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Q()8I"5r=$%ha" 2_ǻ瞜.n+6R 8*y t{{ÙurZo'?pYm͠Nʹzl%*G {69LͲ_arpP%_QfQύe7aeU8r&KJ\6V:*lm[k\ }=yÔB~sM3hۄi+pkOJ%S1t'rpIm9b.23^ل_FH sjHep 3hgԢμKm~&S3?غXQ-mfHʆi 7"[/ )/ ?@$pw> Rd{% Hr}vuԕY(ICav*+RK@sި `Kyy؆| D$$DT5w{*ۙ\6S xL1''ggzd= E-^E+аdSҋ0l- sA++KhEG,=Ѻ4wĪT Rץ1qjY~/#EQuykfv P8j-HYi{_{*'mDи cwwwc=a&qw"s/ocЧ&`IpO 4 J{IJ#_4RgaѦFT/ "ſg[M& *3f E  D  A(??(49'1 'D,'E9) 'D-/J+5H1) 2349'1 'D,'E9) 'D-/J+xxpH  0޽h ? 3fffff3̙3f̙80___PPT10.A" 0@ L0 ZR(  ("   pFY0e0e?Title 1`  Y dExamples ff  D    OY0e0e?,Content Placeholder 2" PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!y;O^Edrs/shapexml.xml\mo9~_O AT R*T}b`g ^V~}xg粒 MdBp^& r}c1$O_Ί)*K(:hT*&}Up9ՃJ2rQUǕe)9mRդIK!%販EM,C%z\CEѴT:2c2":pTxCb|ӄkQY6@LˏN/Tov~]ctO'bJRM3 =fS">JӃziM褆]Α㼩6a2E dS̶dH]A7B{=tA\^ΐ`em kD۱3Kt^ۊo3Uw=TƒLɄK01 Jzͺr mYK^葧rtqƐe6 LI pw3EN-0`UwyK%.x,xnQmܕзeM\7dZ=<<>%s aQb^SDfg^ i@*ͺ?oԿk7R}SQ~JhKbqvA`%Y2 ;!h()2/YYBk!ӆcYrux;xZ g ay-yBҁ~j/8b؉d0L:>EɈkS' RS͙n!NdNVӽL[̱YWH7T.3!Dsؖcae穝/ո%~neڳ4 #& Zc><gKg$YHM:ĞˉOp(~w>k2R+CgjD W[ֶ:ڨh!/X0*ԞYq[![S4ﵬWpskqd'=9-D{2ړIrK潎wp-ʅ'>:6[;EvEr^W%Y+n~w~e;@QAPITjxa?Sz?Q,bmB?Q`90QD jҮ^[BCef!P4Ƅ"pwA#HW™cdX;^<:ھoH-%VUK]|smhK>ٖt(=t7J$ЮA(ٖ.L} =w6?׎UGIw! |8xQR"l8 'KQM;'_9=ܻ/D\-)ϗ)8ZDc>|c$MqPK!E:drs/downrev.xmlDMK@EaxNhKkA"{̼&83Mz˹٢׵hQMnUe u8E5pfՌRe;v !ƧPФRdM~d6;X) uky$RSeCI 7'0e>xlvB韟@xE)1CvtZ[4 ?PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!y;O^E)drs/shapexml.xmlPK-!E:drs/downrev.xmlPK Y If A={a,b}, then P(A)={,{a},{b},A} If B={a,b,c}, then P(B)={,{a},{b},{c},{a,b},{a,c},{b,c},B} Are there any relation between lAl and lP(A)l?  *3f!*3f *3f*!*3f/ *3f E3f        ! "      D  A(??(49'1 'D,'E9) 'D-/J+5H1) 2349'1 'D,'E9) 'D-/J+xxpH  0޽h ? 3fffff3̙3f̙80___PPT10.Aࠔ" 0@ L0  (  2"   phY0e0e?Title 1`  Y nTopological Spaces  ff  D\    mY0e0e?,Content Placeholder 2" PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!PZ5]?drs/shapexml.xml[]o6} up8nQ@n`)%eLIv~}IiM@!K޳wG3t&EL$b"LbDP1\ m]D(,tؘ]drdRrjUzfPr^o6f*1=eCl# Ɏ% K>d*jzMW&]v>KCE`|0C@@lǤ8iULc"@ÖU6A7&LȟX.