熊金城著的拓扑学部分习题解答II

11.(1)R;1,ab∈()()ππ,,,22fgab≅≅⎛⎞⎯⎯→−⎯⎯→−∞+∞⎜⎟⎝⎠()()ππ2xbfxba−=+−()tangyy=,fg()():,,hgfab=→−∞+∞h2b∈()()(),0,1,fgb≅≅−∞⎯⎯→⎯⎯→−∞+∞()exbfx−=()1tanπ2gyy⎛⎞=−⎜⎟⎝⎠,fg()():,,hb−∞→−∞+∞hgf=h3a∈()():,,ha+∞→−∞+∞(2)nnR,.,iiab∈()1,2,,in=()()()()()()()1122,,,1,11,11,1O0,1fhgnnnababab≅≅≅×××⎯⎯→−×−××−⎯⎯→⎯⎯→()12121122,,,21,21,,21nnnnxaxaxafxxxbababa⎛⎞−−−=−−−⎜⎟−−−⎝⎠()()()11211210,0,,0,max0,,,max,,,,max0iinniinniinyhyyyyyyyyy≤≤≤≤≤≤⎧=⎪⎪=⎨≠⎪⎪⎩()()1212π,,,tan,,,2nngyyyyyyy=⋅,,ghfghfϕ=nnRnnR()()O,O0,1gnxψε≅≅⎯⎯→⎯⎯→()()1211221,,,,,,nnnyyyyxyxyxψε=−−−2.YX(),YXX(),(1)YXYA⊂YAX;YA⊂Y()YAUUYλλλλ∈Λ∈Λ==∩∪∪YUλY2UλXYXUYλ∩X()AUYλλ∈Λ=∩∪XAXAAY=∩AY(2)YX,YA⊂YAX;AYXG()AXGY=−∩YXXG−X()AXGY=−∩XAXAAY=∩AY.(3)YX.YA⊂.)(int)(int)(intYAAXYX∩=int()int()int()XYXAAY⊂∩int()XaA∈XXaUXaaUA∈⊂AY⊂XaaUYAYA∈∩⊂∩=int()YaA∈XaaUAY∈⊂⊂int()XaY∈.int()int()YXaAY∈∩int()int()int()XYXAAY⊃∩.int()int()YXaAY∈∩int()XaY∈int()YaA∈,XYaaUU∃..,XYaastaUYaUA∈⊂∈⊂YaUVY=∩VX()()XXXaaaaUVUYVUYVAYA∈∩=∩∩=∩∩⊂∩=int()XaA∈.YAAXY∩)()(∂⊂∂,.XintYintXY;X∂Y∂XY.()()()YYYACACYA∂=∩−()()()()XYYAYAXAYCACXA∂=∩−∩=∩−∩YAAXY∩)()(∂⊂∂.11,,2AY⎧⎫==⎨⎬⎩⎭X=()YAφ∂={}()()0XAYAYAφ∂=∪∩=≠∩()()YXAAY∂≠∂∩.4.YX,Yy∈.(1)LX,YLY;GY,yG∈,XU,GUY=∩.LX12,,,nLLLL∈12nyLLLU∈∩∩∩⊂3()()()12nyLYLYLYUYG∈∩∩∩∩∩∩⊂∩=YLY(2)yWyXyYWyYGyY,XU,GUY=∩.yWyX12,,,nyLLLW∈12nyLLLU∈∩∩∩⊂,()()()12nyLYLYLYUYG∈∩∩∩∩∩∩⊂∩=,yYWyY.5.(,T)X1(,T)YXY⊂.(1)),(1TY),(TX,XYi→:.UT∈1(,)YT(,)XT()11iUUYT−=∩∈XYi→:.(2)XYi→:1YTT⊃.YVT∈UT∈VUY=∩.XYi→:()11iUUYT−=∩∈1VT∈1YTT⊃.6.XY.YXf→:)(:XfXf→()()12,,,XYττU()fX2Gτ∃∈()UGfX=∩.YXf→:,()11fGτ−∈()()()11fUfGfX−−=∩()11()()fGffX−−=∩111()()fGXfGτ−−=∩=∈)(:XfXf→.2Gτ∀∈,)(:XfXf→()()()111111()()()()fGfXfGffXfGXfGτ−−−−−∩=∩=∩=∈,YXf→:.7.XY,AX.:YXf→:,:AfAY→.()()12,,,XYττ,2Gτ∀∈,()()11()AfGfGA−−=∩,()11fGτ−∈.