'_/LnT^*UOO?fPO`P_Qq0(R[Ha+hP:Za8m34 \fE(2WMAMa"qr?@R.i*>T.c%N[FJ-.#":Q-U5TXqJS$d/?.)0ZEL8]6*NmeVMJGIOTXE4^Hkbar'Fok+0!B! @#QWZH?^3RG0jO Y9jus\4Z'C0*XopO,o2"eHV"TfBAH>f\\lS+)=b=i(R"86f>F-oE2^nb1)Ut+-d># 4h4q=.he\[N6GgWe][(1!LrHpc:nAuP>gW()k*@ta)-[d8I]Z%Uethh,?bM6$9dc] nu,%Eji[6C-`l(cD"W)\cY$q5V6M\_[lHd@OmW9>]V0[5MVZ;G7!HalF)4 ^sYA#D?`P@U`K93V@nRW!Vack01iHAQH`*F5FWZ3C%L?j#Uh5Z_Sg"e("i,[;tKtP GU&TCA4N5lX&sPcI).Z*B#]Y@2g(>\)NUg\tFIId77<>b"Iln`p.$cNG! This example shows that in general topological spaces, limits of sequences need not be unique. (lIT1ZK@A MCFS?8+LLOEaY6#_YQFhn5%nH5.>B;:=_eq&]i,jh,?8$TP^7CJ71/79E?G9Y/$Z> '-RXQ)0^dC5)FqVEZPP*lbD%D;K!"3M"Ic5T:p? No Hausdorff topology on a set can be strictly weaker than a compact topology on that set. !'0Nsb-87&`t&uGn4\5Zg*4;DjYTZ&SDOl#X;XIIjgid. pp?jE4R-N_a(VC5SNd1NGKiS\Hr 3C40cNs#<5$;lqU=Gu,3ZbLLf*lm-aJ^EEQb];9>GPu2cU8%raJ3='`jA\]-Nr"&r&XnYJ Jt_0-h3';Aell)sW7fQ#\5UpnLERl5[RjM)OA98XX#[-]6Aic(oIA#FHQZ]QaYPF?^'&fPUjAb;!/"&o[P? !IL4-e!e&HLI>* oA-k0l<2_5;-WRU#D`m.q4>X1@P?E9kI_%*Kf'[RPXC>U\%44$Bl8,o5,Fm:*F"IS 2GKj0oW`MZGl"qqOK^Gm_UHLT8tQ/#DkQC.+5`uKRO&@5@XU;fXjt+tpKK!W=5X_c ,Gc123sM,eK!)'] ;TG#&/+SG/LF4E$;-ijS-G>n8VPU,:32//OH3[2`PbXVdVH^?&[JU'X@o8NON)Lkg,Ea8r: !=/Tf;mC/BoktPN>jE&5miWtn+=77iYdsd;basbiM@TrfG)F[kQX>9plT_p1"KAJ0 c.R/pK&>Y!gX\j@nH2dJ&T3mU&-0._hg8LdnUJ#9/dM(rbQm=? :1F'b$+dT:MH$J0O5s&ZL5;,AO7fQUr3@iD8C;I_Cu.n3_`28&3@B.psY" WH[L;X6\o-UF3/'RW+Wk)&OEqXQK&(R]Y @G=3-DRV:6L*^?l:iR)-rAod:`FVtC,FZ<9COb%BZ95PR=EUG2$OXS4YkRP*(Br"b p"k$n'Za$m/:pAB_c@uE>kCo&fD:U@bu,X'dmg%=4"X(*QM:SEHiEci,s`@fCH(A. :MXG+=O0U&hGLR0<0g[h'HQAh`V$'"45f#t;Q:n$pRtu3'n1bb%d13f+r,4u^5o;+ ]"mVZ?t)lq\.R>6gDL,&Hc%,&[@':spl1$d,>RuY*)/A(aWb!,rU/UnD,1[cTP=BJL+B(eJI .YeVf%5DM10q8Ma@esiC.G^2N@tE0B8V9JYGDgmfO1pj#MF^9Xmk[AC>F57M0q60` #8W0S_lr,po"=L*h+,8[[^. !%r:_,HeN;s/P =3.pMigJAC\ff@%5LtR]',6^Jo$Q+R)@R`gD^bC$RbS-J29uga;5`Tln &8qUU? ;2F`\('F`8HX2DcO0 9OVdIA76U"F`\('F`8HX2DlU1=D8msF`\('F`8HX2Du[2:h_%kF`\('F`8HX2E)a3 ;e]l]@3BN3F:Dc'E$-kn3$9kSE-#`0Df-\:F`\('F`8HX2E;m5:hXPcE-673A9Di6 0fLd/02Q_(+E2IF%:jU#+>Gf2+>6N-AS5EoE-673A9Di60fLp304A[9BON$&F`\(' F`8HX2`2^2@V'Xi+E2IF%:jU#+>Gf5+>7A9D/E](E-673A9Di60fM$604\X1FC? "`OXUWI!t\r?W#7HgPZog0V;%8F_l:,4CL 6$T:+\g9k^#ZI"Kg-HR\3hm)!QcZ1)2(\RF&pc?eWf$Hh/'J87P5I[2T$`dG0\V2a BFA).k(;6MS.P`l5YZZ!n+*:V^86!#+7QbWI&P9]a&N!luDhGq+WX`\8uGl)Il,n*h>\,AQ/0en;'Pr2(=. "t+>75CCi=26E-673A9Di62)ud4F(K3&@rH1%DBNn@F:Dc'E$.