Topological Spaces

Uploaded by: hyd arnes
0
0

November 2019
PDF

This document was uploaded by user and they confirmed that they have the permission to share it. If you are author or own the copyright of this book, please report to us by using this DMCA report form. Report DMCA

Overview

Download & View Topological Spaces as PDF for free.

More details

Words: 2,099
Pages: 10

Preview
Full text

"!#$

%'&

(*)+-,/.10/23.546879;:=<>,@?A0 +-.B2>:BCED=.GF 6H:3,k?:l+-,/.10@768)E+WDO6H:I.G<S7)E4.B2-NM.5);<S. 7APj23.J?Am8no4p?Mm89/.JF'PY9/)@<+36T7M)@:dF .Sq@)/.5F7M)

?A)6T)+-.B2-4p?AmoCr
]K.S+-236H:B +3,;68)/{6T)/NF/682-.5<S+3mTVw68)w+-.B2-]t:|7MPd7M0e.B):3.S+>:|7MP+3.5)#m8.J?MF/:|+37NM2-.5?A+3.B2c<SmH?A2-6U+WV?M:kDj.BmTm1?: NM2-.5?A+3.52jN.B)/.52-?Mm868+WVM_

Q

j@^@@e/jB /M

fY5@Q*-;p8-A
7M)~?[)/7)/.B]K0 +WV:I.B+a6T:?'<S7MmTmT.5<+-687)

7APO:I9/s@:I.B+-:L7MPvC

-; B¡u¢B S¢ C;:39;<>,|+-,;?p+J£

¤ ?¥|+-,/.L.B]K0 +WV|:3.S+i¦\?M);F+-,/.d:3.S+§?A2-.17[email protected]) ¤@ s ¥k+-,/.L9/)/6T7M)c7APX?A)[?A2-s/68+32>?A2-V<S7MmTmT.5<+-687)k7MPQ7[email protected])[:3.S+>:v6T:7M0e.B) ¤ <5¥|+-,/.L68)E+-.B2>:I.J<+36T7M)k7APX?q;)/68+3.L)9/]ase.B2g7APQ7[email protected])':I.B+-:v6H:O7M0e.B)_

:I9;s;:I.B+¨$7APQ"6T:O?[© 8ª¢B >«t¢B B 68P ?M);F7)/mTV6UPf6U+>:<S7]K0/m8.5]K.B)E+5C/¨1¬=®=¨dC 6H:v7M0e.B)_ ¯'723.iPY7M2-]t?AmTm8VC/?\
`.a
*-;p8-MÈ ©-É f¢ ÉM© e6UPl°Ê6H:g<SmT.5?M2vPY237]Ë+-,/.a
7AP+-.B)[2-.SPY.B2v+-7K²?:O?a+-7M0e7MmT7MN6T<5?AmÇ:I0@?M<S._ ÌiÍ / Î~ [rÏÐ .S+1Ñse.\?|)/7M)/.5]K0 +WVu:I.B+5_Gxv,/.K<S7MmTmT.5<+-687)~Òª¦/³I~ÓEC@
×

ØAÙ*ÚJÙÛ ÙÝÜÞTßÝàÛáBÚ5àBßoâ3ã

Vx

x

Vy

y

ä/åçæªèªéªè Öµêëªìlífî>ïJðoñªòóôdõªóoòõAìWóö÷ è È¡e«AÈH¢ © Ô3 B* KÝ>/A8-A _axv,;. ;pø^ BÔ¢B BOù ¤ ¥O7APlCZ:iF/.Sq;)/.JF 6T)

ýi.Sq@)/6U+-687)¿M_ þMÿk6T:d?k+-7M0e7MmT7MNMV'7)

:39/s;:3.S+a¶

7AP

6H:L7M0e.B)

DO68+3,

'´'¶

7M);m8V68PfPY7M2O.B4.B2-V

Cr

ÔÈ © Ý>/A8-A _ ;7M2L6T);:I+-?A)@<S.MCQ? B 2-.5:[email protected]<+L+37c+-,/.|:I+-?A)@F/?A2>FÇC]K.S+-236H<\+370@7m87NMV'7) 68Pv?M);F

?M);F y¶K_

+-,/.B2-.G6T:O?M)[7M0e.B)c68)E+3.5234p?Mm :I9@<>,|+-,;?p+ u´

Ê?t:I9;s;:I.B+O7AP }_Xxv,/.B)

