M2sol - MIDTERM N0 2 Math 440/640 Spring 2010 NAME Qt...

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Unformatted text preview: MIDTERM N0. 2, Math 440/640 Spring 2010 NAME Qt; Lifififlmg Apfil 14, 2010 1. Let X 2: {a313, c} with the topology 7' = {@,{a}, {b,<:}.lX}. List all the open sets in the product topology of X X X and find the connected components of X x X. “W, My, K r' “_ i“ “1&5. fiat/quirk Skit? Min >4vz>a {um )3 >0 3 ‘ ~ ~ ' ”A K k ‘7 ‘ K at???“ >153 K {a} '33 x F 4? Peyt (”f I 71% A f ’3“ «war ' ~ L xi}. C. ’9' w 3 y ,a w, . 3% >< {Moi {1A,}: ? wt 3 n -\ 1&1”th fifijflfivMfi‘Qkfiw-QWQ £41Xfi4m1fifiw‘aimkv’? h R? _ 1 ) » 4 ., W 3: ”n V A i “It E3 3; {if {“3 3;!” « a w“ , u h ‘ “g, ‘ a M imkfizw H bk x713» 26‘1"“ L ~ K a” j fi WW2 mwwajzfimfi WflW/{YUMQXL 63:52 X X" :r/ m E: E’ ,3 k .3 3 it m"; 3: xt @333; 333333 ”3334331334 3: ’ “ ‘ w ’ ' M; - I , Z ., , 'tgm ’W '3 f M, 2 if? ‘4 , fl; ”if/2‘; “PC 32“? )2 MIN} K {533 0:3 5 23%;: XX ) I‘gtfg )3? 13$»; .2 {ab} C‘j 313‘ 3 [@933 j {’01ij X X 2. Consider f : ‘12 —> Elbflxby) : 172 —— 3/2. Describe f—l([—1,1]) and prove the identification map. Describe the corresponding subsets of the partition of E“) . ‘ 1; f is an 2 by 2 y 22 $232322“:in Mittitfi'éiigak ”9W: {’Vdfiwifizkfl’a '2 .2 “ I; “"3 9i ‘i V CW b : w I: H i Q: s Q E: Lgfii 1“ 142:2“: :c‘tfii.:2:2:¥‘<:‘ikb‘~‘~§“§ WW: 2222:”; (27 :2 232:2 2 . “:f 2 w ' j a“? , 2 '20..» {LN .‘ 2., i 1 a 2 f b EL ; ‘9 .? ”€224 ,1.) 2’ im’ Vii-BZTW flag!” @2222 fiiwflzm {‘5} if“? 422 thgm‘ 1 g? A5] “ V Z" 2%” [“2 Him 531 "”3 I!“ M 4i“ if). ‘ n ,, » :2 “in, i {4‘ f é; 35g; WV 3 2 2 2 2 2 $211,225; "3m “1222’: ix’VL’b‘iffiw if 33. 2) “£10“ 22’ 22" i / 2g“ 4 1/? n K Vg KW "d" “it « 3. Consider R z [0, 1] X {0, 1] with the induced topology from E2 and the map 7? : R, 77» 1W which identifies the points {fly} and (171 * y) Let X = R \V ([(L l >< {IV/2}). Prove that: 77(XV) is pajh connected but X 1.9 not, path comlectod. ”I w W , l (y, w,’ fl birmmwmy WAX amid?» W (753 3‘31}? V) 4'? 7R: ( 2‘ “i 313$) “ ’, m1 3 v ‘6 \. K) VE‘EM 7i {\X‘n V x} W?) ‘1’»? ‘9 {x \ 5 *\ i STia‘fifi» 5": [C3 V “M 3 W firgfia¥1¥1€£§4 V517?“ ”W” 13‘ " Iv § :13 a v Vs“! “wart/(Va WY”? 2..“qu ma ‘“ *v > , . . x 36% mfifia ”ii; i i; 3;” J R f 17‘ 77 x? ‘ h 47’» ‘7 YKS'EVL‘ ‘ ‘“ w” .“ 1 }’{J\&, mm A £1ng {““WC’ Q U} A“ ’ ~. A Mama $4M}? ywmm a, . L W MW“ / \‘ (”figmkfkaw .h j} 4. Let X be the infinite cylinder {(3531, 2) E 733 : 3:2 + 3/2 : 1} and define an action of the group Z on X given by the powers of the homeomorphism (r : X —> X, (Mm, y, 2) : (any, 2 + 1). Prove that the orbit space X/Z is homeomorphic to the Klein bottle. » L ~ 3 , "we: 3 w ~ 7 ‘ [{['w.§>kyj ) Vi , ,3 T’ iii» > {7‘5 i3“? ’2’ch i i 356%; Vfljvfl‘fl {i (i K g i”) g t , :3 K37» 1,4! -’ ‘ " ‘ r a 3"». x, E t i M i /"' ¢ (“Wit V - “ I“ 3‘ ’ r )4 xi; {\ wwwwwwwwww f?