对一些弱连续性的讨论

对一些弱连续性的讨论 总39卷第2期 2006年6月 数学研究 JournalofMathematicalStudy Vo1.39No.2 Jun.2006 OnSomeWeakFormsofContinuity }rangErguang (DepartmentofMathematics.AnhuiUniversity,Hefei230039) AbstractInthispaper.thenotionsofweaklysemi-continuousfunctionsandweaklypre— con- tinuousfunctionsareintroduced,andsomeoftheirpropertiesareinvestigated.Also,weinves tigate theinterrelationsbetweensomeweakformsofcontinuity. KeywordsWeak口一continuity;weaksemi-continuity;weakpre-continuity CLChumberO189.1DocumentcodeA 1IntrOductiOn Inthepastseveralyears,someweakformsofcontinuitywereintroducedtogeneralizeconti— nuity.Levine[13definedweaklycontinuousfunctionsasageneralizationofcontinuity.In[2] , Noiriintroducedthenotionofweaka-continuitytogeneralizeweakcontinuity.Inthispaper,w e introducetWOclassesoffunctions.namelyweaklysemi— continuousfunctionsandweaklypre— continuousfunctions.Bothofthemaregeneralizationsofweaka-continuity.Someoftheirpr op- ertiesareinvestigatedinsection2.Insection3,wemakesomecomparisonsbetweencertain weakformsofcontinuity. Throughout,aspacewillmeanatopologicalspace.LetAbeasubsetofaspace(,). WedenotetheclosureandtheinteriorofAbyel(A)andint(A),respectively. AsubsetAofaspace(x,3-)iscalledsemi—open[33(resp.pre—open[4],regularopen [s],ana-set[6])ifACcl(int(A))(resp.ACint(cl(A)),A=int(cl(A)),ACint(cl(int ()))).Thefamilyofallsemi—open(resp.pre—open,regularopen)setsand口一 setsofaspace (,3-)willbedenotedbySO(,)(resp.P0(,9-),RO(X,9-))and3-..AiSsaidto besemi—closed(resp.pre—closed,regularclosed)if—issemi—open(resp.pre— open.regular open).Thesemi-closure(resp.pre-closure)ofasubsetA,writtenscl(A)(resp.pcl(A)),is theintersectionofallsemi-closed(resp.pre-closed)subsetsof(X,)thatcontainA.These— mi-interior(resp.pre-interior)ofA,denotedbysint()(resp.pint(A)).istheunionofall semi—open(resp.pre-open)subsetsof(,)containedinA. PropositionI.I[']LetAandBbesubsetsofaspace(.3-).Then Receiveddate:2004—11—16 Foundationitem:ThisworkissupportedbyNNSFofChina(10071018)andtheEYTPofChina ?146?数学研究2006正 (a)IfA?SO(X,3-)andB?P0(,9-),thenAnB?SO(B); (b)IfA?S0(B)andB?S0(X,),thenA?S0(X,9-). Proposition1.2LetAbeasubsetofaspace(,),then (a)sint(A)=Ncl(int(A));(b)scl(A)=AUint(cl()). Definition1.3Afunctionf:(,)(y,)issaidtobe (a)semi—continuous[3](resp.pre—continuous[4f1)iff一()?SO(,)(resp.f—l ()?P0(,))foreach?; (b)weaklya-continuous[2](briefly,w.口.c)(resp.weaklycontinuous[1])ifforeachz ?XandeachV?containing厂(z),thereexistsU?(resp.?9-)containingz,such that厂()Ccl(); (c)almostsemi—continuous[9](briefly,a.s.c.)