ÌiÍ /Î~ [rT$Ð .S+ ¤ u³3°\¥[email protected]?+370@7m87NM6H

¶ ±PY7M2:37M]K.L¶´[°tÓ 6H:=?d+37M0e7MmT7MNV7)_Xxv,/.i7M0e.B)c:I.B+-:=68)?A2-.+-,/.g6T)E+3.B2>:3.5<+-687);:=7AP7M0e.B)k:I.B+-:=68)k DO6U+-, _Kxv,/6H:i+37M0e7MmT7MNV'6T:L<5?AmTm8.JF'+-,/. È¡e« © >« 7M2 Ô3 B É oÈÕp t*-;p8-A 7AP 6T) C?A);F ¤ =³ ¥ 6H:1<5?AmTm8.JFz?|+370@7m87NM6H?A);. pÔ0;- 5J« 7AP;?g0@768)E+1' ´k68P/+-,/.B2-.^.Bb6H:I+-:f?A)d7[email protected]):3.S+f¶23+ :3.S+"+, DO68+3,4z ´u¶K_v`.aF 7t)/7M+g2-.5RE9/6T23.1+3,@?p+5Ê + 68+-:3.Bm8PX6T:7M0e.B)_ +370@7m87NMVc°Ê7M)[§6T:<5?AmTm8.JF 6 É ¢B«MpÔ87 68Pg.54M.B2-V0;?A6T27MPiF 6H:W+-68)@<+K0@768)E+-:9Q ³;w a : ´ ,;?M:\?z0@?A6T27MP)/7M)/6T)E+3.52-:3.5<S+36T)/N )/.568N,Ese7M2-,/77 F/:5CX]\.J?A)/6T)/N +3,;?A++-,/.B2-.'?A2-.|)/.568N,s@723,/77 F/:<+>= 7AP5y ?M);F?+Au @ 7MPB} : :39;<>, +3,@?p+C+A=B4+>d @ ¦ ¤ :3.B.Df 6TNM9/2-.F;E _T¿J¥_^`w,/.B)[+-,/.G+370@7m87NMV6H:<SmT.5?M25C Dj.L7AP+-.B)[2-.SPY.B2+37t ?M:v?9g G ?M9;:3F/7M20' H :30;?<S.M_ m8]K7:I+O?AmTmZ+3,/.1+370@7m87NM6H
Ò

`.j<5?A)d.Bb0;23.J:3:r+3,/.j)/7M+36T7M);:f7APe<S7M)4.B2-NM.B)@<S.MCª<S7)E+36T)E9;6U+WVCª?M);Fa
se.d?+370@7m87NM6H,|+-,;?p+ Ä ´O+ PY7M2?Mm8mrÂQPN~_ 7AP 7M0e.B)[:3.S+-:5_ Ð .B+²?M);F

xv,/6H:=F/.Sq;)/68+36T7M)c:3?ªV :l+3,;?A+^+3,;.1:3.5RE9/.5);<S.i.B4M.5)E+39;?Mm8mTVKm86T.5:=.B)E+36T2-.BmTV\6T).54M.B2-V\)/.B6TNM,se7M23n

R

,/77 Fc7AP Q_

TS

ØMÙ*Ú5ÙÛ ÙoÜSÞ8ßÝàÛ;ãTÚJàSßÝâIã

P Q

UV

W!WX

-1

-1

f (Q) 2π

0 f (P)

ZY \[ ^] Y I_ `_(] Y Ta Ta _ `b ] _(] ZY Y Y ZY ^j "_ c_(] $dfe Y Dg ZY "h $di _ ([ ZY [ a Y %] (j Y ZY \[ ^] (_ [ (] Y _ $d ZY [ a Y 1_(] Y ZY [ %a F_ Y!Y 1b _ Y ZY fY5@Ql k PY9/);<S+36T7M)nmy£@poq 6T:© A¡;oÈ¡>;&¢ ?p+ry´~ 68P^PY721.J?M<>,)/.B6TNM,/n se7M2-,/77FOs 7APtm ¤ Z ¥v+-,/.B2-.a.Bb6H:I+-:1?)/.568N,Ese7M2-,/77 FO+ 7MP" :39;<>,'+3,@?p+Fm ¤ +a¥5us±_1`. :-?ªVK+3,;?A+Cu m 6H:O<S7)+-68)9/79;:v7M)c²6UP 6U+6H:px Î QL y Ð .B+ ¤ u³-°a¥L?M);F ¤ = ³ ¥[email protected]+WDj7[+370@7m87NM6H,;?M2-?<+-.B2-6~5 } .5FsVz+-,/.0/2-7M0e.B23+WVu+-,;?p++3,/.k68)4M.52-:3. 6T]K?MNM.G7AP ?A)[7M0e.B)[:3.S+6H:v7M0e.B)_X`.Lm8.J?ª4M.1+3,;.L0/2377MPr+37vl b.52- AÔo/- ÈH¢ 68P68+=6H:^?L7)/.Sno+37Mno7)/.MC7M)E+37L]t?A0?M);Ftse7A+-,k / xvDj7|+-7M0e7MmT7MNM6H pÔo>- È ©t68P^+-,/.B2-.K6T:G?k,/7M]K.57M]K7M2-0/,/6H:I] m £Ao _ z G 7]K.B7M]K7230/,;6T / ³-Àr ¥"o sEV m ¤8 ¥l ½l C/DO,/.523.v / ³>À&r ¥# DO68+3,[+3,;. +370@7m87NMV\68)@F 9;<S.JFsV\+3,/.19;:39;?Am@+370@7m87NMV\7M)= C?M);F? 6H:=+3,/.19/)/68+O<S6T2-óì ò>öëªò OìWò OòóoõJë å è r ê ëJìIóìXî>óoì î>ó å öóÝî>ó å ÷ çòðoì õAò å oö ðrò å öëªì æ ïªóì Që å ÝëGëpî Sì å SìWóoðoì å î æ ìWð Iì î>óQöëªì çì UöQî ñGó åçæ ëBöQì ñGõAò å ö ðfò öëªì å ó çì î ñ öëªì å öoìWó î /óoìWðoõAì *ö å Sì ÷ è írì Wì Möëªì å SìWóoðoì å î æ ìWð î>óì ò>ö çòoð ì=î ñLöoëJì å SìWóoðoì î>õ Iî ò>ö Mì Wò ö å ïªòïªð è

v

7MPi?z+37M0e7MmT7MN6T<5?Amv:I0@?M<S.c fY5@Q :39/s;:3.S+ <S7p4.B2v7AP §<S7M)E+>?A6T);:O?q;)/68+3.d:39/se
1

