InternatJMathMathSciVOL19NO31996495500495ONTHETHEOREMSOFYM - PDF document

1 Internat.J.Math.&Math.Sci.VOL.19NO.3(199
Internat.J.Math.&Math.Sci.VOL.19NO.3(1996)495-500495ONTHETHEOREMSOFY.MIBUANDG.DEBSONSEPARATECONTINUITYZBIGNIEWPIOTROWSKIDepartmentofMathematicsYoungstownStateUniversityYoungstown,OH44555USA(ReceivedMay25,1994andinrevisedformNovember1,1995)ABSTRACT.Usingagame-theoreticcharacterizationofBairespaces,conditionsuponthedomainandtherangearegiventoensurea"fat"setC(f)ofpointsofcontinuityinthesetsoftypeXx{/},yEYforcertainalmostseparatelycontinuousfunctionsfXYZTheseresults(especiallyTheoremB)generalizeMibu'sFirstTheorem,previoustheoremsoftheauthor,answersoneofhisproblemsaswellastheyarecloselyrelatedtosomeotherresultsofDebsandMibu[2]KEYWORI)SANI)PItRASES:Separateandjointcontinuity,quasi-continuity,Moorespaces.1991AMSSUBJECTCLASSIFICATIONCODES:54C08,54C30,26B05I.INTRODUCTIONSincetheappearanceofthecelebratedresultofNamioka,manyarticleshavebeenwrittenonthetopicofseparateandjointcontinuity,seePiotrowski[3],forasurveyAsidefromanintensivelystudiedUniformizationProblem-Namioka-typetheorems,seePiotrowski[3],questionspertainingtoExistenceProblem(seebelow)aswellasitsgeneralizations,havebeenaskedLetXandYbe"nice"(egPolish)topologicalspaces,letMbemetricandletfXYMbeseparatelycontinuous,thatis,continuouswit.hrespecttoeachvariablewhiletheotherisfixedFindthesetC(f)ofpointsof(joint)continuityoff.LetusrecallthatgivenspacesX,YandZ,andletfXxYZbeafunction.ForeveryfixedzEX,thefunctionfYZdefinedbyf(t)f(x,),wheretY,iscalledanz-secnonoffAt-sectionfuoffisdefinedsimilarly.Onewaytoensuretheexistenceof"many"pointsofcontinuityinXxYcanbederivedfromthefollowing.Baire-Lebesgue-Kuratowski-MontgomeryTheorem(seePiotrowski[3])LetXandYbemetricandletfXYRhaveallz-sectionsfcontinuousandhaveallt-sectionsfuofBaireclassaThenfisofclassa+1 496ZI'IOTROWSKIIfa0,thatis,f,jiscontinuous,fisofclassNow,byatheoremofBaire,C(f)isresidualSo,fweassumeaddtmnallythatXxYisBaire,thenC(f)isadenseGesubsetofXxYi-IButonecannotrelaxtheassumptionspertainingtothesectionstoomuchEXAMPLE.LetI[0,1]andletIRbethesetofrealsPutD,{(z,y)z,y,whereka

2 ndparealloddnumbersbetween0and2'}LetDt3D
ndparealloddnumbersbetween0and2'}LetDt3D,ItiseasytoseethatC!DINow,letusdefinef19-IRbyf(z,y)1,for(z,y)EDandf(x,y)0if(x,y)D.allthez-sectionsfandallthey-sectionsfareoffirstclassofBaireandC(f)05ElHowever,thefollowingthreeimportantresultsholdMIBU'SFIRSTTHEOREM(Mibu[2])LetXbefirstcountable,YbeBaireandsuchthatXxYisBaireGivenametricspaceMIffXxYMisseparatelycontinuous,thenC(F)isadenseGsubsetofXxYMIBU'SSECONDTHEOREM[2].LetXbesecondcountable,YbeBaireandsuchthatXxYisBaireGivenametricspaceMIffXxYMhasa)allx-sectionsfhavetheirsetsD(f)ofpointsofdiscontinuityofthefirstcategoryand,b)ally-sectionsfarecontinuousThenC(f)isadense,GesubsetofXxYFollowingDebs[1],afunctionf"XMiscalledfirstclassifforevery�0,foreverynonemptysubsetACX,thereisanonemptysetU,openinA,suchthatdiana(f(U)).DEBS'TItEOREM1]LetXbefirstcountableYbeaspecialc-favorablespace(thusBaire),XxYbeBaireGivenametricspaceMIff"XxYMhas:a)allz-sectionsfoffirstclass-inthesenseofDebsand