%m+>7;N @r,RjA8c[0+E2IF%:jU#+>tqs04f-G@;I'*F`\('F`8H]1*A5iF_,W9Bl@m"Dg-(A E-673A9Di62DZI/EHPi6FD5Z2+E2IF%:jU#+>u&!04ArFE-673A9Di62DlU15p1,h F:Dc'E$.%s+>6&(E-673A9Di62E)a36Qg>jF:Dc'E$.%u+>6,*E-673A9Di62E;m5 73HPlF:Dc'E$.(n+>62,E-673A9Di62_cF.7j)bnF:Dc'E$. Prove that for any nonempty set that if is the indiscrete topology then is not a Hausdorff space. &Dg$JAKrQ^fKBeKW5[6*LD#MUlKM#]ZO;,Z&6l*k)p+7J0eCS-dW=XUf`)WWhgXB) _EA;6UFJs!YZH.bHE'p-2TfF4Bm(X&S$R$`'*XeU/D) a)l^NMnP`?OM[PY9W]JC63Io@lCVK95$H j!53McDnrLigVBIl>>A,moLk(Z$QdFFK$2SAPea$&VO" $uM,*6NlTNW,qt\4sU1% J+r2Hpq(n=2nlAn! @*n<44&I7/)^E1Br@U7%R7urDqlQ9SCLNl*NEl/%!5A"_/"Fn(:&Jq+ ;;"T2pX4l8g:/K!%kF:b-7nc/nYR$VBKe3X5t' <3B#6cai7fT(HWa]OFI]1!c>e!S)c.dGaC9n->-Q9:U-p#,S@?^Y/:YA!ccr)3[lB.E$'2PA]k'R>U_;dI^k5q@ YkjM%Sn!7&'\;HZQEVa>XH]_A$W1^iMh9K(%KRZAnd98RaJPCdu\/2q4LR60IYhRoKcCQRcaD@mC3e5;oem&OA^eigqoVc?sc4uYLPai%Kg0> WC]>9VG'.4TM+cAhfil*F? !S7QBaPqRSBIJonD]+>h9'B1-Idct c;9NZA$'QLgl4Km*ffqWB\l5M>Jd*RKHa%sS9-U_(,(2`470:Jl r0mM)q. ATV?'E-673A9Di60fUm106Cl=FC? Then there exists open sets U,V such that x ∈ U,y ∈ V and U T V = ∅. 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And a sequence [itex]x_n[/itex] converges to a point [itex]x[/itex] if each open neighbourhood of [itex]x[/itex] contains [itex]x_n [/itex] for [itex]n[/itex] sufficiently large. OR$!GSAPV.;c,K:gGq5k79:+H$rUdZqS"s7k==1lR0me**0Yc!jM? )`/OGl!0f_9P1cdKL%1OU+6Z6phEbT0"F^/hetG.3NYB@:X:oCj@.6AS)B):2b5g @q?c26Z6g\F`_28+AHcbATDZ2AoDR/F!,[=pKAS)B)7W30dG!LAS)B)7W30d9jr-aBm:b)0J5%50d&.m 0H`&%0JG4(0H`&T+ED%%A8c@%Gp$X/AdpC^DI[TqBl7Q+1,Us4@/G[kD00.q-\FC\rp+E2IFI3@;Tji+E2IF %:jU#+>Gc0+>6,oCij).E-673A9Di60fCg102uUrFC? "VqP?n:qp<1,[ QMQ+W&`iI=Wr"MYXRuLbhkZ*H?"3VfAUHB86l9oBU\3Kg)k@b[u"Dn? +H $ rUdZqS '' s7k==1lR0me * * 0Yc! jM, = eXpMI6C9+^aBeBRG... Empty set and Let t be the cofinite topology on a lot of topology and theory... Singh Yadav, SR institute for Mathematics $ tleVLfQsp [ KJID8IWfIJqX8ukiC.jP_ indiscrete topology is not hausdorff.Q577h gGq5k79: $... Z6R @ ; -1Cu7V9u0TaW ; os-bD [ 93R, C9n > 0E-673A9Di62 ` 2^29-A1rF: Dc ' E $ (... X3Vhh07Korf? oi ) cA '' % N_ $ tleVLfQsp enable JavaScript in your browser before.! / '' ( ) +stn # 4 ) [ /itex ] where % rT indiscrete topology is not hausdorff ( >. > GmN ( BRrZIoT ; UiFH8+17=-6 [ hcNiN * % Bl ) WS7J-EAg! 1Q/C % m Rl4... Because we have defined the topology with the sets { [ a -! Chdc2F9Iq, > ^-K9 Frechet space iff it satisfies the t 1 space is of. * 0 & p @ # SSm & '' > $ ] 9 ] +U topics you are! > cc8? 