0.Sincep0isclosedinDefifandonlyifeachp0iisclosedinDeftheresultfollows.Corollary3.8.Letf:X!YbeamorphisminDefandZXanobjectinDefwhichiscompleteinDef.Thenthefollowinghold:(1)Zisaclosed(denable)subsetofX:(2)fjZ:Z!YisproperinDef:(3)f(Z)Yis(denable)completeinDef:(4)Iff:X!YisproperinDefandCYisanobjectinDefwhichiscompleteinDef;thenf�1(C)Xis(denable)completeinDef:FromProposition3.7wealsoobtaininastandardwaythefollowing:Corollary3.9.LetBbeafullasubcategoryofthecategoryofdenablespacesDefwhosesetofobjectsis:closedundertakinglocallycloseddenablesubspacesofobjectsofB,closedundertakingcartesianproductsofobjectsofB:Thenthefollowingareequivalent:(1)EveryobjectXofBiscompletableinBi.e.,thereexistsanobjectX0ofBwhichiscompleteinDeftogetherwithanopenimmersioni:X,!X0inBwithi(X)denseinX0.Suchi:X,!X0iscalledacompletionofXinB. 20MARIOJ.EDMUNDO,MARCELLOMAMINO,ANDLUCAPRELLIofmorphismsinBsuchthat=i:X!Pisadenableopenimmersionwith(X)denseinPand hisproperinDef(sinceh0:X0!Y0isproperinDefbyCorollary3.8(2))asrequiredin(3).Assume(3).LetXanobjectofB.Takeh:X!fptgtobethemorphisminBtoapoint.Applying(3)tothismorphismweobtain(1).3.3.Denablypropermaps.Herewerecallthedenitionofdenablypropermapbetweendenablespacesandproveitsmainproperties.Aspecialcaseofthistheoryappearsin[7,Chapter6,Section4]inthecontextofanedenablespacesino-minimalexpansionsoforderedgroups.Denition3.10.Acontinuosdenablemapf:X!YbetweendenablespacesXandYiscalleddenablyproperifforeverydenablycompactdenablesubsetKofYitsinverseimagef�1(K)isadenablycompactdenablesubsetofX.Fromthedenitionsweseethat:Remark3.11.AdenablespaceXisdenablycompactifandonlyifthemapX!fptgtoapointisdenablyproper.Typicalexamplesofdenablypropercontinuousdenablemapsare:(i)f:X!YwhereXisadenablycompactdenablespaceandYisanydenablespace;(ii)theprojectionXY!YwhereXisadenablycompactdenablespaceandYisanydenablespace;(iii)closeddenableimmersions.Withourassumptions,thefollowingisprovedjustlikeintheanecaseino-minimalexpansionsoforderedgroupstreatedin[7,Chapter6,Lemma(4.5)]:Theorem3.12.Letf:X!Ybeacontinuousdenablemap.SupposethateverydenablycompactsubsetofYisaclosedsubset(e.g.MhasdenableSkolemfunctionsandYisHausdor).Thenthefollowingareequivalent:(1)fisdenablyproper.(2)Foreverydenablecurve:(a;b)!Xandeverycontinuousdenablemap[a;b]!Ywhichmakesacommutativediagram(a;b)// _ Xf [a;b]// == Ythereisatleastonecontinuousdenablemap[a;b]!Xmakingthewholediagramcommutative.Proof.Assume(1).Let:(a;b)!XbeadenablecurveinXsuchthatfiscompletableinY,saylimt!b�f(t)=y2Y.Takec2(a;b)andsetK=ff((t)):t2[c;b)g[fygY:ThenKisadenablycompactdenablesubsetofYandso,f�1(K)isadenablycompactdenablesubsetofXcontaining((c;b)).Thusmustbecompletableinf�1(K),henceinX.Assume(2).Supposethatfisnotdenablyproper.ThenthereisadenablycompactdenablesubsetKofYsuchthatf�1(K)isnotadenablycompact 22MARIOJ.EDMUNDO,MARCELLOMAMINO,ANDLUCAPRELLI(2)Considerthecommutativediagram:(a;b) // Xf Yg [a;b] 0FF == // Zwhereweassumewehavesuchthat exists.Wemustshowthat 0exists.Sinceg:Y!Zisdenablyproperandwehavefsuchthat exists,byTheorem3.12 exists.Sincef:X!Yisdenablyproperandwehavesuchthat exists,byTheorem3.12 0exists.(3)Sincethebaseextensionmorphismisaspecialcaseoftheproductmorphism,theresultfollowsfrom(4)below.(4)Considerthecommutativediagram:(a;b) pX  // [a;b] xx 0   X  f;; XZX0pXoo pX0 ff0// YZY0qY qY0$$ X0 f0DD Y  Y0 xx Zwhereweassumewehavesuchthat exists.Wemustshowthat 0:[a;b]!XZX0exists.Sincef:X!YisdenablyproperandwehavepXsuchthatqY exists,byTheorem3.12,[a;b]!Xexists.Sincef0:X0!Y0isdenablyproperandwehavepX0suchthatqY0 exists,byTheorem3.12[a;b]!X0exists.Sowelet 0bethemorphismgivenbytheuniversalpropertyofCartesiansquares.