()1fGA−∩A,:AfAY→.8.XYAX(1)YXf→:)(:AfAfA→;()()12,,,XYττ,()UfA∈()fA2Gτ∃∈()UGfA=∩()()1AffA−⊂()()()()()()()11111()AfUfGfAAfGffAAfGA−−−−−=∩∩=∩∩=∩()1()AfU−AAf1()fAf−4)(:AfAfA→(2)XYXY.XY:fXY→():fXfX→AX(1))(:AfAfA→XY.9.2R{[,)[,)}Labcd=×},,,,,|dcbadcba∈R2RTL.}1|),{(2=+∈=yxyxAR.A2(,)T?(:(,)AAT).(1)[)[)21,,nnnnn∞==−×−∪,[)[),,nnnnL−×−∈2L.()2LL,11111{[,)[,)}Labcd=×,12LLφ∩≠.()312max,aaa=,()312min,bbb=,()312max,ccc=,()312min,ddd=33333[,)[,)Labcd=×12xLL∀∈∩312xLLL∈⊂∩.,2RTL.A2(,)TA.ALA(),xyA∈1yx=−[)[),1,1Bxxyy=+×+BL∈(){},AxyBAL=∩∈(,)AAT10.:XXf→Q:.QR.1X{}1012,,,,,,,,nnXxxxxxx−−=Xf→Q:()(),2,21nftxtnn=∈+++∩()1()2,21nfxnn−=+++∩Xf→Q:Xf→Q:.2X{}01,,,nXxxx=Xf→Q:()()(){}()0,,,221,2,1,2,,1,21,inxtftxtiiinxtn⎧∈−∞∩⎪⎪=∈+−+∩=−⎨⎪⎪∈+−∞∩⎩(){}10,1,,ifxin−=Xf→Q:Xf→Q:.11.:(1)?5(2)?(),fXτ(),cXτ{}{}cfAAXτφ=∪{}{}ccAAXτφ=∪1X2Xfcττ==()():,,fcidXXττ→X2X2Xfcττ==()():,,cfidXXττ→X.1.),(ρXR→×XX:ρ.()(),,nnxyxy→()()()()(),,,,,0nnnndxyxyxxyyρρ=+→()(),,nnxyxyρρ−()(),,0nnxxyyρρ≤+→R→×XX:ρ.2.),(11ρX),(22ρX,R→×××)()(:,212121XXXXdd212121),(),,(XXyyyxxx×∈==,1111222(,)(,)(,)dxyxyxyρρ=+)},(),,(max{),(2221112yxyxyxdρρ=(1)1d21XX×;1()1111222,0(,)0,(,)0dxyxyxyxyρρ=⇔==⇔=2()()11112221112221,(,)(,)(,)(,),dxyxyxyyxyxdyxρρρρ=+=+=31111222(,)(,)(,)dxzxzxzρρ=+()()11111122222211(,)(,)(,)(,),,xyyzxyyzdxydyzρρρρ≤+++=+1d21XX×.2d21XX×.1()2111222,max{(,),(,)}0dxyxyxyρρ==111222(,)0(,)0xyxyxyρρ⇔==⇔=2)},(),,(max{),(2221112yxyxyxdρρ=111222max{(,),(,)}yxyxρρ=()2,dyx=32111222(,)max{(,),(,)}dxzxzxzρρ=}{{}111222211222max(,),(,)max(,),(,)xyxyyzyzρρρρ≤+()()22,,dxydyz=+.2d21XX×.