(sٕWhNyv~wQ8Y6wi=%eLoun:QW1)Uƹ%φ#phq`l)KL5oӮɜaNpґbԧBG1mXb hѵ¿ƨGL$$\ʃƜG]C VIk>xM5Hhv e^W0r]DأfMzR EAÅO&A[lY(^3(oAyp X8͞D-^Bkv }7K 7ϻ/ƳWHs78}@Ђ4'egYn^Xv#^ѮB6(1#}~1^ut\SA)2hAEVc/1' *ZP~UQb )K6bbأ <0!CƋUr{s 7)! W $`^e#ҼӣzKGrz2Ksz6ۣ{a"k=ãK O}+/ouqPK!pdrs/downrev.xmlDMK@Eaxb';-"HD(#?3IhM&wp˽˙/Gӊo,+& K5I<,s̴xC6T"BgˤeMvı;Xg0*nZy$`Ǝ^k*ۓQ0)W;|v}?Y+u}5<4)+QZA ␟\79-FKPK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!PZ5]?)drs/shapexml.xmlPK-!pdrs/downrev.xmlPK Y A topology on a set X is a collection  of subsets of X, satisfying the following properties: ,X T . The union of any number of sets in  belongs to . The intersection of any two sets in  belongs to . 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Z  3fr! 3f  3f PEMEMPEME3f  B  P   D  A(??(49'1 'D,'E9) 'D-/J+5H1) 2349'1 'D,'E9) 'D-/J+xxpH  0޽h ? 3fffff3̙3f̙80___PPT10.Bp " 0@ L0 @ W(   2"   pY0e0e?Title 1`  Y nTopological Spaces  ff  D    Y0e0e?,Content Placeholder 2"W Q PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!Ӊ+drs/shapexml.xmlZ[o9~_i4(AJPHUSdgmC!.e4,rl}3Af\TɈ4(2VT"c2$_ K#:6o&sDYTknbLԛI=cJyOR|=i#zDd(i1X!פIuMWSĞPo7D:t =?%kE1G-];vhbX8::93 qSaNy-)" o i$f/e͛VF%U1-yLKٝRNp .rD+$Omvr&4c"CH5}=TT[채cUT E!+]?`lDZ`Ic[մ+onE٫4: #}& ڀS%>6gkdɹؐXb6!aEPK!@jdrs/downrev.xmlDKK1!\Mi3Z2AF|`vf$̣Bs|9Zt|e $&2~<F8bNp SB(ChR)}^&? btT׵M{2񡤆JOV#ٮ}8==~6]7Z#^_ G?N۷E()cvrڐ[4 ?PK-![Content_Types].xmlPK-!1_a ._rels/.relsPK-!Ӊ+)drs/shapexml.xmlPK-!@jAdrs/downrev.xmlPKF Y dExample 2. Let X be any set. Let 2=P(X), the power set of X. Then 2 is a topology on X called strong topology or discrete topology and the pair (X,2) is a topological space. >  3fs! 3f  3f "EM!EMQEME3ff! ! Q   D  A(??(49'1 'D,'E9) 'D-/J+5H1) 2349'1 'D,'E9) 'D-/J+xxpH  0޽h ? 3fffff3̙3f̙80___PPT10.B" 0@ L0 C;P$(  $2" $  pY0e0e?Title 1`  Y nTopological Spaces  ff  D  $  Y0e0e?,Content Placeholder 2" PK![Content_Types].xmlJ0*miG]`Hm6 Xwt?. g#Wi3E).+7 >ON ޑ1ˋz "+R RhB.} /<^ITխ%rHK4uK~I0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!ˌZ=N>drs/shapexml.xml[mo9~_OBHҠl)J4B!Uu"kنB~=\Fidn,dʵɔLA.S0_n*(1>Jι/~h4"#kfjϙ9P((3V Y jV;,]ihJM~B(,ǐ-%-Z`))i2̦ұ)^2f߰΍>j, PYLMbfnbڨ5NNgePV1X]5{%0ʢC0׽/݆6ocmCGLDi)LvRKmX<Aɵ|.xnY}j|؇u]^YڛdV;::9%s OJSDf^ iP*z6oZ5觪,Ւa-H{v%=+l ewBPRd6]<# Ƕx!ڶƿv0ϚÍaƓ6'joUW|LmJC,biUm?_(bOy "rJ=bZ? 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Let 1 and 2 be two topologies on a set X. Then 1)"2 is also a topology on X. Let 1 and 2 be two topologies on a set X. Then 1U 2 need not be a topology on X. Example. Let X={a,b,c} and 1={,{a},X} 2={,{b},X}. Clearly 1U2 is not topology. 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