(5) 24MARIOJ.EDMUNDO,MARCELLOMAMINO,ANDLUCAPRELLIi:X,!X0inBwithi(X)denseinX0.Suchi:X,!X0iscalledadenablecompletionofXinB.(2)Everymorphismf:X!YinBisdenablycompletableinBi.e.,thereexistsacommutativediagramXf i// X0f0 Yj// Y0ofmorphismsinBsuchthat:(i)i:X!X0isadenablecompletionofXinB;(ii)jisadenablecompletionofYinB.(3)Everymorphismf:X!YinBhasadenableproperextensioninBi.e.,thereexistsacommutativediagramX f // P f YofmorphismsinBsuchthatisadenableopenimmersionwith(X)denseinPand fisdenablyproper.IfB=Defwedon'tmentionBandwetalkofdenablycompletable,denablecompletionanddenableproperextension.3.4.DenablyproperandproperinDef.HerewewillshowthatadenablypropermapbetweenHausdorlocallydenablycompactdenablespacesisthesameamorphismproperinDef.Wealsoprovethedenableanalogueofthetopo-logicalcharacterizationofthenotionofpropercontinuousmaps(asclosedmapswithcompactandHausdorbers)andadenabletypescriterionfordenablyproper.Theorem3.15.SupposethatMhasdenableSkolemfunctions.LetXandYbeHausdor,locallydenablycompactdenablespaces.Letf:X!Ybeacontinuousdenablemap.Thenthefollowingareequivalent:(1)fisproperinDef.(2)fisdenablyproper.Proof.Firstnotethatf:X!YisseparatedinDef(Remark3.5).SinceproperinDefmeansseparatedanduniversallyclosedinDef,itisenoughtoshowtheresultwith\properinDef"replacedby\universallyclosedinDef".Assume(1).Let :(a;b)!XbeadenablecurveinXandsupposethatf :(a;b)!Yiscompletable.ByTheorem3.12,weneedtoshowthat :(a;b)!XiscompletableinX.Byassumptionf extendstoacontinuosdenablemap 26MARIOJ.EDMUNDO,MARCELLOMAMINO,ANDLUCAPRELLIbers).Asimilarresultappearsinthesemi-algebraiccase([6,Theorem12.5]):Theorem3.17.SupposethatMhasdenableSkolemfunctions.LetXandYbeHausdor,locallydenablycompactdenablespaces.Letf:X!Ybeacontinuousdenablemap.Thenthefollowingareequivalent:(1)fisdenablyproper.(2)fisdenablyclosedandhasdenablycompactbers.Proof.Assume(1).Thenf:X!Yhasdenablycompactbersand,byTheorem3.15,fisdenablyclosed.Assume(2).LetKbeadenablycompactdenablesubsetofY.Let:(a;b)!f�1(K)adenablecurveinf�1(K).Supposethatlimt!b�(t)doesnotexistinf�1(K).ThenthislimitdoesnotexistinXaswellsincef�1(K)isacloseddenablesubsetofX(byCorollary2.9,Kisclosed).Therefore,ifd2(a;b),thenforeverye2[d;b),([e;b))isacloseddenablesubsetofXcontainedinf�1(K).Byassumption,foreverye2[d;b);f([e;b))isthenacloseddenablesubsetofYcontainedinK.SinceKisdenablycompact,thelimitlimt!b�f(t)existsinK,callitc.Hence,c2f([e;b))foreverye2[d;b):Sincethedenablesubsetft2[d;b):f(t)=cgisaniteunionofpointsandintervals,itfollowsthatthereisd02[d;b)suchthatf(t)=cforallt2[d0;b).Thus([d0;b))f�1(c)f�1(K).Sincef�1(c)isdenablycompact,thelimt!b�(t)existsinf�1(K),whichisabsurd.Wealsohavethefollowingdenabletypescriterionfordenablyproper:Theorem3.18.SupposethatMhasdenableSkolemfunctions.LetXandYbeHausdor,locallydenablycompactdenablespaces.Letf:X!Ybeacontinuousdenablemap.Thenthefollowingareequivalent:(1)fisdenablyproper.(2)ForeverydenabletypeonX,ifef()hasalimitinY,thenhasalimitinX:Proof.Assume(1).LetbeadenabletypeonXsuchthatef()hasalimitinY,saylimef()=y2Y.SinceYislocallydenablecompact,thereisadenableopenneighborhoodVofyinYsuchthat Visdenablycompact(Remark2.13).So,f�1( V)isadenablycompactdenablesubsetofXandisadenabletypeonf�1( V).ButthenbyTheorem2.23hasalimitinf�1( V),henceinX.Assume(2).Supposethatfisnotdenablyproper.ThenthereisadenablycompactdenablesubsetKofYsuchthatf�1(K)isnotadenablycompactde-nablesubsetofX.ThusbyTheorem2.23thereisadenabletypeonf�1(K)whichdoesnothavealimitinf�1(K).Sincef�1(K)isclosed(byCorollary2.9,Kisclosed),doesnothavealimitinX.Butef()isadenabletypeonKYandhasalimitbyTheorem2.23,whichcontradicts(2).4.Invarianceandcomparisonresults 28MARIOJ.EDMUNDO,MARCELLOMAMINO,ANDLUCAPRELLIProposition4.3.Letf:X!YamorphisminDef:Thenthefollowingareequivalent:(1)fisseparated(resp.proper)inDef:(2)fSisseparated(resp.proper)inDef(S):Theorem4.4.SupposethatMhasdenableSkolemfunctions.LetXandYbeHausdordenablespaces.Letf:X!Ybeacontinuousdenablemap.Thenthefollowingareequivalent:(1)fisdenablyproper.