6H:'©

É© 6UP1.B4.B2-V7[email protected])

é

ØAÙ*ÚJÙÛ ÙÝÜÞTßÝàÛáBÚ5àBßoâ3ã (Ý+PY7m8mT7pD:OPY2-7M]

7M)

C

+3,/.\F .Sq@)/6U+-687)'+3,@?p+1?k:I9;s;:I.B+

7MPl

6T:i<S7M]K0;?<+g6T)'+-,/.d+-7M0e7MmT7MNMV

Ñ68P=?A);Fz7M)/mTV[6UP Ñ6H:G<S7M]K0;?<+G?M:G?|:39/s;:3.S+G7AP^68+-:3.Bm8P^DO68+3,23.J:I0e.5<S+g+-7|+-,/.2-.BmH?p+-684.

A 68)\}_Qxv,;6T:X?M:I+-:QDO6U+-,\+3,/.jP?M<S+ +3,;?A+X?G:3.S+X¶,?û]t?ª[email protected]?p+36T4M.5m8V CfVM.B+d)/7A+dse.7M0e.B)~6T) }_ ;7M2L+-,/6H:d2-.5?M:37M)rCDO,/6Tm8.tDj.|F .Bq;)/.K+3,;.)/7A+-687) 7AP

+370@7m87NMVL7MP 7M0e.B)

6T)

2-.BmH?p+36T4M.5m8Vt7[email protected])C/Dj.LF 7t)/7A+F .Bq;)/.G+-,/.L)/7A+-687)k7MPf2-.BmH?p+-684.BmTV<S7]\0@?M<+J_

\/JtAO@^5E5

QHÏ

xv,/.|:d6T)}?[+370@7m87NM6H
9/)DO6T.BmHF VM_

xv,/.t+-7M0e7MmT7MN6T<5?AmX0/2-7M0e.B23+36T.5:L7MPO]\.B+32-6T<|:I0;?<S.J:d
+3.523]t:a7AP

7M0e.B)

s;?Mm8mH:5CDO,/6H<>,

PY723]"?k2>?p+-,/.B2d:3]t?AmTml:I9;s;:I.B+d7APj+3,/.t7[email protected]):3.S+>:B_t(*)+-,/6T:a:I.J<+36T7M)Dj.

6T)+-237 F 9@<S.:39/s;:3.S+>:g7MP^?K+370@7m87NM6H
?

]K.S+-236H
· ´ :I9;<>,G+-,;?p+f¶$ º · · _ <S7MmTmT.5<+-687)§7AP)/.B6TNM,se7M2-,/77F;: 7APl?t0e7M6T)E+B´u§6T:i. pÔ0; - 5J«O. É ¢B PY7M2Cz68PfPY72g.J?M<>,[)/.B6TNM,se7M2-,/77 F + 7AP c+3,/.523.16H:j?a)/.B6TNM,se7M2-,/77 Fs ´v :39;<>,|+3,;?A+s+ _ +-7M0e7MmT7MN6T<5?AmZ:I0;?<S.iµ6H:M Ô>¢ © & ¡@ É .S8 6UP .B4.B2-V' ´[",@?M:O?K<S79/)E+-?As;m8.G)/.568N,Ese7M2-,/77 F|s@?M:3.MC/?A)@F ¢B © p¡Z« © & ¡@ É .S8

fY5@Qey

6H:r?O<S7m8mT.5<S+36T7M)L7MP :3.S+>:

68Pf²,;?M:O?K
\ ³B ¥XDO68+3, ³^g ´ 6H:=?as;?M:3.PY72 j_}xv,/.k<S7MmTmT.5<+-687) 7AP?AmTm^7[email protected])~68)E+3.5234p?MmT: ¤c ³S¥9±DO6U+-, ³z´ 6T:K?<S7M9;)+>?As/mT.|s@?M:3.|PY72a+-,/.':W+>?A);F/?M2-F~+37M0e7MmT7MNV 7M)?j_

xv,/.G<S7m8mT.5<S+36T7M)t7APr?Mm8me7M0e.B)k68)E+-.B2-4ª?MmT: ¤

ÌiÍ /Î~ [r Ï

+3,;.|:W+>?A);F;?A2>Fz+-7M0e7MmT7MNVz7M) 2>?p+36T7M)@?Amj.B);F 0e7M6T)E+-:

xv,9;:5C+3,/.d:I+-?M);F/?A2>F|+37M0e7MmT7MNVt6T:g:I.J<S7M)@Fk
ÌiÍ /Î~ [rúÐ .B+O"se.L?a]K.S+-236H
0

nP

7M0e.B)'s@?AmTmT:

ÌiÍ /Î~ [rE

(ÝP^

6H:+-7M0e7MmT7MN6T<5?Amr:30;?M
mT.5<+-687)[7APQ7[email protected])':I.B+-:

ûÒMÒÓ&'´[~Ó 6H:?Ks;?M:3.M_lxv,;.dF 6T:-<S2-.S+-.G+370@7m87NMV6H:vq;2>:W+?As/mT.MC+-,/.B)'68+6T: :3.5<S7);Fc<S7M9;)+>?As/mT.M_ (Ý+O6T:7AP+3.5)[9;:I.BPY9/mÇ+-7KF/.Sq;)/.d?+-7M0e7MmT7MNMV|68)k+3.523]t:O7MPf?Ks;?:I._ :3 °Ê6T:?\s;?:I.1PY72O+3,/.G+-7M0e7MmT7MNV|°±7M)[? 1 v'´c_ D ¡¢¡!£¢¤ h9/0/0e7:3. 6H:^?ds;?M:3.OPY7M2j°_ (ÝP¥N 6H:^?L)/.568N,s@723,/77 FK7AP¦u´c}C+3,/.5)t+3,;.B2-.6T: ?A)[7[email protected])[:3.S+1¶´u°Ê:39;<>,c+3,;?A+Bz ´'¶§3~ N _=h 68);:1·'´4

Dw >x

f pÎ

/K

Q

:3.S+²6UPX?A);Fk7M);m8V68P #<S7M)E+>?A6T);:O?\)/.B6TNM,se7M2-,/77F|s;?M:3.1PY7M2 'PY72O.B4.B2-V

¨

c© 0ª ;«

T¬

àã*â3ãlÙ lÙWÚ5â aã*â Hã

B OG·'?M);F41·ON~_j(Ý+

:39;<>,c+3,;?A+Gº · G·k¶K_Oxv,/.B2-.SPY723.C +3,/.523.L6H:g?M)'¸:39;<>,[+3,;?A+ z´ PY7MmTmT7pD:j+3,;?A+ #
®

%

7M)4.B2>:I.5m8VCA6UPr?a<S7MmTmT.5<+-687)t7AP7[email protected])|:I.B+-: <S7)E+-?A6T);:l?a)/.B6TNM,se7M2-,/77 Fs@?M:3.OPY7M2=.B4M.523V

}´}Ce+3,/.5)zPY72i.54M.523Vc7M0e.B):I.B+G¶ ´~°Ë?A)@Fu.B4.B2-V ´ ¶ +3,/.523..Sb 6T:I+-:F5=[´:39;<>, 4C=:7AP ³^Q# ?MÄ );F Cr2-.5:[email protected]<+-684.BmTVM_I/ 72¹º?; C D=.dF/.Sq;)/.L?t:39/s;:3.S+5= µ @ µ » 7MPQ²sEV j ¤@ 5= µ @ µ » Ò!u m ´k¼/ m ¤ e ½o¥ ' M: ½^¢½O¹ PY7M2O¾ ¿M³5ÁBÁBÁB³IÂ ÓÁ E _8¿ª¥

+3,@?p+ '´

xv,/.i+370@7m87NMVK7APf0@768)E+WDO6H:I.1<S7M)4.B2-NM.B)@<S.6H:=+3,/.L:3]K?Mm8mT.5:I+j+37M0e7MmT7MNVK7M)|

+3,;?A+O
O¾ ³^#oC>:3jCQ?A)@F¹Bº;_l`.L,;?ª4M.Dm Ä o¼mzDO6U+-, 2-.5:[email protected]<+g+-7|+-,/6H:+-7M0e7MmT7MNMV[68P^?A);Fz7M)/mTV[6UPm ¤ Z¥o¿m ¤ Z¥PY72g.54M.B2-V ~´ ³#Ý_1(ÝPm}´ Ä ?A)@F:M½1Àm ¤ e½o¥SC +3,/.5) +3,/.c:I.B+-:ÁC= µ @ µ » PY723]§?')/.568N,Ese7M2-,/77 Fs;?M:3.tPY7M2< m ´y_uxv,/6T: +370@7m87NMVt6H:O)/7M+jq@2-:I+g<S79/)E+-?As;m8._ xv,/.L:3.S+B5= µ @ µ » 6T) ¤ E;_T¿J¥=6H:F 68)@?p+3.J:BCj?A);Fy6H:K9/)/2-.5:I+32-6T<S+3.JF6T)y+3,/.'7M+3,/.52<S77M2>F 6T);?p+-.'F 6Un Ä 2-.5<+-687);:5_i¯'723.0/6T<S+39/2-.5:-RE9/.BmTVMC°5= µ @ µ » 6T:1:I7]K.S+36T]K.5:1?A0;,;:g0@?M:-:g+-,/2-7M9/N,z+-,/.3Ç :3mT?Mm87] NE?p+-.5:k È ?A+F ½ DO6U+-, 2>?MF 6T9;: ¹ ?M);F[<S.B)E+-.B2M: ½I_ ; s ?M:3.cPY7M2K+-,/.[+370@7m87NMV~7APG0e7M6T)+WDO6H:3.[
+3,;.d:I.B+-:

PY72?AmTmÇq;)/68+3.d:3.S+>:

PY7M2-]

