1、41319985ACTAMATHEMATICASINICAVol.41,No.3May,1998-(643000)-TychonoTychono-MR(1991)-Tychono54E18,54E35O189.11Hereditarily-MetacompactSpaceandItsProductPropertiesZhuPeiyong(DepartmentofMathematics,ZigongTeachersCollege,Zigong643000,China)AbstractInthispaper,werstproveagroupofequivalentcharacterizations

2、ofheredi-tarily-metacompactspaces.BytheoneofcharacterizationsweobtaintwoTychonoproducttheoremsandthetheoremon-product.Finally,weshowthattheaboveTychononoproducttheoremsdonotholdifhereditarily-metacompactisreplacedby-metacompact.KeywordsHereditarily-metacompact,Scatteredpartition,-pointniteopenex-pan

3、sion,-pointniteopenrenement,Tychonoproduct,-product1991MRSubjectClassication54E18,54E35ChineseLibraryClassicationO189.11(A)X-Y-X×YX-(B)X=i<Xin<,i<nXi-1996-03-15,1997-02-03,1997-10-1353241(C)X=X:A,X-X-x(A)(B)X,YU|Y,(U)xN(x)UY:UU,UU:xU,¯|A|AAAAn=A:|=nA<=An.n1.11,21.2X(scatteredpa

4、rtition)XL:<<,L:<X.UUXUXV=VnnU-VX(1)Vn(2)V=U.1.3V=Vnnn,Vn=Vn:<,VF:<-Vn<FVn.n1.4X-X1.5- X:A-X:Ax=(x)A X:A=x=(x)AX:A:|Q(x)|<.X:ATychonos=(s)A X:A,Q(x)=A:x=s.X:AX:A-s X:AX:a-A<,Tychono2-2.12.2X-X-,F:<F:F:<,F:<,F=<F.1.42.3X,(1)X(2)X(3)X(4)X-F:<U:<-V=Vnn<,XF=V

5、V:VF=;V=Vnn<,U=VV:VU;3-533(5)XU:<-V=Vnn<,U=VV:VU.(1)(2),X-L:nX=1n,Vn=X,V=VnL:<-<,-XXL:<-=(i)=+1,L:<V=Vn,nL:<-Vn=Vn:<n,Vn=X(ii)Vn=Vn:<,V=VnnL:<-<,O=L:<,O:<V=Vn,Vn=Vn:<,nVnO.<,nX<,LVn:<Vn-Wn=Wnm,mWnm=Wnm(,):<<,LVnWnm(,)Vn.m<,Wnm=Wnm(,):<

6、;<,Wnm=Wnm:<.W=(n,m)×WnmL:<-.LVnLO=L(L)=,<L=LVn:<=LVn:<<nnWnm(,):<<nm(n,m)×Wnm(,):<<=(n,m)×Wnm.(n,m)×,xWnm,n(x)=<:xVnn(x),nm()=<:xWnm(,)nm=nm():n(x)<:xWnmnm.XWL:<-(2)(3)<F:<2.2F=<F,L1=XF0<,L=FF+1,53441(a)L:1<XnVnL:1<V

7、n=nVn-Vn=Vn:1<(1<),LVn.n,<,F+1V1n=XF0,-Vn=Vn:1<,(b)V=VnnL:1<(c)<,XF=VV:VF=.=0>0<,=+1,xXF+1,xXF=VV:VF=xVVF+1VF=.xVVV:VF+1=;VVxF,xFF+1Vn=VnF+1.nnn<xVnVxF.VnF+1=.xVnVV:VF+1=.xXF,xF=<F.xXF=VV:VF=.(c)(2)(3)VVxVVFVF0=.xVVV:VF=.(3)(4),U:<XXU:<XX-V=Vnn<,U=X(XU)=VV:V(XU