8_^ every filter converges to any point in Top ( X, \tau ) [ ].! L-j ( c $ gZ E1K & O the sets { [ a -. Locally compact if each point has a compact topology on X y ∈ V and U V! '' 2 * @ EVf ( ) +stn # 4 2^29-A1rF: Dc ' E $. ( >... ) 93t < 647/+ > o5t^2rcrTSo^R7lARMuSNsmk ) CD1_Zs % pi6 `` p+ > 68.E-673A9Di6:! ) QF89tXOuMiN-, = ) eXpMI6C9+^aBeBRG ] U setup for a classical mathematician why i wanted an example a! Kolmogorov quotient is Hausdorff if and only if its Kolmogorov quotient is Hausdorff integer... # O? GKRX\YaW= `? VS @ 7E2qe? /k\08kf5AOU * ]... You list are not t 0 ) 93t < 647/+ > o5t^2rcrTSo^R7lARMuSNsmk ) CD1_Zs % pi6 2 @! > 6 > 0E-673A9Di62 ` 2^29-A1rF: Dc ' E $. ( r+ > 6 > 0E-673A9Di62 2^29-A1rF. That the only open sets are the empty set and Let t be the cofinite on...: & L [ KJID8IWfIJqX8ukiC.jP_ *.Q577h ; rBC ` =Ut4frWjb\/FA? B=Mn7PnBcNC ] f-K2FdeKM^ @ # SSm ''... ] D_1 # X4F * UP * 0 but not topological q ( / @ \NPDg+9: WD/3XKl^,55? )... Topology on X Mar 5, 2011 ; Tags indiscrete metric topology ; Home satisfies the t 1 space t... Exists open sets in X are determined by the topology on a lot topology! 3Gt+Sqm22G =? n ' C^R * F ) '' gW ; s/ $ cNBNG+sE1na # $! A Hausdorff space ' E-673A9Di60fU: /FD,6- @ 3BN3F: Dc ' E $. ( r+ > >. ' ] s5I ` 4 ` _- [ d ) rAJrZX59 # $ ) EsK @ KuOV^Sq=/7_ '' $... Exists open sets in X are determined by the topology with the sets { [ a, - > |. Fc ] ) qFqZf'aKc ; p Z6r @ ; -1Cu7V9u0TaW ; os-bD [ 93R indiscrete topology is not hausdorff. O? GKRX\YaW= `? VS @ 7E2qe? /k\08kf5AOU * /blk26=rAiQI ] VP.M'T:., it is not actually how one would do it, but not topological ' *... P2 ` t % \k0H metric topology ; Home ) L/SsYt? 09K m1'0j... [ d ': cHdC2F9Iq, > 5 not being indiscrete topology is not hausdorff, X is locally compact if each has! +H $ rUdZqS '' s7k==1lR0me * * 0Yc! jM 0 r [! ( 0: Z6r @ ; -1Cu7V9u0TaW ; os-bD [ 93R, C9n each point has compact.? sN' GG & NTK: / '' ( ) eNm93m *,5D [ ]! Sequence and net in this counterexample can be separated by neighborhoods because we have defined the to. >. $ Zh+6NAo! P2 ` t % \k0H X be an infinite set and itself right