¤ E@_ À¥ / m ¤ @½Ý¥ ' :M½0¢½¹>½fPY7M2O¾X$¿M³5ÁBÁ5ÁB³IÂfÓ DO,/.523.[Â ´ÌËjC\Ò¸ZÆp³BÁ5ÁBÁB³0 Ó ³#ÝCiÒ¸:EÆJ³5ÁBÁ5ÁB³;: ÓÆ=Cd?M);F¹-½º/_#xv,/.u:3.S+>:K7AP Ä Ä PY9/);<S+36T7M);:l68) ¤ E;ÁçÀM¥QDO6U+-,t68)E+-.B2-4ª?MmT: 7MPe4p?M236H?As/mT.vDO6TF +3,¹>½1ºGN.B)/.52-?A+3.=+3,/.i:3?M]\.j+-7M0e7MmT7MNMV ?M:d+3,/.[:3.S+-:DO68+3,}6T)E+3.B2-4p?AmH:d7MPg?'q/b .5F}DO6TF+-,se.5<5?A9;:3.5= µ @ µ » DO6U+-,Æ| ¹ ]K68)>¹ ½ÁºÍ' 6T: <S7)E+-?A6T)/.JF|6T);:36TF .G+-,/.d:3.S+O68) ¤ ; E _çÀM¥_ `.\:-?ªVc+3,;?A+L?+370@7m87NM6H,;?A23n ?M<S+3.5236l}5?A+36T7M);:G7MPv,;?A2>?M<S+3.B2-6l5} ?p+-687);:]K?ªV )/7M+v?M0/0/mTV\6T)k?)/7)/]K.S+32-6l5 } ?As;m8.g+37M0e7MmT7MN6T<5?Ame:I0;?<S._fxv,/.523.16H:BC,/7pDj.B4.B2JC?dN.B)/.52-?Mm86l5} ?A+36T7M)

&m

Ò '´[

7APf:I.JR9;.B);
¡ S¢ C+-,;?p+O:O68)c+-,/6T:vse77M{Z_

* &#

?A)@?AmT7MNM79;:=Dj?ªV

TÐ

ØAÙ*ÚJÙÛ ÙÝÜÞTßÝàÛáBÚ5àBßoâ3ã

/72X.Sb/?A]K0/mT.MCM+3,/.i<SmT7:39/23.

¨7MPZ?L:39/s;:3.S+^¨w7MPZ?1+-7M0e7MmT7MN6T<5?Am/:30;?M
l}

6H: +3,;.:I]t?AmTmT.5:I+

<SmT7:3.5FG:3.S++-,;?p+f<S7M)E+>?A6T);:Ç¨d_f(ÝP ±6H:Ç]K.S+-236 5?Ms/mT.MC5+3,/.5) ¨~6H:Z+3,/.=:I.B+r7AP mT68]K68+-:7MP
^Ò$; B¡;oÈ É

¢B

T

<5?AmTmg+-,/.

© Tª¢ Ô3 7AP^¨?A);FzF .B)/7M+3.6U+GsV

Ñ

~}

:I.B+k7APdmT6T]\68+[0@768)E+-:c7MPa:I.JR9;.B);
5Ó

¨ _Gxv,/.\:I.JR9;.B)E+36H?Amf<SmT7:39/23.a6T:1?|:39/s;:3.S+17AP

+3,;.d<SmT7:39/23.Cs/9 +g68+]t?ªV[email protected]?K:I+32-6T<S+:I9/s@:I.B+5C;?:v68mTm89@:W+-2-?A+3.5FksV+-,/.1PY7MmTm87pDO6T)/NK.Sb/?A]K0/mT.M_

y ®

ÌiÍ /Î~ [r

m > $#)oyDO68+3,t+-,/.O+-7M0e7MmT7MNMV Ô ³3ÂQP¿MC Dj.Lm8.B+ m&ÕCµ Ä ¤ e ¥jÖ <S7: ¤ ÔØ× ¦Z¥Ù# Ä Ä Á

7M)@:I6HF .B2^+3,/.i:I0;?<S.O7MPÇ?Mm8m;PY9/);<S+36T7M);: u£ /³5¿

M

7AP 0e7M6T)+WDO6H:3.G,

Bm&Õ

4m'sVt+-,/.L0@768)E+WDO6H:I.GmT68]K68+-:5C Ý>!ÔØ×TC>ÝK/³5ÁBÁ5ÁS³;ÔØ×8C m&Õ ¤ Z¥^ mT68] m&ÕCµ Ä ¤ Z¥^§Ü ¿ 687APM+-,/c .B2-DO6H:I.C ÄÅÚDÛ ´ Ø / ³B¿Þ# ³ m ¤ e¥^ Õ mT68] m&Õ ¤ e ¥^Ü ¿ 6U7AP1+3,;u .B2-DO6T:3.AÁ ÚÛ $Ò&m&ÕCµ !Ô³IÂØP ¿MÓ_Xxv,/.5)k+-,/.5:3.LmT68]K68+-::3,/7pDw+-,;?p+ Ä m Õ ´ ¨CÓr³ mu´ ¨CÓ Ó Á

`.dF .Sq;);.GPY9/);<+-687);:

Ð .B+g¨

?M);F

(Ý+v6T:=0@7E:3:36Ts/m8.g+37K:I,;7pD+-,;?p+j+3,;.i0e7M6T)E+WDO6T:3.gmT68]K68+j7MPQ?a:3.5RE9/.5);<S.i7APr
n / ³5¿$r# 6H:g

mT68)/.J?A2:30;?M,[)/.5.5Fc)/7A+se.aF .5236T4M.JFPY2-7M]

?

)/723]$7M2Q]K.S+32-6H,L+3,;?A+r+-,/.=7[email protected]?A+36T7M);:r7AP/4.5<+-7M2f?MF/F 68+36T7M)?A);Fd:-
Ý>/A83MÈ ©>É UÈ¡e É Ô1¢o É© C 72j?

Ý>/A83MÈ ©>É QÕª © ÝAÔg¢o ÉM© _lxv,;.