(resp.almostpre—continuous[10] (briefly,a.p.c.))ifforeachz?XandeachVffRO(Y,)containing厂(z),thereexistsUE S0(X,9-)(resp.U?PO(X,))containingz,suchthat厂()CV; (d)almostweaklycontinuous[11]iff一'(V)Cint(cl(f一(c())))foreachVff; (e)weaklyquasicontinuous[12]fforeachz?XandeachG?containingzandeachV ?containing厂(z),thereexists?suchthatj2『?UCGandf(U)Ccl(V). Definition1.4Afunctionf:(X,)(y,)issaidtobeweaklysemi—continuous (briefly,w.s.c.)(resp.weaklypre=continuous(briefly,w.P.c.))ifforeachz?Xandeach VEcontaining厂(z),thereexistsU?SO(X,)(resp.U?P0(X,))containingz, suchthat厂(U)Ccl(). 2Propertiesofweaksemi—continuityandweakpre-continuity Inthissection,weinvestigatethepropertiesofw.s.c.functionsandw.P.c.functions.To simplifyournarration,weonlystatethepropertiesofw.s.c.functions.Thepropertiesofw.P. c.functionscanbeobtainedbyslightlymodifyingthoseofw.s.c.functions. Theorem2.1Forafunctionf:(X,)(y,aed),thefollowingareequivalent (a)fisw.s.c.; (b)f一()Ccl(int(f一(c())))foreachVff; (c)int(cl(f()))Of_1(cf())foreachV?; (d)scl(f一())C广(c())foreachV?; (e)广(V)Csint(f_1(cf(V)))foreachV?. Proof(a)(b)LetV6.Foreveryz?f_1(),wehave厂(z)?V.Sincefisw.s. c.,thereexistsU?SO(X,)containing工suchthat厂(【,)Ccl().Thatis,UCf_1(cl ()).HenceUCcl(int())Ccl(int(f_1(c()))),whichimpliesthatf(V)Ccl(int(f(cl ()))). (b)(c)ForeveryVff,wehaveY\c()?,andsof_1(y\c())Ccl(int(厂_1 (c(y\cl()))))Ccl(int(厂(cl(y\)))=cl(2,(厂_1(y\)))一\int.(cl(厂_1())). Therefore,int(cl(:f一()))Cf一(c(V)). 第2期YangErguang:OnSomeWeakFormsofContinuity?l47? (c)(d)ThiscanbeseenfromProposition1.2(b),sinceforeachV?,wehave厂 ()c,((V)). (d)(e)Supposethatscl(f-1())cf一(())foreachV?.Then,byProposition 1.2(b),int(cl(f()))cf(()).ForeveryV6,wehavey\()?,soX\cl (int(广(cl())))一int(cl(厂(y())))广(cl(y\()))cf-1(y\)一\广 ().Therefore,厂()c(int(厂-1(cl()))).ByProposition1.2(a),oneeasilyverifies ' hat(e)holds. (e)(a)Supposethat(e)holds.Thenforeach?XandeachV?containing厂 (),wehave?厂(),andso?sint(广(())).LetU=sint(广(())).Then? SO(X,)containingandUCf(()).Hence厂()(),whichimpliesthat厂isw. SC. Theorem2.2If厂:(,)一(y,)isw.s.c.andA?P0(,),thentherestriction 厂IA:(,I)一(y,)isw.s.c. ProofForeach?ACXandeachV6containing(厂I)()一厂(),since厂isw.s. c.,thereexistsU?SO(X,3-)containingsuchthatf(U)Ccl(V).PutW=UNA,sinceA ?PO(,),byproposition1.1(a),wehaveW?SO(,I)containingand(厂IA) ()一厂()c厂()().Thisindicatesthat厂Iisw.s.c. Theorem2.3Let厂:(,)一(y,)beafunctionand{U,:5?S)acoverofXbyse— mi—opensetsof(X,).If厂I【,,:(【,,,I【,,)一(y,)isw.s-c.foreachs?S,then厂is W.S.C. ProofForeach?X,thereexists5o?Ssuchthat?USince厂IUisw.s.c.,for eachV6containing(fl,.)()=厂(),thereexistsW?SO(U,.,I,.)containingz suchthat厂(,.)一(厂I.)