“ 3‘ r W m "V 1‘ E i 1 E i , ‘9 '3 r 7, \x , _, w 3» (it LNMM‘gr Mfr/l , ’47, , v, If a \ 1’ r» R%%QQ%,. anti/1 172,.qu.._ri<\ 7 # “Ad/‘1‘!) X 'I V 2 We 5. Let X , Y be topological spaces, and oonsider_X x Y with the product topology. Prove that for A C X a13d B C Y we have (A X B)’ = (A’ X B) U (A X B“), WhOI‘C C" denotes the set of limit points, and C denotes the Closure. ' ‘ ”Alma , " ”)3 ~ W , 2 A}? 223} A If 35 .. A A, Awéihm‘j” Hml’ A a“) A ‘\ {WA A $43 ‘ ~¢ A M _. I, 3 «I {A} k ._ I M x .25 M , WWI , ”1 :1 [5,5 m" I 5“"; IA f" K4 EMA»; L3,, H 12’) ‘5 A 3} I/ {4, y a w" (a gt m: m w .r i :5er . , M. L m . of Q 3A ayaug. AA” {3 Ag 13% AL A: A A“ (A a. “3 M {A 3 wvflwx AAQM g, A 7’: ,1? RA C, ‘- e’f’A‘x f A). ,A A, IA ”a /,a I f ,3“ ’ ~ ' " ~74: ['5’, s ' A/ H Eli) k} if}: yfi; 6?; ) (7w, fiffltkmhw A A": {flyk A . ? t, ,, w, I. v “Ab ,: ‘X‘ \e ‘13 “(3, ME“ VA if 6: WW A‘L’“ A A A A A ’ ,. w , I x A: ’ ; r» (“A r“ M , mum y "a A \H‘w ; ‘{ AA A MerV/KXML vi 7f N Way-J, ‘5’)" x , aw .~ _ A/‘a’fwgé': 11A A rxemzw%>\WaA%%,am Wghmw ; , I 6. Let f : X ——+ Y be continuous, and suppose we have partitions 73, Q of X and Y respectively, such that if a? b E X he in the same member of 73, then Ha), f (b) lie in the same member of Q. If X ., Y are the identification spacee given by these partitions, Show that f induces a, continuous map f:X—~>Y. ‘ t“! ; . ,fi‘fl .N ,, i we HJ"? Ely/ZWMMCJLLJX «livéij’wtmm» 52mg” Whimi‘flfi up»): i‘ _ 1..“ N A 1 ”3 ’ k4. _H 2) w, ,2 5 :5 U, r; f , , 5 i & ,‘tianv ** / i" i ‘ _ . J? '1 , I, 'm n ‘ g‘ngs 74‘?” km», 22;:- Cm Cw k g Qéi £5; ’Ki ’V ,5 A? 5‘14““ i i: L i [J at?) vti‘iff i,» Qfiviwflvk~ {:MVK g‘ *4“ 3 ‘ ~ CV ".«WJ {1.5%, k 35 “<— -- ' W V» {Claw W Nf‘ ' \ \f “j i} i 7< 2 ‘K i" (,9; if ‘3 i “t W ":1 £3}, “ i“ " '4? i n yam-«at 61%“ijV‘“’““*’ 3;} 5) i w , '5" i , N. :61» N ‘ g ‘imiizW-a 3 , ‘ 4» ii i»: _ 7 i u “My Y i a . J» mfifv mph/Ly»: 1'" I um «, mime Lv z Lu :1 m. flu WMéx‘ r? 9-K! i3 [pm§:',§f§flgf.;a~7£v‘w’_.) I 0 a 3”, F' ‘5,“ . ”,1“ "' 3K {3 IV ’1 W \ the” nu— ‘M « F \.— ,, ‘ ~_ \ '1, |, w v “\mm/u W123 x“: mum-2%) \{ lfi LA, MMW ii g ”324‘wa iii/9:35 £3 7. Consider X 2 Hz \ {(0, 0)} and the paths 04718: [0,1] ~—> X, 04(5) 2 (cos 773,5in 775), [3(8) 2 {cos ws,38in 7T5). Prove that a and {3 are homotopic relative to {0, 1}. Deduce that the loop Oz - ,8‘1 is homotopic to the trivial loop at (1, 0). l l l: 3 lm ll 76 lm 33 W X «i ”FUR ii} lLL/m .4» @1790 SW Li‘s; .. 3:”? L‘ \ m _ if?" t g L} 3‘ WllULm “l: M ”WW/Uh “(*me 1 “WA \» l. 5:: ‘9) “L U) ) I L 5 9 l > l l )’ . h :7 \ l f . If ”t: 1" . p} ‘3, t . ) «v ; » .5; ”Sf 5% (xi 3%; (kmgls‘ ,jfi . {3 W A L LA. MA .. (l y f: , » - 2? \qu 3)?“ W V (EWLQ. (X """"" l ’2 f l, ’Ll W ‘ L j ,r'”) Ilka“ ‘ a“. b; HEM: ‘l‘n‘L/W‘L’LQ is“??? Ml l l L (J) . § v: lifikkfljflé. (:203 ,Q I) 8.1101, 51— — (:05 67115 — 1'. sin 6m ”1091); .f*(<€1>\ :: 1:) {z ‘ C: t. Fin Cal (for):>. 4 :1}, Ietf: 51 6] fl: ) a 23,a11dlet3 @1001) o 111 S1 based at 1 such that fig ((01))— (mm m W 7: :1» 7 1 11111 ; {Cf/V» 4 51? t “' {AWN fgwfi'ifi :) i ‘3; C123 (335,. i — (13) Whexe f* . 10,1] 4) 51 6(1) 2 711(153) fl ...
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