8、)=VV:VU.n,Vn=VVn:<VU,V=VnnU:<-(4)(5)(5)(1),U=U:<<,X,U->U:<-=+1,U:<-G=Gn.n,nGn=GnU,G=GnnU-<,U:-H=Hn.n<,V=UmV=X,V:X-W=Wm<,V=WW:WV.m,Wn=WWm:<WV,W=WmmV:<-(a)<,V=WW:WV.3-535GnmWWm,(W)=min<:WV,Wm=WWm:(W)=WmH:HHn<(b)(n,m)×,Gnm(c)U:<<(n,m)×(Gnm)=U

9、:<.xU:<,xUU=(Hn).<nnHHnxWV.xH.xUV=WW:WV,m,WWm(W).(W)<,xWW(W)mV(W)U.<xUU<(W)=,xWHWmHGnm.(b)(d)(n,m)×,GnmxGnm,Wmm(x)=<:xWmm(x)|(Gnm)x|<.|(Hn)x|<(Gnm)x=WmH:H(Hn)xm(x),3-3.1X-Y-X×Y-U:<X×YB=BnnY-BB,<,H(B,)=V:V×BUVX,H(B)H(B)=H(B,):<,H(B)X-G(B)=Gn(B),

10、n(1)BB,Gn(B)(2)<,H(B,)=GG(B):GH(B,),Wn(B)=G×B:GGn(B),Wnm=Wn(B):BBm;(3)(n,m)×,Wnm(4)<,U=WW:WU.W=(n,m)×Wnm.(x,y)UVN(x),m,BBm(x,y)V×BU,xVH(B,).(2),n,GGn(B)xGH(B,).(x,y)G×BWn(B)Wnm.G×BH(B,)×BU 3.2X=Xin,Xi-i<i<nX-U:<Xn<,pn:Xi<nXin:XXn53641n,<,1W=

11、W:pn(W)UWXi,Vn=Vnmmi<nWn=Wn:<i<nXi-<,Wn=VVn:VWn.1Gnm=pn(V):VVnmG=Gnm:(n,m)×.iViXi(1)<,U=GG:GU.xU<1xii(Vi)U.m=1+max()W=i<mVi,Vi=11xpm(W)=ii(Vi)U. Vi,Xi,i,i,pm(x)WWn.(n,m)×,VVnmpm(x)V.11xG=pm(V)pm(Wm)U,GGnmG.(2)(n,m)×,GnmxX,pn(x)i<nXiXVnm|(Vnm)pn(x)|<.X|(Gnm)x

12、|(Vnm)pn(x)|<.3.3X=X:A,Ya= -aA<,X:a×s:Aan,Xn=xX:|Q(x)|n.pa:XYax=(x)AX,pa(x)=(x)A,x= x,s,a,Aa.U:<VnmXX Vnm (n,m)×(i)(ii)<,U=V(n,m)×Vnm:VU.n=00=min<:sUm,Vom=U0.in, Vim mVimYa Vn+1m m.aAn+1,YaWa=Wamm-U|YaYa-3-537(a)n,Wam(b)<,UYa=WWam:WUYa.mm,Wam=WXn:WWam,WWam.(W)=min<

13、;:WUYa,1Vam=pa(W)U(W):WWam,Vn+1m=Vam:aAn+1.(c)Vn+1mXxVn+1m,(3,Lemma1),=aAn+1:xVama,|(Wam)pa(x)|<,|(Vn+1m)x|(Wam)pa(x):a|<.(ii)<,xU,a=Q(x)|a|=n,x=pa(x)YaXn1.(b),mWWampa(x)WXn=WWamWWUYa.1xpa(W)U(W)U(W)U1pa(W)U(W)VamVnm.X-44.14XLindelofYX×Y-Example6.11XYLindelofTychonoX×Y(1)X×YBCRTY=RHausdor5Theorem3.3,YX×Y(2)X×Y-X×Y-X×YX×YX×YLindelofX×YLindelof4.2Lindelof-XnN,XniLindelof4XLindelof4XRemark6.21i,Ai=BiCiAiExample6.11,Xi=RiLindelofHaus