高等数学(4)

922A.N.DRANISHNIKOV,STEVENC.FERRY,ANDSHMUELWEINBERGER

Lemma2.2.Let(X,d)beacompactmetricspacewhichislocallyk-connectedforallk.Foreachn,theopenconeonXhasacompleteuniformlyn-connectedmetric.WewilldenoteanysuchmetricspacebycX.1

Proof.Wewillevenproduceametricwhichhasalinearcontractionfunc-tion.Itsconstructionisbasedontheweightedconeoftenusedindi erentialgeometry.Drawtheconevertically,sothathorizontalslicesarecopiesofX.

Chooseacontinuousstrictlyincreasingfunctionφ:[0,∞)→[0,∞)withφ(0)=0.LetdbetheoriginalmetriconXandde neafunctionρ by(i)ρ ((x,t),(x ,t))=φ(t)d(x,x ).

(ii)ρ ((x,t),(x,t ))=|t t |.

Wethende neρ:OX×OX→[0,∞)tobe

ρ((x,t),(x,t))=inf

wherethesumisoverallchains

(x,t)=(x0,t0),(x1,t1),...,(x ,t )=(x ,t )

andeachsegmentiseitherhorizontalorvertical.Itpaystomovetowards0beforemovingintheX-direction,sochainsofshortestlengthhavetheformpicturedabove.ThefunctionρisametriconOX.ThenaturalprojectionOX→[0,∞)decreasesdistances,soCauchysequencesareboundedinthe

[0,∞)-direction.ItfollowsthatthemetriconOXiscomplete.WewritecXforthemetricspace(OX,ρ).

Itremainstode neφsothatcXisuniformlyn-contractible.Wewillde neφ(1)=1andφ(i+1)=Ni+1φ(i)fori∈Z,wherethesequence{Ni}willbespeci edbelow.Fornonintegralvaluesoft,weset

φ(t)=φ([t])+(t [t])φ([t]+1).

“c”notationincXreferstoaspeci cchoiceofweights.Thereprobablyshouldbean“n”inournotation,butweleaveitoutforsimplicity.1The i=1ρ ((xi,ti),(xi 1,ti 1))

搜索“diyifanwen.net”或“第一范文网”即可找到本站免费阅读全部范文。收藏本站方便下次阅读，第一范文网，提供最新高中教育高等数学(4)全文阅读和word下载服务。