(,.)CcI(V).SinceU?SO(x,9-),byProposition1.1(b), ?O(,).Therefore,fisw.s.c. Theorem2.4Afunction厂:(,)(y,)isw.s.c.ifandonlyifthegraphfunction g:(,)一(×y,×)definedbyg()一(z,厂())forevery?Xisw.s.c. ProofSupposethat厂isw.s.c.Let?Xandg(z)?WE9-×,thenthereexistUl? andV?suchthat(,厂())?l×.Since厂isw.s.c.,thereexistsU2?O(, )containingsuchthatf(U2)c().PutU=UlnU2,wehave?U?O(,)andg ()cl×f(U2)cl×()().Therefore,gisw.s.c. Conversely,supposethatgisw.s.c.Let?Xand厂()?V6,thenwehaveg()= (,,())?X~VE~-×.Asgisw.s.c.,thereexistsU6SO(X,.)containingsuch thatg(U)Ccl(XXV)一X×().Thusf(U)Ccl(V),whichimpliesthat厂isw.s.c. 3Comparisons Inthissection,vceinvestigatetheinterrelationsbetweentheweakformsofcontinuityprevi ^systated. 数学研究2006矩 Theorem3.1Letf:(X,)一(y,)andg:(y,)一(Z,)befunctions.Then thecompositiong.f:(X,)(z,)isw.s.c.iffandgsatisfyoneofthefollowingcondi— tions (a)fissemi—continuousandgisweaklycontinuous; (b)fisw.s.c.andgiscontinuous. ProofWeonlyprovethecaseof(a).(b)canbeprovedanalogously. ForeachxEX,let=(z)andz=g()=g(,(z)).ForeachV?containing,since gisweaklycontinuous,thereexistsUEcontainingysuchthatg()Ccl(V).Thusf() ?S0(X,),becausefissemi—continuous.LetU----f()thenW?SO(X,)contain— ingand(g.,)()=g(f(W))Cg(U)Ccl(V).Therefore,g.fisw.s.c. Lemma3.2[Forafunctionf:(X,)(y,),thefollowingareequivalent (a)fisw.口.c.;'' (b)f(V)Cint(cl(int(f(cl(V)))))foreachV?; (c)cl(int(cl(f_1())))CT_f(())foreachV?. Since'CSO(X,)and.CPO(X,),bytheirdefinitions,itisobviousthatweak a-continuityimpliesweaksemi-continuityandweakpre— continuity.However,theconverseisnot trueingeneralasshownbythefollowingexamples. Example3.3LetX==:{a,b,c),={,{a),{c),{a,c),X)and={(z),{a), {b,c),X).Oneeasilyverifiesthattheidentityfunctionf:(,)一(X,)isweaklysemi— continuousbutnotweaklya-continuous,sincethereexists(b,c)?suchthatf1((b,c))一 {b,f)isnotcontainedinint(cl(int(厂(({b,c))))))={c). Example3.4Letx={口,b,c,),一{,(c),{口,),{口,c,),x)and= ,{a),{b,c),{a,b,c),).Letf:(,)一(X,)beafunctiondefinedasfollows:f (口)=,(6)一,(c)=口and,()=6.Thenfisw.P.c.butnotw.口.c.sincethereexists{a)? suchthatf({a))={a,b,c)isnotcontainedinint(cl(int(广(cl({a))))))={c). Lemma3.5LetAbeanopensubsetofaspace(X,).Thencl(A)isregularclosed. ProofItisdearthatcl(int(c(A)))C(A).Soweneedonlytoshowthatcl()C (int((A)))alsoholds.SinceAisanopenset,fromACcl(),wehaveAC(cl(A)). Then,clearly,cl(A)Ccl(int(cl(A))). Lemma3.6c.Forafunctionf:(X,)(y,),thefollowingareequivalent (a)fisa.p.c.; (b)广(F)ispre—closedin(X,)foreveryregularclosedsubsetFof(y,); (c)f一()ispre—openin(X,)foreveryregularopensubsetVof(y,). Theorem3.7Forafunctionf:(,)一(y,),iffisbothw.s.c.anda.P.c.,then ,isw.口.c. ProofForeachV?,cl()isregularclosedbyLemma3.5.Sincefisa.P.c.,by Lemma3.6,广(())ispre—closedin(X,9-).Hence,cl(int(f-1(())))Cf(()). Sincefisalsow.s.c.,byTheorem2.1,wehaveint(cl(f1()))c厂1(()),whichshows 第2期YangErguang:OnSomeWeakFormsofContinuity?l49? thatcl(int(cl(f一())))Ccl(int(f一(()))).Hencecl(int(cl(f())))cf(()). So,byLemma3.2,厂isw.