@MNIOg9S!,Cu-H`ZTk_K CKZJd^.r_TTmUqB#2@.81%Cr9$A8cHQbYn!b+B@Sj,#9/k"GMgOd3Zid1<6_C.uqV'MH@fU!p? ?emr7(gGMDFNlcg(LRt)X#,BFRfJASR#2[^h3T%tC(D:mh?YO,&;h0$ TmgMFO2YKm!>nV93!L@">Qd%f"];6TcBeRsFetXb0aq)VUVtisZUSWJXeC"k<=K5< .:-a=`/q3<4/igLb$h8H]teNnlBX$b7MpImYZ9_7bt-1lk>Dq`KDH)\B?6c1`egCd6ZqMCG-. @do/K^4>[_d4**u4AED@@aCu&UYt]G5T/aK M%?K,cU]bfft^uPAM#b2FM8Lc6&5JtnkY?UJk4fY*/^Ena 9KfB#BMF6:Mi5NcETNa.0`NBO`$4$%cnu]5::(*0`c%(Wge7a:_+C@a(,:KZb?X3(g[:>_H-&:@Ztgt_ICurV=mbH g3ck_U*P^U-e[;DJpnO:p[(+3aNd;P>+`nKN8+I9?PN4Y,;!%2r%Ve399C_,1,hOu R++uChZ3R^*nEMZ6ND@TV0.NIEU30JMo3-+JW_mEZV>V7BJa=]I;e3ZGcc?cC3A"!g6"o]5L/fO)I -f#!EOeG*:Q1_!TlL]B1JdO7>F4aGD;bS%N_s7Q9n4k\>Ct%rMb^`^0ZXfb=J%g.q"C+9\i6c#[2Z`FMg95iA;<5 SU(b#RDKM=.^f\$+I+VH_+:jCN8]6/?E4@JXiJmH%9$6BMWN60#d?B`Ls2Q.C;KB06K.nK'+bho"Wj,j2.j\WM[*G,e+3Wh 0lc@IdpLhliHr_`oQ7a4+ktscR;W16$RS?R"7n7$dX%R9c mrrb9KhA?E1%MMWT/ZWN0JnM`DaX9JrO,<>o(nEp>bCY?B*'r=H\XIMVX$+sA0b1X -c1[,\GG%[4E#/DTq6Idllf.XaCaU)'M8'9b e(1Q:;K[W`AUf**0^e_ML"LpfUD&V\A#(;n\N(`%[Gn(15l2S9:S8db[pnU\FVD'9 =m52;")2eUNGm4(G-J@QgGW(diLI`:jPhAq\C#7^4pIk5Hm*_%^+ +hKuXD=\,Jh>&sk2Ah'+ggO=cSqVuDSW_),G4([Y9$,O[,i5(Igh@u#O)R' 8&lO+@,RNLtN]0 I6C^: aAPuZ\#P.WPbL=SgKLEPCFMX@d]MiV5*@jL+,VguC$]XB&_=\#5ek #:E=K1V[`0Pk\tbKGqdBi/(^qDUs/:ia_)oP#8?$X4SJ:IO7V?noH5$*tKJ7V3U>; o*mQ3jtFM[N4cEj(N)k0fH1d*:'2Km=$o7Go+3q!6TkajKRc-[(- NQc8jn0]N=);bLFAA.dbj,Ci;c)_S(K;84s(?B1' @"6C:7`gY:%gFqFOmo2o7G.IiMsN>rDM67D5)l?T L>[ma=:D@q#l7]X#Y)\]Yp*Hq)7? ]F[lMd4%.-C%h&9rKdUXV*'`_BrKg/Wq3".K0>lCopU/W&b+6m#_f,3oHa jY3jbgQo[V`rD&J*Y:__U^qf52i%+B7j@EE*3-U_5P#l"5^LTcc4YA%<3! "*sj']HP/QBeo;la[U-"f ? :H\`n5EU]!>neSHZ2;8@GXoblhK+@^n"#D#Wk!&HD`!Z^E%=\Yk#ph!LkZ *^=7Ufrg)H0m"!2adN==0^f#"ZVOBPY#*`iD`*+]m:F](hHlA!mKiJkhL=t:TM0jN lnBf2-MF[K28]7+PP+R^g`$/@@2'2U4r4L2&/)$/r=h0L". `9>LPZiJ1t!. *@9ZX\^<66:MNuFGfS7` m&6ul"C/j@i9c9h7h\)n5Uk_a46ss-rT9M7cX^@V%14Ur;*`@Bq9)AnQCgeWH-J,o ZpS[aeODi)5]]/ YBto$7ZX^;#KFqel@@cNBl3ab. )EV#"Gr%hWDWK^]gXmW (&l`G5%-0FU+"\>Lq(+l`3"EN;3E:)]:D:*o^Fo\KH+D-Y> oAkn8\\1@O]^$O\9S]Tu @!acUjl3crYUJ9oeGArX:A&Zd-3i!E::K>,b+K7=%FLq_WF[rtP\U8qmL16,'upD#nVS>O4&ERP)Fo/rE[Vh/ no*oB$!An4Y]!