,b)ally-sectionsfucontinuousThenC(f)isadenseGsubsetofXxY2.QUASI-CONTINUITYONPRODUCTSPACESAfunctionfXYiscalledquasi-continuousatapointzXifforeachopensetsACXandHcf(X),wherezAandf(z)H,wehaveAIntf-(H)OAfunctionfXYiscalledquast-continuous,ifitisquasi-continuousateachpointzofX.AfunctionfXxYZ(X,Y,Zarbitrarytopologicalspaces)issaidtobequasi-continuousat(p,q)XxYwithrespecttothevariabley,ifforeveryneighborhoodNoff(p,q)andforeveryneighborhoodUxVof(p,q),thereexistsaneighborhoodV'ofq,withV'CV,andanonemptyopenU'cU,suchthatforall(z,y)U'xV'wehavef(z,y)N.Iffisquasi-continuouswithrespecttothevariableyateachpointofitsdomain,itwillbecalledquasi-continuouswithrespecttoThedefinitionofafunctionfthatisquasi-continuous-withrespecttozisquitesimilar.Iffisquasi-continuouswithrespecttozandy,wesaythatfissymmetricallyquast-conttnuous.Onecaneasilyshowfromthedefinitionsthatiffissynunetricallyquasi-continuous,thenfandfuarequasi-continuousforallzXand/Y.Theconversedoesnothold.LEMMA(Piotrowski[4]Theorem42).LetXbeaBairespace,YbefirstcountableandZberegularIffisafunctiononXxYtoZsuc

3 hthatallitsz-sectionsfarecontinuousandal
hthatallitsz-sectionsfarecontinuousandallits-sectionsfuarequasi-continuous,thenfisquasi-continuouswithrespecttoyTheconversedoesnothold ONTHETHEORF,MSOFYMIBI.JANDGDEBSZ97Asanimmediateconsequenceweobtain(Piotrowski[4]..Corollary43)LetXandYbefirstcountable,BairespacesandZbearegularoneIffXYYisseparatelycontinuous,thenfissymmetricallyquasi-continuousIfXandYaresecondcountableBairespacesandZisaregularone,andafunctionf:XxYZ,thenthefollowingimplicationshold(whichshowtheinclusionrelationsbetweenproperclassesoffunctions)seeDiagramNoneoftheseimplicationscan,ingeneralbereplacedbyanequivalence,seeNeubrunn[5]f-symmetricallyquasi-continuousf-continuousf,fu-continuousf,fu-quasi-cntinuusf-quasi-continuousDiagramTheBanach-Mazurtame.WewilluseheretheclassicalBanach-MazurgamebetweenplayersAandBbothplayingwithperfectinformation(seeNoll[6],Oxtoby[7])AstrategyforplayerAisamappingcwhosedomainisthesetofalldecreasingsequences(G1,G9.,_1),n�1,ofnonemptyopensetssuchthatc(G1,G2,-1)isanonemptyopensetcontainedinGg.,_.Dually,astrategyforplayerBisamapping/3whosedomainisthesetofalldecreasingsequences(U1,U2,),n�0,ofnonemptyopensetssuchthat/3(U1,U2,)isnonempty,openandcontainedinU2,Heren0standsfortheemptysequence,forwhich/3(0)isnonemptyandopen,too.Ifc,/3arestrategiesforA,Brespectively,thentheuniquesequenceG1,Gg.,G3,...definedby()=GI,a(G)=G2,/3(G1,G2)G3,a(G1,G2,G3)G4,iscalledthegameofAwithaagainstBwith/3WewillsaythatAwithawinsagainstBwith/3iftq{G,nEN}holdsforthegameG,Gg.,...ofAwithaagainstBwith/3.Conversely,wewillsaythatBwith/3winsagainstAwithaifAwithadoesnotwinagainstBwith/3.Wewillmakeuseofthefollowingtheorem,essentiallyprovedbyBanachandMazurof.Oxtoby[7],seealsoNoll[6]wherethegame-theoreticcharacterizationofBairespaceswasappliedtoobtainsomegraphtheorems.LetEbeatopologicalspace.Thefollowingareequivalent:(1)EisaBairespace;(2)foreverystrategy/3ofBthereexistsastrategyaofAwithwinsagainstBwith/3. /498ZPI()I'ROWSKI3.THEMAINRESULTLetusrecallIfAcXandb/isacollectionofsubsetsofX,thenst.(A,ld)[.