口.c. ItiseasytoshowthatthefunctioninExample3.4isa.P.c.,but,aswehavealreadyseen, itisnotw.口.c.Thefollowingexampleshowsthatweaka-continuitydoesnotimplyalmostpre— continuityingener~,too. Example3.8LetX;{a,b,c!d}and;{(2j,{b},{c},{b,c},{a,b},{a,b,c}, {b,c,d},}.Let厂:(X,)一(X,9-)beafunctiondefinedasfollows:厂(口)=c,厂(6)=, 厂(c)=6and厂()=口.Oneeasnyverifiesthatfisw.口.c.Butitisneithera.P.c.nora.s.c., sincethereexists{c}?RO(X,3-)suchthat,-1({c})一{a}neitherbelongstoP0(X,)nor belongsto0(X,) Theorem3.9Forafunctionf:(X,)(y,),if厂isa.P.c.(resp.a.s.c.),then厂 isw.P.c.(resp.ws.c.) ProofThatthetheoremholdsforana.P.c.functionfcanbeseenfromLemma3.5, Lemma36andTheorem2.1.Analogously,wecanprovethatthetheoremalsoholdswhenlis aS.. Generallyspeaking,weakpre—continuity(resp.weaksemi—continuity)doesnotimplyal — mostpre—continuity(resp.almostsemi— continuity).ThiscanbeseenfromExample3.8.The functioninExample3.8isw.口.c.Henceitisbothw.s.c.andw.P.c.But,aswehavealready seen,itisneithera.s.c.nora.P.c. Lemma3.10c.Afunction-,-:(,)(y,)isweaklyquasi--continuousifandonly if广(V)Ccl(int(f(())))foreachV?. Theorem3.11Afunctionf:(X,)(y,)isw.s.c.(resp.w.P.c.)ifandonlyif fisweaklyquasi--continuous(respalmostweaklycontinuous). ProofThisfollowsfromLemma3.10,Definition1.3(d)andTheorem2.1. AcknowledgmentTheauthorwouldliketothankhistutorprofessorYanPengfeiforhis valuablesuggestions. References [13LevineN.Adecompositionofcontinuityintopologicalspaces,Amer.Math.Monthly,1961,68:44— 46. [23NoiriT.Weaklya-continuousfunctionsfInternat.J.Math.Math.sci.,1987,10:483--490. [33LevineN.Semi— preopensetsandsemi-continuityintopologicalspaces,Amer.Math.Monthly,1963, 70:36—41. [43MashourAS.HasaneinIAandEl?DeebSN.a-continuousand口?openmappings,ActaMath.Hungar., 1983,41:213—218. [53DugundjiJ.Topology.AllynandBacon(Boston,1972). [63NiastadO.Onsomeclassesofnearlyopensets,PacificJ.Math.,1965,15:961—970. [73MashourAS,HasaneinIAandEl?DeebSN.Anoteonsemi?continuityandpre-continuity ,IndianJ. ? 150-数学研究2006矩 [8] [9] [10] [11] [12] PureAppl?Math.,1982.13:1119—1123. AndrijevicD?SemiPreopensets,Math.Vesnik,1986.38:24—32. 雌sanDS?Alm0stSemi-cont.nu0usmappings,Math.Student'l981,49:239--248. NaefAAandNo.rT?Someweakforms0falmostc.ntinuity,ActaMath.Hungar''1997,74:211 一上. JankovicDS?regu1arsPaces,Intel'nat. J.Math.Math.Sci.,1985,8:615—619. PoparandStanG()nadec.mpos.tIon0fquasi--continuItyint.p.l0g.calspaces(R.manian),stu d. 摘要 进行了探讨. 关键词 对一些弱连续性的讨论 杨二光 (安徽大学数学系,安徽合肥230~39) 引入了弱半连续及弱准连续性的概念,讨论了他们的一 些性质,并对某些弱连续性之间的关系 弱a一连续;弱半连续;弱准连续