(2)21XX×21,ddρ,ρ.61()[][]22111222,(,)(,)xyxyxyρρρ=+()()[][]()2211112222,,(,)(,),dxyxyxyxydxyρρρ≤=+≤12ddρτττ⊂⊂12ddττ⊃.2dAτ∈A122(,)XXd×xA∀∈0ε∃2(,)dBxAε⊂{}{}2211122212(,)|(,)max(,),(,),dBxydxyxyxyyXXAερρε==∈×⊂11112222(,)(,)(,)2(,)dxyxyxydxyρρ=+≤12111122212d(,)|(,)(,)(,),(,)22dBxydxyxyxyyXXBxAεερρε⎧⎫==+∈×⊂⊂⎨⎬⎩⎭121(,)XXd×xAxA121(,)XXd×1dAτ∈12ddττ⊃.121dddρττττ⊂⊂⊂12ddρτττ==.2()[][]22111222,(,)(,)xyxyxyρρρ=+()()[][]()()22111122221,,(,)(,),2,dxyxyxyxydxydxyρρρ≤=+≤≤()()()()121,,,2,dxyxydxydxyρ≤≤≤21,ddρ3.2n.),(11ρX),(22ρX(),,nnXρn,121212,:()()RnnddXXXXXX×××××××→121212(,,,),(,,,)nnnxxxxyyyyXXX==∈×××,()1111222(,)(,)(,),nnndxyxyxyxyρρρ=+++2111222(,)max{(,),(,),,(,)}nnndxyxyxyxyρρρ=(1)1d2d12nXXX×××;(2)12nXXX×××21,ddρ,ρ.()[][]22111,(,)(,)nnnxyxyxyρρρ=++()()()()121,,,,dxyxydxyndxyρ≤≤≤721,ddρ.4.1X2X,21XX×1XA⊂2XB⊂(1)BABA×=×;ABAB×⊂×()12,xxAB∈×iiixUX∈⊂1212UUXX×⊂×()()12UUABφ×∩×≠()()12UAUBφ∩×∩≠12,UAUBφφ∩≠∩≠12,xAxB∈∈()12,xxAB∈×ABAB×⊃×()12,xxAB∈×12,xAxB∈∈()1212,xxVXX∈⊂×iiixUX∈⊂()1212,xxUUV∈×⊂1xA∈1UAφ∩≠2UBφ∩≠()()12UAUBφ∩×∩≠()()12UUABφ×∩×≠()VABφ∩×≠ABAB×⊃×.(2)oooBABA×=×)(;()oooABAB×⊂×(),()oxyAB∀∈×,xyUU∃(),xyxyUUAB∈×⊂×,xyxUAyUB∈⊂∈⊂(),ooxyAB⊂×()oooABAB×⊂×()oooABAB×⊃×(),ooxyAB∀⊂×,xyUU∃,xyxUAyUB∈⊂∈⊂(),xyxyUUAB∈×⊂×()oooABAB×⊃×(3)()(())()ABABAB∂×=∂××∂∪.AAA∂=−AAAXA−=∩−.AAAXA−⊂∩−.xAA∈−,,xAxA∈∉.Ux,UA⊄,()UXAφ∩−≠,xXA∈−,xAXA∈∩−.AAAXA−⊂∩−.AAAXA−⊃∩−.xAXA∈∩−,,xAxXA∈∈−,xA∉xAA∈−.,xA∈,xxU,xxUA∈⊂,()xUXAφ∩−=,xXA∈−.xA∉,xAA∈−,AAAXA−⊃∩−.,AAAXA−=∩−,AAA∂=−.()()ABABAB∂×=×−×;()()()()(())()ABABAABABB∂××∂=−×∪×−∪()()()()()ABABABABAABBABAB=×−×∩×=×−∩×∩=×−×;()(())()ABABAB∂×=∂××∂∪.5.1X2X,1A2A1X2X21AA×821AA×21XX×.21AA×1τ21AA×21XX×2τ1Gτ∀∈1,2,GGGααα∈Λ=∪×121,2,,AAGGααττ∈∈()()()()1,12,21,2,12GUAUAUUAAαααααα∈Λ∈Λ=∪∩×∩=∪×∩×2Gτ∈.,2Gτ∀∈,()()()()1,2,121,12,21GUUAAUAUAαααααατ∈Λ∈Λ=∪×∩×=∪∩×∩∈.,21AA×21AA×21XX×.6.321,XXX(1)21XX×12XX×;(2)321)(XXX××123(