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Extra resources for Essays on Topology and Related Topics: Memoires dedies a Georges de Rham
Example text
Y v q SÆqq SÆ Vlqfh wkh uljkw phpehu ri wklv htxdwlrq jrhv wr }hur dv q $ 4> lw iroorzv wkdw S SD y h{lvwv dqg lv htxdo wr v= @ > ij D q q iru doo Ohw =@ ^R qdR q d frxqwdeoh vxevhw ri D= Wkhq iru 5 q dqg khqfh [ y nyd n @ 4@q $ 3 dv q $ 4= Sij Wkhuhiruh y @ 3 iru doo 5 D q = +5, Ohw eh dv lq Gh qlwlrq 43155 dqg D vxfk wkdw = Wkhq [ [ W y nW n v y ? nW n W v SÆ SÆ S zklfk vkrzv wkdw SÆ W y h{lvwv dqg lv htxdo wr W v= 441 Gxdo Vsdfhv [ Æ Qrwdwlrq 44141 Li [ lv d qruphg vsdfh zh ghqrwh wkh Edqdfk vsdfh O+[> I, e| [ Æ dqg uhihu wr [ Æ dv wkh +frqwlqxrxv, gxdo vsdfh ri [1 Sursrvlwlrq 44151 Ohw [ eh d frpsoh{ yhfwru vsdfh ryhu F1 Li i 5 [ Æ dqg x @ Uhi 5 [UÆ wkhq +4414, x 5 [Æ i +{, @ x+{, lx+l{,= Æ Frqyhuvho| li U dqg i lv gh qhg e| Ht1 +4414,/ wkhq i 5 [ dqg nxn[UÆ @ ni n[ Æ 1 Pruh jhqhudoo| li s lv d vhpl0qrup rq [> wkhq mim s l x s= Surri1 Ohw y+{, @ Lp i +{,/ wkhq y+l{, @ Lp i +l{, @ Lp+li +{,, @ Uhi +{, @ x+{,= Wkhuhiruh i +{, @ x+{, .
E|> }l @ dk{> }l . ek|> }l l1h1 { $ k{> }l lv olqhdu1 51 k{> |l @ k|> {l1 61 n{nS k{> {l 3 zlwk htxdolw| n{nS @ 3 l { @ 3= Qrwlfh wkdw frpelqlqj surshuwlhv +4, dqg +5, wkdw { $ k}> {l lv dqwl0olqhdu iru {hg } 5 K> l1h1 k}> d{ . e|l @ d k}> {l . ek}> |l= Zh zloo riwhq qg wkh iroorzlqj irupxod xvhixo= +4514, n{ . |nS @ k{ . |> { . |l @ n{nS . n|nS . k{> |l . k|> {l @ n{nS . n|nS . 5Uhk{> |l Wkhruhp 4515 +Vfkzdu} Lqhtxdolw|,1 Ohw +K> k > l, eh dq lqqhu surgxfw vsdfh/ wkhq iru doo {> | 5 K mk{> |lm n{nn|n dqg htxdolw| krogv l { dqg | duh olqhduo| ghshqghqw1 Surri1 Li | @ 3> wkh uhvxow krogv wulyldoo|1 Vr dvvxph wkdw | 9@ 3= Iluvw r qrwlfh S wkdw li { @ | iru vrph 5 F> wkhq k{> |l @ n|n dqg khqfh mk{> |lm @ mm n|nS @ n{nn|n= l Pruhryhu/ lq wklv fdvh =@ kn{>| |n5 = Qrz vxssrvh wkdw { 5 K lv duelwudu|/ ohw }{ k{> |l |= n|nS P DW K 573E OHFWXU H QRWHV= WRSRORJ\ DQG I X Q F W LR Q D O +Vr } lv wkh ruwkrjrqdo surmhfwlrq ri { rqwr |=, Wkhq S S k{> |l @ n{nS .
Ek|> }l l1h1 { $ k{> }l lv olqhdu1 51 k{> |l @ k|> {l1 61 n{nS k{> {l 3 zlwk htxdolw| n{nS @ 3 l { @ 3= Qrwlfh wkdw frpelqlqj surshuwlhv +4, dqg +5, wkdw { $ k}> {l lv dqwl0olqhdu iru {hg } 5 K> l1h1 k}> d{ . e|l @ d k}> {l . ek}> |l= Zh zloo riwhq qg wkh iroorzlqj irupxod xvhixo= +4514, n{ . |nS @ k{ . |> { . |l @ n{nS . n|nS . k{> |l . k|> {l @ n{nS . n|nS . 5Uhk{> |l Wkhruhp 4515 +Vfkzdu} Lqhtxdolw|,1 Ohw +K> k > l, eh dq lqqhu surgxfw vsdfh/ wkhq iru doo {> | 5 K mk{> |lm n{nn|n dqg htxdolw| krogv l { dqg | duh olqhduo| ghshqghqw1 Surri1 Li | @ 3> wkh uhvxow krogv wulyldoo|1 Vr dvvxph wkdw | 9@ 3= Iluvw r qrwlfh S wkdw li { @ | iru vrph 5 F> wkhq k{> |l @ n|n dqg khqfh mk{> |lm @ mm n|nS @ n{nn|n= l Pruhryhu/ lq wklv fdvh =@ kn{>| |n5 = Qrz vxssrvh wkdw { 5 K lv duelwudu|/ ohw }{ k{> |l |= n|nS P DW K 573E OHFWXU H QRWHV= WRSRORJ\ DQG I X Q F W LR Q D O +Vr } lv wkh ruwkrjrqdo surmhfwlrq ri { rqwr |=, Wkhq S S k{> |l @ n{nS .