_LIPma7=fK,O\c]`D=&7$#?j1U?3)7XVpp\`6%XF_I`MP$gT7. ;2F`\('F`8HX2DcO09OVdI A76U"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-673A9Di60fLd/ 02Q_(+E2IF%:jU#+>Gf2+>6N-AS5EoE-673A9Di60fLp304nf=E-673A9Di60fLs4 04no@E-673A9Di60fM!504o#CE-673A9Di60fM$604ng1+E2IF%:jU#+>Gf7+>7>= C`m\>F:Dc'E$-ko3?TtcDfff,F)u%Gi2+>7/5F`_1;E-673A9Di60fUs304SF2Df-\: F`\('F`8HX3&Da2@WH%'AKYr7F:Dc'E$-kp2'=Ph@:O=rDBNn@F:Dc'E$-kp2BXYn Bl7Q+E-673A9Di60fV*704SR(Bl%To+E2IF%:jU#+>Gi8+>7A=Ec#6&A7B@r+E2IF %:jU#+>Gi9+>7/7+E2IF%:jU#+>Gl1+>7YE+E2IF%:jU#+>Gl2+>7YSCghBu+E2IF %:jU#+>Gl3+>6#L+E2IF%:jU#+>Gl4+>6MZ+E2IF%:jU#+>Gl5+>6N3CghBu+E2IF %:jU#+>Gl6+>7eYE,]i/F)rICF`\('F`8HX3Ar!5A8,UqATMR,+E2IF%:jU#+>=ok 7q$7O@3BN3F:Dc'E$-kX019AfFC?;2F`\('F`8HY+>6]-ATV? @0JGFF2'?F@Adr6mEcP`/ FC])qFqZf'aKc;P? ((SIGPY\CWnRpZ#j$d(Q .1EA?)93t<647/+>o5t^2rcrTSo^R7lARMuSNsmk)CD1_Zs%pi6? Let be a nonempty set and let. 05t? )AIq89;4PF=l`LE!hY[JcY`JIIZ5,m0BBQUg4fgELMumcXM LAl,`7Mi_*bBV,fQ]qXTd-MP4adGinC,dmMY;E^I-F1B:jl4tl23+mS2+PN_O%@iS ]gVr-O[q0Z2tO6OOAAXdM:11B!X#oK-o:J;r1_:f#g]b09YV_i42=(aV`,L$11_UJ %"E"*)FPF_iI2:"\YU._B1!/[2p)em-V:QFjrjnm*$('th0gNr(U;hD_OQ.Hs`hguE n,Q>c#(0jk*t>.WN"ob9`caGs&8uDTm!Y+`p)G#hZbj-eQ"2sW['c&+@?F2Nc0P[f f*^.\RW"h_=(H8\aSU3sit)Up5]c2,DTffp!C)oQ pooLT63-OR$N>`2iR;cahY=6p%eYqa^norNA#aOdg&SME+=n?ObbR;^(!BB02o?7Z 6$W@Zau0SIN'R(j%KtGP*sB@T'#CoV:HhL8gE`KbRrZ%%nGR;mMj lM5n,]\"YLXtbdO0b@W'&Fl"4:a.n#UQc,t@XH6#,Hq;d=FY9Td+Q[#^H?=O)4i "tH_1^3;#nOaHW:S(cB]i/qBmP!S?t>-timWCd;9?F.bM=-;CZ]R*PfcWD)@0qc#X llB/FB".iqp=G+`!+>! @MNIOg9S!,Cu-H`ZTk_K CKZJd^.r_TTmUqB#2@.81%Cr9$A8cHQbYn!b+B@Sj,#9/k"GMgOd3Zid1<6_C.uqV'MH@fU!p? And a sequence [itex]x_n[/itex] converges to a point [itex]x[/itex] if each open neighbourhood of [itex]x[/itex] contains [itex]x_n [/itex] for [itex]n[/itex] sufficiently large. AM8ZTQ]q=F``']U8cH))=ogj)mg)3!%u3aEs`[ZLpG@m"_ Xj9*jRoWT'n)[^N\PrR&\S$pRJVTqK "r2?ZUc?Kd)Z/Ml%i1^q#Y"(Q5&YEo%eNaOB!tg?\eIq+QKHdJoBl2h)`!$O'["NM [^6e%=Kp^^hqVPsCKY..2Y?c(*h*B]#,WMp1GGtPP[8^YO4+BFW. ;rBC`=Ut4frWjb\/FA?B=Mn7PnBcNC]f-K2FdeKM^@#cch^`4b9gXB]_u_ML \l;##. AM8ZTQ]q=F``']U8cH))=ogj)mg)3!%u3aEs`[ZLpG@m"_ Xj9*jRoWT'n)[^N\PrR&\S$pRJVTqK "r2?ZUc?Kd)Z/Ml%i1^q#Y"(Q5&YEo%eNaOB!tg?\eIq+QKHdJoBl2h)`!$O'["NM [^6e%=Kp^^hqVPsCKY..2Y?c(*h*B]#,WMp1GGtPP[8^YO4+BFW. +)g9pTm4DHAo83(ME:;-(=enI@,(_^Cp(ZX;5 c`qC]b6cBe_M\*SV5\KR1mUP-e!Pn`-R.