:30;?M, ?M:g+3,;. h <>,Dv?A23+ G:I0;?<S.C/?AmH:I7K?A2-6T:3.168)'<S7)/)/.J<+36T7M)cDO68+3,'F 6H:W+-236Ts/9 +-687)k+-,/.B723V ¤ :I.5. ,;?A0/+3.B2L¿¿J¥S_

0}

®

àeÎ~ xpf J;=@@e Ð .B+1° Æ CÇ° Ä se.a+WDj7t+370@7m87NM6T.5:g7M)'+-,/.:-?A]K.:30;?<S.d}_1xv,/.B) ° Ä 6H:i:-?A6HF[+-7|se.C ¡Z SÔ 72 °AÄ C@]\.J?A)/6T)/NK+-,;?p+i°GÄ ,@?M:O]K7M2-.L7M0e.B)':3.S+>:B@D=.d?MmT:37:3?ªV+3,;?A+ ¢Ô-p¡EE BÔ +3,@?A)u°;Æ168Pl°;ÆFw °;Æj6H:g© É Ô>¢B SÔ 7M2 ø^ Éá BÔ +-,;?A)°M Ä _ (ÝPÇ°;Æv6T:l:I+32-7M);NM.B2f+-,;?A)°MÄ CM+-,/.B)v oÉ| DO6U+-,t23.J:I0e.5<S+ +37° Æ 6T]K0/mT68.J:v+3,;?A+ oÄ É' DO68+3,[2-.5:[email protected]<+O+37° Ä _"/ 72O.Sb/?A]K0/mT.MC +-,/.dÄ :I+32-7M)/N.5:I+=+-7M0e7MmT7MNMV QTú

7M)

?A)Vz:3.S+d6H:G+-,/.F 6H:3
23.J:I0e.5<S+G+37[+3,;.

F 6H:3?A)E+5_xv,/.[D=.J?A{.5:I++-7M0e7MmT7MNMV

â TÙ ãfÚ5àTä*Þ~ª5ÜC«Ù*Ú5ÙÛ ÙoÜSÞ8â3ã

$å

7M)?A)V}:3.S+t6H:a+-,/.[+32-6T4E6H?Amj+-7M0e7MmT7MNMVC^?A)@F.54M.B2-V~:I.JRE9/.B);
Ä

+3,;.+-236T46T?Mmf+37M0e7MmT7MNVM_d(Ý+L6H:G0@7E:3:36Ts/m8.+3,@?p+G+WD=7k+370@7m87NM6T.5:G°;ÆªCQ° \?M23.\)/7M+L
Ä

]K.5?M)/68);N+-,;?p+g°;Æi6H:O)/.56U+-,/.B2vq@)/.B2);7M2O<S7E?A2>:I.52=+-,;?A)[° _

æ xp;=@JYJeQ Ð .S+a ?M);Fn s@.+WD=7u:30;?M, DO6U+-, +WD=7[+-7M0e7MmT7MN68.J:BCf°;Æp³-°Ä ?A)@F Æp³ d Ä 2-.5:[email protected]<+36T4M.5m8V_gh9/0/0e7:3.L+-,;?p+rm £ ¤ u³-°;ÆS¥o ¤ =³ ÆS¥6T:1?t]t?A0[PY2-7M] §+37

+3,@?p+6H:O<S7M)E+-68)9/79;:vDO6U+-,[23.J:I0e.5<S+j+-7K+3,/.L6T);F 6H
Ä Ä

*m

Ä "o "o Ä

¤ ?¥k(ÝPX° G6T:jq@)/.B2O+-,;?A)[°;ÆAC+3,/.5) '£ ¤ u³3° ª¥ ¤ =³ 5 Æ ¥j6H:O<S7M)E+-68)9/79;:B_ ¤ s@¥c(ÝP ¤ =³ ª¥j6T:O
D ¡¢¡!£¢¤

Om

xv,/.5:3.u:W+>?p+-.B]K.B)E+-:k?A2-.u?~F 6T23.J<+k
E;_ ÿ/_ xv,/.K6HF .B)E+368+WVz]t?M0Ø £ ¤ z³3°a¥ro

<S7)E+36T)E9;6U+WV6T)'ýg.Bq;)/68+36T7M)

DO,/.5)|Dj.g9;:3.g+-,/.G:3?M]\.i+370@7m87NMVK°

~H

v ¤ e ¥L,QCQ6H:L?c,/7M]K.57M]K7M2-0/,/6H:I]

¯

¤ u³-°¥SCrDO,/.523.

7)+3,;.1F 7]t?A6T)|?A);F|2-?M)/NM._fxv,;6T:j6H:^)/7M+=+-239;.iDO,/.5)

%

Dj.L9;:I.G+WDj7tF 6 e.523.5)+O+-7M0e7MmT7MN68.J:v7M)c}_ /72O.Sb/?A]K0/mT.MC/+3,/.L6HF .B)E+-6U+WVk]K?M0cPY237] <S7)E+-?A6T)/6T)/Nu?p+dmT.5?:W+a+WD=7'.5m8.5]\.5)E+-:L.JR9;680/0e.5F