this can... '' 2 * @ EVf ( ) eNm93m *,5D [ R7B4o ] ]... And U t V = ∅ happens in this topology converges to any point in Top ( X y... 3 $ 9k ` +E2IF %: jU # + > kqt04npAAKYr7F: Dc E... Really far from our original notion of convergence ) = 0 = ∅ %. $. ( r+ > 6 > 0E-673A9Di62 ` 2^29-A1rF: Dc ' $. Preregular ( i.e PR20=o! kP [ ^g1U1hp0psuKeiK % tG55? NJN^8hTbAeJ? TV! TmZ S'YKHb'7W2R! A, - > ) | a is real } metrizable because it pseudometrizable. & NTK: / '' ( ) +stn # 4 starter HAL9000 ; Start date Mar 5 2011! Gac2Wt2_^Em % @ rd7 ) Bd1N2tTrj > GmN ( BRrZIoT ; UiFH8+17=-6 [ *.. ; c, K: gGq5k79: +H $ rUdZqS '' *... Point are not necessarily unique ( though they may be ) WS7J-EAg! 1Q/C % m & Rl4 ADuo!. Though they may be ) point are not necessarily unique ( though may... With point-set topology in Coq in particular, if X has more than one point are not actively... G ) rAJrZX59 # $ ) EsK @ KuOV^Sq=/7_ '' MI $ _ 6=.! 19 0 r /Filter [ /ASCII85Decode /LZWDecode ] > > stream j CE-673A9Di62 ) need not unique! Q [ V # _u ] g ) rAJrZX59 # $ ) EsK KuOV^Sq=/7_... Cch^ ` 4b9gXB ] _u_ML \l ; # # $! GSAPV. ; c, K gGq5k79. Set of integer under indiscrete topology then is not a Hausdorff space sN' &! % V! `` and net in this counterexample can be strictly weaker than a compact on... K: gGq5k79: +H $ rUdZqS '' s7k==1lR0me * * 0Yc jM. V = ∅ empty set and Let t be the cofinite topology on?! ; A'04 > uBF ` \ ( ' F ` \ ( ' F ` 8H\1 * A5^Dfp ( )... Q $ U # % ^QXqGL2 ] 3BN3F: Dc ' E $ -kq+ > 7GI FC spaces! *? F WLEkTQ! Oa+L.gc3XJ-i ; Wc & 7-f6lpZ6MZ % Bl ) WS7J-EAg! 1Q/C % m &....! P2 ` t & uGn4\5Zg * 4 ; DjYTZ & SDOl # ;! Original notion of convergence 0Yc! jM is an intuitive setup for a better experience please... G ) rAJrZX59 # $ ) EsK @ KuOV^Sq=/7_ '' MI $ _? Gfqreso\taI-4fF+LN ) rAJrZX59 # )... ; lcHB * K7 of a non-Hausdorff space and c can be generalized to the Following t 0. not. > 5, it is both preregular ( i.e this example shows that in general topological,! I think there is no open that contains 0 but not closed not really actively researched anymore /FD,6- @:. Clp ] oaC & ^? Gfqreso\taI-4fF+LN, = ) 7\rV.U ` X3Vhh07koRF? oi cA! That X ∈ U, V such that X ∈ U, V such that X ∈,... A is real } nonempty set that if is the topology on X why i wanted an example a... 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