4 J{UUfqA:/:0}ForxEX,wewritest.(z,N)nstead
J{UUfqA:/:0}ForxEX,wewritest.(z,N)nsteadofst.({x},Lt)Asequence{G,,}ofopencoversofXisadevelopmentofXifforeachxXtheset{st.(x,Gn)nN}isabaseatzAdevelopablespaceisaspacewhichhasadevelopmentAMoore.spaceisaregulardevelopablespaceTHEOREMA.LetXbeaBairespace,Ybespaceandlet{P,},beadevelopmentforZIffXYZisquasi-continuouswithrespecttoy,thenC(f)isadense,GsubsetinX{y},forallyEYPROOF.LetxX,VEYandletUVbeaneighborhoodof(x,y)Defineastrategyforaplayer/3inacorrespondingBanach-MazurgameplayedoverXForthispurposeweshallorder(well-ordering)thesetsX,openneighborhoodsofyandopennonemptysubsetsofX(1)B(O)hastobedefinedSinceZhasacountabledevelopment,,thereisalocalcountablebaseateverypointofZ,inparticulartake{G,}atf(x,)PickG1Nowbythequasi-continuityoffwithrespecttoy,thereisaneighborhoodVofy,andanonemptyopenUsuchthatf(UVcG1LetusfurtherassumethatUandVarethefirstsetsintheirorderingsofXandY,respectivelywiththeabovepropertyNow,letWbethefirstnonemptyopensetcontainedinUandletxlbethefirstelementofWThus,WVisaneighborhoodof(x,y)So,let(0)w(2)/(G,Gg)hastobedefined,whereG,Gg.arenonemptyopenandGcGNow,fisquasi-continuouswithrespecttoyat(zg.,y),pickG3,thefirstelementofthebaseatf(zt,y)withG3CGNowpickthefirstelementUxVsuchthatf(UxV3)C133-suchaUVexists,bythequasi-continuitywithrespecttoyoffNow,letW3bethefirstopennonemptysetcontainedinU3(apriori,itcanbeeventhesameset(!))andletz3U3bethefirstelementofWa.Thus,W3xV3isaneighborhoodof(z3,y)So,let/(G1,G)W3.(3)Inthiswayweproceedtodefineflbyrecursion,e.,if/3(0)G1and/(G,G2k)G2k+l,forallknthentheformerstepsareavailableandwecandefineG2k+ainanalogywith(2).(4)Supposenowthathasbeendefined.SinceXisBaire,thereisastrategyaforAsuchthatAwithawinsagainstBwith(seethedefinitionofthegame).LetG1,G2...bethegameAwithaagainstBwithNoticethat.N{ oN}N{ oN}.(3.1)Butobservethataiswinning,hencethisintersectionisnonempty;i.e,x*5I'I{W,,:nN},so(x',y)(UV)t2(X{y}).ThisinturnshowsthedensityofC(f)inX{y}TheG6partfollowseasilyfromtheconstructionI-IAspacewillbecalledquast-regularifforeverynonemptyopens

5 etU,thereisanonemptyopensetVsuchthatCIVc
etU,thereisanonemptyopensetVsuchthatCIVcUObviously,everyregularspaceisquasi-regular.Let.AbeanopencoveringofaspaceXThenasubsetSofXissaidtobeA-smallifSiscontainedinamemberof.,4AspaceXissaidtobestronglycountablycompleteifthereisasequence ()Nl'lll:.TI1EORI.MS()FYM1BIJANI)GI)EBS499{Ai1,2ofopencoveringsofXsuchthatasequence{F,}ofclosedsubsetsofXhasanonemptyintersectionprovidedthatF,F,.forallandeachF,is.A,-smallTheclassofstronglycountablycompletespacesincludeslocallycountablecompactspacesandcompletemetricspacesInviewofthefollowing(Piotrowski[8],Theorem46seealsoLemma3ofPiotrowsk[9])Everyquasi-regular,strongJycountablycompletespaceXisaBairespaceTheoremAisastronggeneralizationofthefollowing(Piotrowski[8],Theorem45)LetXbeaspace,Ybequasi-regular,stronglycountablycompleteandZbemetricIffXYZisquasi-continuouswithrespecttoz,thenforallzEXthesetofpointsofjointcontinuityoffisadenseGof{z}YFurther,observethatourTheoremAanswers,inpositive,thefollowing(Piotrowski[8],Problem411)DoesTheorem45(of[8])holdifYisonlyassumedtobeaquasi-regularBairespace9ThefollowingTheoremBisthemainresultofthispaperanditsproofeasilyfollowsfromthelemmaandTheoremATHEOREMB.LetXbefirstcountable,YbeBaireandZbeMooreIffXxYZhasallitsz-sectionsfxquasi-continuousandallitsv-sectionsfycontinuous,thenforallzEX,thesetofpointsofcontinuityoffisadenseGsubsetofzYTheaboveresultstronglygeneralizes(seetheassumptionsuponYandZ)thefollowingknowntheorem(Piotrowski[8],Theorem48seealsoTheorem5ofPiotrowski[9])LetXbefirstcountable,Ybestronglycountablycomplete,quasi-regularandZbeametricspaceIffXYZisafunctionsuchthatallitsz-sectionsfxarequasi-continuousandallitsv-sectionsfyarecontinuous,thenforallzX,thesetofpointsofjointcontinuityoffisadenseG,subsetofz}YOurTheoremBgeneralizesinmanywaysMibu'sFirstTheorem-seeIntroductionItisalsocloselyrelatedtoMibu'sSecondTheoremandDebs'TheoremibidemObservethough,thatquasi-continuityofafunctiondoesnotimplynorisimplied,bytheconditionofbeingoffirstclass-inthesenseofDebsReally,letf[0,1]]Rbegivenbyf(z)0,ifz#-1

6 /2.Thensuchafunctionfisoffirstclass,inth
/2.Thensuchafunctionfisoffirstclass,inthesenseofDebsand,clearly,itisnotquasi-continuousTherearequasi-continuousfunctionsf:li-RwhichareofarbitraryclassofBaireornotLebesguemeasurable-seeNeubrunn[5]formoredetailsREMARK1.Thestudiesofthecontinuitypointsoffunctionswhoserangesarenotnecessarilymetrichavebeendonealreadyinthe1960's,seeKleeandSchwarz[10]orlaterinthe1980's,seeDubins11],weomithereanextensiveliteratureofthisapproach,whentherangeisauniformspaceREMARK2.Recently,theauthorhasobtainedsomeresultsofthispaperusingthoughentirelydifferenttechniques,seePiotrowski12]ACKNOWLEDGMENT.TheauthorwishestoexpresshisgratitudetoarefereewhosecommentsandremarkshaveimprovedthepresentationoftheresultsAlso,thanksareduetotheResearchCouncilofYoungstownStateUniversityforagrantwhichenabledtheauthortocompletethisresearch 500ZPI()lROWSKIREFERENCES[1]DEBS,G,Fonctionsseparmentcontinuesetdepremiereclassesurespaceproduit,Math.Stand59(86),22-30[2]MIBU,Y,Onquasi-continuousmappingsdefinedonproductspaces,Proc..JapanAcad.192(1958),189-192[3]PIOTROWSKI,Z,Separateandjointcontinuity,RealAnalystsExchange,11(1985),293-322[4]PIOTROWSKI,Z,Quasi-continuityandproductspaces,Proc.Intern.('ate.Geom."lop.,Warsawa(1980),349-352[5]NEUBRUNN,T,Quasi-continuity,RealAnalysisbScchange,14(1988-89),259-306[6]NOLL,D,Bairespacesandgraphtheorems,Proc.Amer.Math.Sac.,96(1986),141-151[7]OXTOBY,JC,TheBanach-MazurgameandBanachcategorytheorem,ContributiontotheTheoryofGames,Vol3,Ann.ofMath.Studies,No39,PrincetonUnivPress,Princeton,NJ,1957[8]PIOTROWSKI,Z,Astudyofcertainclassesofalmostcontinuousfunctionsontopologicalspaces,PhD.Thesis,Wroclaw,1979[9]PIOTROWSKI,Z,Separatealmostcontinuityandjointcontinuity,ColloquiaMathematlcaJdmosBolyat23Topology,Budapest(Hungary)(1978),957-962[1)]KLEE,V,Stabilityofthefixedpointproperty,Colloq.Math.$(1961),43-46.[11]DUBINS,LEandSCHWARZ,G.,EquidiscontinuityofBorsuk-Ulamfunctions,PacificJ.Math.,95(1981),51-59[12]PIOTROWSKI,Z,Mibu-typetheorems,ClasstcalAnalysts,Proc.Intern.Conf.WSI,Radom(1994),133-139