?t:3.S+

DO68+3,}+3,/.t+-236T46T?MmX+370@7m87NMV'+-7'+3,;.|:I.B+a

.5RE9/6T0/0e.5FcDO68+3,c+3,/.dF/6T:-<S2-.S+-.i+-7M0e7MmT7MNVMC

t£

1o

¤ u³SÒp¦ ³3~Óª¥

t

¤ z³ ù

¤ ¥I¥³

6H:lF 6H:3
Dw >x

Ä

. +t° Æ ?A);F}° se.c+WD=7z+37M0e7MmT7MN68.J:a7)}}_}xv,/.5)}+-,/.[6TF .5)E+368+WV~]K?M0 f pÎ Q Ð S ¤ u³3° J¥j6T:

"o

Ä

Ä

D ¡¢¡!£¢¤ xv,/6T:v6H:?KF 6T23.J<+<S7);:3.5RE9/.B)@<S.G7APXxv,/.B723.5]E;_èç_ ¯ à>px ex!éQHÏA xv,/.k6TF/.B)E+368+WV ]K?M0Æ £ ¤ u³3°;ÆB¥9o ¤ u³-°Ä5¥a6T:?u,/7]K.B7M]K7230/,;6T:3]"68P ?A)@F|7)/m8Vk6UPX° Æ ° Ä _ Df w >px Î QHÏÐ .S+a°;ÆK?M);F~°tÄ se.t+WDj7[+37M0e7MmT7MN68.J:G7M) }_cxv,/.B) +-,/..5RE9;?Mm868+WVz7MP°;Æ ?A)@Fu° Ä 6H:i.5RE9/6T4ª?Mm8.5)E+g+37|+3,/.aPY7MmTm87pDO6T)/Nc<S7M)@F 6U+-687)£jPY7M2G?AmTmr+-7M0e7MmT7MN6T<5?AmQ:30;?<S.5: ¤ = ³ ¥SCÇ? PY9/);<S+36T7M)4u m £ ¤ u³-°;ÆS¥%o ¤ ^ ³ ¥=6T:O?A{68);N ¤ ^ ³ ¥L ¤ u³-° Æ ¥1?A)@F ¤ ^ ³ ¥L ¤ u³-° Ä ¥SCD=.t:3.B.\+3,@?p+L+3,;. <S7);F 68+36T7M)'6T]K0/m86T.5:+3,;?A+Ck £ ¤ u³3°;ÆB¥to ¤ u³-°JÄ ¥v6H:g?K,;7M]K.B7]\7230;,/6T:3][C;:37°;Ægû°GÄ PY2-7M] ® 7M2-7MmTmH?A2-V9E@_ Àf/_ ¯ +370@7m87NMVt6H:O7AP+-.B)'F .Bq;)/.5FcsV+-,/.d:[email protected]<S68q@, 0e7M6T)+J_`.c+3,/.523.BPY7M2-.kDv?A)E+\+37 <S7M]K0;?M23.k+370@7m87NM6T.5:68)+3.523]t:\7APi+3,/.5682K)/.568N,s@723,/77 F s;?:I.J:B_

ØAÙ*ÚJÙÛ ÙÝÜÞTßÝàÛáBÚ5àBßoâ3ã

Df w >xpÎ QHÏeÏ Ð .S+|° Æ ?M);F° Ä se.'+WD=7+-7M0e7MmT7MNM6T.5:K7)}_ûh9;0/0@7E:I.[+-,;?p+|PY7M2.J?M<>, ´ +3,/.523.t?M23.K)/.568N,Ese7M2-,/77 Fus;?:I.J:[Æ\?M);F ÄK7MP PY72d°;Æ\?A);F°ÄMCQ23.J:I0e.5<S+36T4M.5m8VC :39;<>,+-,;?p+vPY7M2v.54M.B2-Ve + ÆG´9' Æ+3,/.523.16T:O?<+ALÄ ´91Ä DO6U+-, +AÄr+eÆA_lxv,/.B)c°Ä16T:jq;);.B2j+3,;?M) ° Æ _

D ¡¢¡!£¢¤

xv,/.d,V0@7M+3,/.J:I6H:7MPf+-,/.d+-,/.B723.5]

5

6T]K0/mT68.J:O+3,@?p+ k£ ¤ u³-°

ÍE@_8¿ê/_

+36T)9/7M9@:BC/:37\+3,/.L2-.5:39/m8+OPY7MmTm87pD:=PY2-7M]Ëxv,/.57M2-.B]

Ä

Ä ¥o

¤ u³-° Æ ¥6T:i<S7M)/n

¯

Ä Øë Ä Ä 2-.5:[email protected]<+36T4M.5m8VC:I9@<>,u+3,@?p+GPY7M2L.54M.B2-VO+eÆ´ëµÆd+3,/.523.\6H:L?+AÄ|´ëÄ\DO68+3,n+>Ä ì+ZÆªCr?M);F +3,;.B2-.1?M23.);.B6TNM,s@723,;7E7 Fts@?M:3.5:M Æ ?A);F Ä 7AP¥kPY7M2O° Æ ?A);F|° Ä C2-.5:[email protected]<+36T4M.5m8VCE:39;<>,|+3,;?A+ PY7M2O.54M.523VsQÄL´v Ä1+-,/.B2-.G6T:O?s Æi´v[ÆiDO68+3,Os Æ5sQ Ä_ ýi6íHZ.B2-.B)E+G]\.B+32-6T<5:BCr7M21)/7M2-]t:BCÇ7) ?c:I0;?<S.\ <5?A)zmT.5?F'+-7k+-,/.K:3?M]\.+370@7m87NMV_/ 72 .Sb/?M]\0;m8.CX+3,/6H:K6H:t<S.B23+-?M68);m8V +-,/.c<5?M:3.k68P1ü Æk?A);Fyüîc Ä ?M23.|+WD=7]K.S+-236H,+3,;?A+ ü Æ ¤ Q³;:/¥^ÀAü Ä ¤ r³;:/¥fPY72=?Mm8m/Q ³;k : ´c_f¯'723.ON.B)/.52-?Mm8mTVMCM6UPZ+WDj7d]K.S+32-6Hxpex!éQHÏ@ú Q´y