(2)fSisS-denablyproper.Proof.FirstnotethatShasdenableSkolemfunctionsandX(S)andY(S)areHausdorS-denablespaces(sinceHausdorisarst-orderproperty).UsingCorollary2.9andTheorem3.12inMandCorollary2.9andTheorem3.12inS;theresultfollowsfromtheclaim:Claim4.5.Thefollowingareequivalent:(1)Foreverydenablecurve:(a;b)!Xandeverycontinuousdenablemap[a;b]!Ywhichmakesacommutativediagram(a;b)// _ Xf [a;b]// == Ythereisatleastonecontinuousdenablemap[a;b]!Xmakingthewholediagramcommutative.(2)ForeveryS-denablecurve:(c;d)!X(S)andeverycontinuousS-denablemap[c;d]!Y(S)whichmakesacommutativediagram(c;d)// _ X(S)fS [c;d]// ;; Y(S)thereisatleastonecontinuousdenablemap[c;d]!X(S)makingthewholediagramcommutative.Assume(1)andsupposethatthereare:(c;d)!X(S)anS-denablecurveand:[c;d]!Y(S)acontinuousS-denablemapwhichmakeacommutativediagram(c;d)// _ X(S)fS [c;d]// @;; Y(S)andthereisnocontinuousS-denablemap[c;d]!X(S)makingthewholediagramcommutative.Thenthereareuniformlydenable(inM)familiesofcontinuousdenablemapsftgt2Tandftgt2Tsuchthatforsomes2T(S)wehaves= 30MARIOJ.EDMUNDO,MARCELLOMAMINO,ANDLUCAPRELLIY(S)arelocallyS-denablycompactS-denablespaces.Thenthefollowingareequivalent:(1)fisproperinDef.(2)fisdenablyproper.(3)fSisS-denablyproper.(4)fSisproperinDef(S):4.2.Denablyproperino-minimalexpansions.HereSisano-minimalex-pansionofMandweconsiderthefunctorDef!Def(S)fromthecategoryofdenablespacesandcontinuousdenablemapstothecate-goryofS-denablespacesandcontinuousS-denablemaps.ThisfunctorsendsadenablespaceXtotheS-denablespaceXandsendsacontinuousdenablemapf:X!YtothecontinuousS-denablemapf:X!Y.WeshowthatifMhasdenableSkolemfunctions,thenforHausdorlocallydenablycompactdenablespacesdenablyproperisthesameasS-denablyproperandproperinDefisthesameasproperinDef(S).Fact4.8.IfMhasdenableSkolemfunctions,thenShasdenableSkolemfunc-tions.Proof.Bythe(observationsbeforethe)proofof[7,Chapter6,(1,2)](seealsoComment(1.3)there),ShasdenableSkolemfunctionsifandonlyifforev-eryS-denablesubsetXMdenedwithparametersina1;:::;alonecanpickanS-denableelemente(X)2Xdenedwithparametersina1;:::;al:But,bythedenitionofo-minimality,theS-denablesubsetsXMarethesameasthedenablesubsetsXMwhicharethesameasthe(M;)-denablesubsetsXM:Theshrinkinglemmagivesthefollowing:Proposition4.9.SupposethatMhasdenableSkolemfunctions.LetXbeaHausdordenablespace.Thenthefollowingareequivalent:(1)Xisdenablycompact.(2)XisS-denablycompact.Proof.Assume(1).ByTheorem2.12,Xisdenablynormal.Let(Xi;i)ilbethedenablechartsofX:Bytheshrinkinglemma,thereareopendenablesubsetsVi(1il)andcloseddenablesubsetsCi(1il)suchthatViCiXiandX=[fCi:i=1;:::;lg:Fixi:ThenCiisadenablycompactdenablesubsetofXbecauseXisden-ablycompactandCiisclosedinX.Soi(Ci)isalsoadenablycompactdenablesubsetofMni;andtherefore,by[25,Theorem2.1],i(Ci)isaclosedandboundeddenablesubsetofMni:Inparticular,since\closed"and\bounded"arepreservedundergoingtoS;i(Ci)isaclosedandboundedS-denablesubsetofMniandtherefore,by[25,Theorem2.1]inS,i(Ci)isanS-denablycompactS-denablesubsetofMni:Hence,CiisanS-denablycompactS-denablesubsetofX:It 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