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1 Basic nsn metric inite Re on teu need on etc hen J Review o opology concepts o what a ology? 3 JCPC UEJ ti tlu 3 U UZEJ U NUZEJ Examples space RLP C[ 0 ]y complement RE lry R air CEO ]pt com 2 How to generate a ology? B and sub 8 s B B B sin sies}u } UEJ i txe U BEB EBCU 3 i Examples lr B a b acb a be Q } B

2 i criterion J UEJ Un on 4 closed subsets limit points closure t closed xe U txe U txea # t n B Y?i # i Yi } + $ nbhd U o x U to so xct t 7 x mbhd U o UN > UC Examples 5 Subspace ology by universal B and S C subset U UEJ property ii continuous h s continuous 7 " h ioh ioh continuous Z Example C C C then [ N s the closure C n o 6 product ology Jz Bx Ux E EJZ } universal property 7 Quotient ology h /ti Z oh C i x r i s continuous x ~ l i 3 Y s surjeetive UEY i U c x C ~ i cx C x [ ] /~ a Y h continuous oh s conti he e ail quotient an ti Y cx gluing points in [ x ] together

3 Quotient connected such locally Y on the inest that continuous universal property Y g i yzl9 g s continuous coat c got Continuity Y continuous i UEJY U C Jx ; n terms o ; or or closed subsets read notes Y cont k FL z Y continuous YUL a C td i i C closed ti i y Ud Y cont Y cont # # ly lai i Ua Y continuous Y continuous homeomorphm y and onto are continuous embedding Example x an embedding 0 x x 2 op 0 o z * normal 2 Connected path * * components path connected R [ RC [ 0 ] # txt } C s closed components read the notes connected 3 Compact Lo ] inite complement metric spaces uniorm continuity extreme value theorem Read the notes 26 7

4 Jr rom closed UK Jk lu Review the property Prove 3 a that t K C compact 2 2 C s 3k xez i check Proo o space s the set o U C satying in K a ology on 3 such that 3 C Jk c z n in 3 K n } C 2 x n n in 3 by ZNK closed in K in C } 2 n K xn x xe Znk xe 2 C compact z Y z Y are continuous maps o spaces i let xy i E i x zcxz } 4 Given y i the subspace xy r ix the product hen a prone that the projections x i s continuous j 2 Prove that the map ixy s i z s Y called an map i t UC CY Proo al xxz# in i U yuz 2 o xy i s given by U Uz U c Uzt z t suices to prove that U y Uz C s U xyvz YU xu U Uz y Check U Uzn y * U i C UZ z s z U s continuous Uixy Uz C

5 let Prove suppose Write i } Let Prove Review 2 have the with subsets & 2 } be a that subsets o x i 2 } are o the Ux u orm 2 space where C U and U are n 2 B C C are connected and Bnc t $ that 3 U C 2 s a connected subset o x 2 } Proo n element in a o 2 Ux 2 or ik } where U E x so subsets o 2 are W!±Ui 2 } U ;Yyjx i } U ±Ui W ; W U Yu i jej } u U u so W Ux U 2 U C U U are n czl as a djoint union o subsets win u WZ where W U U ix 2 Wz Uz U i 2 i CU ; n ti Win BU u C 2 WZ Bauzlxl U ki 2 hus WW BU u 3027 u nv u Cnn 2 Bx u C2 B BNU UCBU c v Cnw cnv C connected suppose CN Cnz C C C z then o ± CNBC BC UZB BNU BnUz B BC Uz hus Win$ connected

6 Need C Y F Review example 2 Y s a projection between spaces Prove that i Y s compact then closed a closed i t C closed FC CY closed map Y called Proo C Y closed C C closed ^ C txe tiwxyc*gyjn By the ube Lemma? Y 7 an nbhd U o x in Ux Y C xy C Y s UyUy yxuy C xy x Y C n YEY C xxy U yi C Uyi Y i t YEY xxy E x Y mbhd Uy o y and y o such that compact 7 Y ;YnEY n ake ny U # Uyi Y xy C xy C Y i hus C Ux Y Y x xe U C El

7 suppose Review example 3 Y a map x cxi xe } C Y between spaces Y Hausdor # and compact Prove that continuous C i Y s closed Proo Need Y to x y E Y cxl ty in Y which s Hausdor 7 djoint sets U and the U ye Un x y e i U C s Y Gg unv x Y s a closed map since Y compact CY closed need to prove that C C s closed Y Y in t d x ix eicly # the a C n x t t closed closed d closed n