such T Etr ics I Topology * Product ology Recll tht for spces Y we defie product ology Y Ty o B {UV Ue3 ve Ty 2 y y C 4 i ( y Ty Y y Ty corsest tht projectis Tl Tly re ctiuous Note Ue J ( U U Y VE Jy Thi ( v V 5 { Ttjllu UEJ U { Th ( v VET is bsis Now how bout product YZ? Ty Suppose ; re spces wh s J respectively Wht product ology defie Csider i th Og product projecti It such tht Tht mes tht t Uje Jj IJTU IT j ( li corsest ology projecti d such elemets geerted product tht S µ { Ttw ; UJET ; j lejs is ctiuous for y j j It Ujj Jii bsis product
R( i lr ll Rmk If V ; ti U U Ti ( Ui I product ii E lrc lrt typicl set ( b C c d I e f HEIL! this is bo such boes become bsis to geerte product which sme s obtied by blls I geerl 2 lr IRC 23 IRL I ology Two suppose z s is Ifie fmily ologicl 7m we c defie II { ( ili ie ti Wht is product? Eve more geerlly let be empty ologicl spce for y iet T is Ide set TYTI { ( li ie tie T Wht is product Tel? Ide tjel IT i like j j product ITI corsest ology such tht j th projecti ctiuous tj ET Rmk this mes t Uje F d ll such elemets geerte i e S ( Uj ITI ietllj product { TJYU ; Uje Jj bsis Ti iet Uj Tl
Let i let El Rmk If T is Ifie set Vie Ji tie ttv ; tiiwi NOT I product ology ctully sets IT re forms IT Ui IT ie TI where C T y fie set d U ; tie be spce tiet i C ti IT T TIT Pro let Ihm il e IT T iet Ee IF lie t i e ielti tie T We bhd U U t bhd U ± U is form U TV where C T is fie sice ie T ti Uii to tie Tke YIEUII I ( z Yie i ie H suppose ± ( so UNIT ; t $ iletl# we pre IE prove tht O Yile UII ETTI IT ; IEH tht ±lil Iti lijej t bhd U bhd i y Ti ( U ti if i i so ± I;Yu ; U ; I c TH ti Vj J to je T tj
{ { 2{ Thm if f ; ; ctiuous tie T f( file IT ctiuous ( give iet lfii i Product im?k Pro elemet I form bsis TII ( Uj for some JET Uj E Tj IT f ( Uj UjT ( Ij ( Ujl HJ ( Ttj ( Vj f IT fj i tj fj ( U ; C is j sice fj ctiuous tj f is ctiuous Eh Emple i 23 { 2 B{ { i 4 S{ 3{4443 sets re? { i 2 2 2 { 2 { 2 {242 {242 { 2{442 W I Z{ 23 CZ fie il typicl set is IT { 2 TI IE iczl Rmk 0420 E lrc Prop let ie T be fmily spce product tyi i Yi C where spres Teti Y i C sme s Y C ; csidered s spce
( bsis ie bsis J y * metric ology If ( d metric spce B{ Bl e E > o is which geerte J This metric is spce wh is clled metrizble if eists metric d such tht 3 is sme s from d I this cse is Clled metric spce wh metric ( d ology E lr B{ ( b c b lr metric spce wh d( y Thm If Y is metric spce wh metric dy f Y homeomorphism is metrizble ( this mes tht metrizble ologicl property Pro d Y ( e i z d( dy ( fc flz Esy to check tht d is metric which iduces sme Thm d metric spce t IR I ( y mi { dc y El d lso metric which Iduce sme s d Pro Ic y zo o y I ( 4 I ( y d ( y s d ( z t ( z y t y ZE ( if I ( z 2 or d ( z g 2 true ; If I ( z < I d ( z y c d ( y s d ( g d( Y Z d ( y e t ( g d ( y z l 2 Whey y iduce sme ology? B{ Bdc e oyy for ( d
D D Ei bl Yill ( IR lrdp dp ( Eli i i pe d ( m{ li Yil ieie d IR ologies lr dp ph ll pb re sme IR dz Euclide metric TIR Rw hm IR be IR defie I ( b mi { I For ± IE defie D( ± g This bouded metric sup { dkili izi D metric tht iduces product Pro is metric ( 20 o D ( 7 trigle Iequly I t s DC z Dlz dyig izl izl dig dye DC I e D ( 2 D( I I 2 3 product Jz is metric ( i # IR j H U E Tz t±( ; e U let IE By ( E CU choose N lrge eough such tht µt < E clim E V IT li e IF Pro Ie V Dc < E e Idiiyi < e { ye Bgcz!R C B < E iz ee IEN El C E t VET csider bsis elemet U TIUI for ology where Ui CIR for i i N Zt product d Ui lr for tiztl e U 7 o< E < leis N such tht ie ( ii Ei C Ui ie E mi { Eili IEIEN
I Ei Clim Bp ( c C U Pro t I ( iilil ye Bp i e tdus < E ti lei EN ( i Yi < IE < Ei < I Hi il < Ei Y ; e ( i t Ei C Ui lei < N is N iinil y e ITUI ITIR Or defie metric by f ( ± sup { dciiyi izi metric f clled uiform metric LRW 2 bo is geerted by bcis * Ui where Ui CIR is ti Thm uiform is fier th product ology