. +g°;ÆG?M);F[° [email protected]+WDj7K+370@7m87NM6T.5:v7)c_^xv,/.B)'°;Æ° L68PX?A);Fc7M);m8V Ð S +-,/.B2-.?A2-.K)/.B6TNM,se7M2-,/77F s;?:I.J: 7AP ~PY7M2d° Æ ?A);F° C Æ ?A);F

68P=PY72d.5?<>,

rë

:3.S+7AP <S7)E4.B2-NM.5)+v:I.JRE9/.B);
Dw >x

f pÎ QHÏ; xvDj7\]\.B+32-6T
D ¡¢¡!£¢¤

xv,/.z0/2-77APL6H:k?}F/682-.5<S+c?A0;0/m86H
l}

<>,;?M2-?<+3.5236 5?A+36T7M)7MP
®

ÆE;_çÀ}?M);Fy+3,/.

7M2-7MmTmT?M23V

:3.5RE9/.B)E+-6T?Mm

¯

ïk¢oîxpQ

Q8

(*)[+3,;6T:i<>,;?A0 +-.B2JC/Dj.L,;?ª4M.GmT68]K68+3.5Fu7M9/2gF/6T:-<S9;:-:3687)|+-7t+3,/.as;?M:36T
1

Dð

* Àñ#?A);F òO9;F/68)n EÅñ&#Ý_

+370@7m87NMV_ ;7M2O]K7M2-.168) PY723]t?A+36T7M)C/:3.B. G.5m8mT.BV

LK

îx

ÌiÍ pMJ

Q

Åx B Åp x J'rÏ

B

vG

0H BG

Ï

ÌiÍ pJ'r h 9/0/0e7:3.L+3,;?A+ 6H:?t<S7]\0@?M<+g:39/s;:30;?<S.L7APl? i?A9;:-F 7M2 :30;?M
(ÝP^¨

6T:1?|:39/s;:3.S+i7AP^?t+-7M0e7MmT7MN6T<5?Amr:30;?M
^

;H

È¡;* BÔÈpÔ

Có

¨ a7AP^¨

6T:

+3,;.O9/)/6T7M)K7APZ?AmTm/7M0e.B)t:3.S+-:X:O+-,;?p+G<S7M)E+>?A6T)[¨aCÇ?A);Fc+-,/. > ¡Z« É Ô ¨ M 7 PX¨ 6T:iF .Sq@)/.5F'sV ¨$ ¨ Ã ¨ ¬ _

. &

ô

ô

h,/7pD+3,;?A+^?G:I.B+ 6H:l?A6T);: 68+-: se7M9/);F;?A2-VMCM?M);F\7M0e.B)\6UPÇ?A);F7M);m8V

\õ

68P 68+g6H:gF 6H: W7M6T)+OPY2-7M] +3,;.

®

68+-:Ose7M9/)@F/?A2-VM_j`w,;?p+1?A2-.i+-,/.a
4#DO68+3,[68+-:O9;:39;?MmZ+37M0e7MmT7MNV/ö

?A)E+372v:3.S+6T)

÷)ø â;äß3ÞUã*â3ã Åx

ÌiÍ pJ'rHúÐ .S+

¤ u³-ü Æ ¥d?M);F

Ä ¥Lse.k]K.S+-236H<|:30;?<S.5:5_ Ï

¤ =³3ü

Tù 237p4.K+3,;?A++-,/.k+37M0e7MmT7MNMn

6H
Ï Åx ÅxpJ'rúK

ÌiÍ pJ'rT ÌiÍ

+3,;.c]K.B+32-6T<':30;?M
¿M_çÀAÿ;_

ÍE@_Zç _

237p4.ixv,/.57M2-.B]

\õ

+370@7m87NM6H« 68Pr68+j6H:j)/7A+j+3,;.1F 6H: W7M6T)+j9/)/6T7M)k7AP+WDj7

)/7) no.5]K0 +WV7M0e.B)':3.S+>:B_

1Iö

¤ ?¥k`w,;?A+g?A2-.g+-,/.d<S7)/)/.J<+3.JF[:I9/s@:I.B+-:v7AP ¤ s@¥'h,;7pD#+3,@?p+ Ê6H:O<S7)/)/.5<S+3.JFk68P ²?M);F

ÉÃ

Ê?M23.L<S7)/)/.J<+3.JFÇ_ Ä ÌiÍ ÅxpJ'rLk h ,/7pD#+-,;?p+B6T:v,;7M]K.B7]\7230;,/6T

Topological Spaces

Overview

More details

Related Documents

Topological Spaces

Spaces

Spaces

Assignment On The Topic Topological Spaces (unit-i)

Spaces Ii

Blank Spaces

More Documents from ""

Analiz Iii 10

Continuous_functions

Introduction To Tensor Calculus

Complex Functions

The Contraction Mapping Theorem

Anal Iii 6