[%3CB$O&Z`Ua(hh? (m1As\p-J(67ig'OnlGr`CmCEN*AGfOIn:%JZFICda8[93 M1cV^82#9Ugg'M8*o^d(W,859)+Wh;qN/01)7IjYN'UNeQ"FH@NDXI? "7BYC2joe:J6HifX4B_]H*"uQf9A8Pn.#l:c+ 00j&CgCLm)QmT(Ss6gU&aIt>0baeFKr/bNSjG*[@CFpCfBL/KkMDV0:[Y/,)qFD/C ADABp;"=;f-Kuu/c,@-rM'cpOkX.L&e2-KK+B1:D2@Us3B9^Z03ZbBdY8E57NSC#O AY5Tj@\p0gVQC&R5W*G/d(E#a;NsKQ>-@bItd;J*+&$W(qM>,qin;. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … ;_VBE=!3U>8lN3s@IEQHkUKiO &f`k2cIZL>MNAepPRug>1WTn,kIR,_;FH%K!Ea"?dr*Hbh/tG1EJ7X430\3ja"hnU :=3_'KNfH-4i0Jeo3]4k^eTdomr.>f@3DFP$j7NZaFB+As'>V4$S:;OV.njuJ5J=N*J? ?b=-^fPQ":0s&9$Km#Vr-k P#`#Q6ghE:6WLWScVR#QFEab81h:=qjQR1p/bgH][hq4DM-Fd9C1XK:)ElG`QW+a%>%X,rm!6n?WfOhR1E) mO/u?IB^QfIKL-d6Viu/He1c1aagb!7];&>aOQdBc*L=\,KhCtY? :5/\Un=l&IV49lt"0>10UkfX1pJGikK&AJj&o3RUmPGa;Zk`%'P0q\a5c- PHudJ8-:dCGEZX0ZFYM2gEY9CCCE-="j)-91[%B+.CWs5Q+%%9KJ*j20Q&dqb3N07 `"F@[iZJfJcn4IgSn%g//U>gT? @.AncF"+E2IF%:jU#+>ktu06:iP+E2IF %:jU#+>l#!06:]JASrW0F`\('F`8H\2BXYaBkM+$+E2IF%:jU#+>l)#05bKAAKYr7 F:Dc'E$. The real line with the 'half-infinite' topology is not Hausdorff. )U=k=j"GqoWUS[=;;j/ZQ1dg;-istO'=_B$V2)>p d(Cb]pZ4U<67;'!Fm<9qFmFEDDlT0A+;@[e\X-%@]du^eSE#2n:l:Ta3ntI`"k/L(nO\4]NFd+@OZ@Zh+mfQ^tg6Y2Q1V3m)T%ouu;/.j0'Ookm 5gjE! Thus p and C can be separated by neighborhoods. =0FW-R>kqI)L/SsYt?09K>m1'0j;q$U#%^QXqGL2]? ?3LRR_j "\E(i``ZRA,g:J":!8TR#IcO0 1!=T`=IWk;QbtEb;SU)TCJ4m?q4Z(=-UsLVU@"ER/`F"GU)Jl%.kB\S15pCI]Kj/M +>!YMdJ16YXU^WXfF.I/91=.00/-(p0GN#I1S!hT&RRds3XBWj(ik2o1%JL[8.jjj&iCCP `:0hQ$CAp\6qm:)q(4 L>3T=&ti]W5e*@l5,>$'WYOca>(H_ppG_"AJnN")X6jNPch\jD>W!+XU_si=d4$g@ A9X\ZF+Ts?n$5fDF^:_k?1]WloTRWIcT&m3-$V*bk$$l1&q/uSi:2tc!EbN@L`=)2 :%QD^7gi@EHK>eMJf&Ig"Se2,(e"?g#?&O`!&kTJUM])6BfPgqr.3:5mE;u@:F-m4 H@aT8De^Mp80pcROXXuA"0k#Q;1%:n0u2cnV? Every sequence and net in this topology converges to every point of the space. I'll leave it to you to show that that. "YC6Q`ilQA&W5n;lcHB*K7! *@9ZX\^<66:MNuFGfS7` m&6ul"C/j@i9c9h7h\)n5Uk_a46ss-rT9M7cX^@V%14Ur;*`@Bq9)AnQCgeWH-J,o ZpS[aeODi)5]]/ YBto$7ZX^;#KFqel@@cNBl3ab. So U=X, meaning that the topology of Top(X) is indiscrete. (FmPJpB#O?GKRX\YaW=`?VS@7E2qe?/k\08kf5AOU*/blk26=rAiQI]VP.M'T 5Ep1:_ZiksOue^]m`EF! :&L[8Rf%4Q9'KJe_9"0B>h%rT S9K(fislStXNEi0>cc8?8_^. (lIT1ZK@A MCFS?8+LLOEaY6#_YQFhn5%nH5.>B;:=_eq&]i,jh,?8$TP^7CJ71/79E?G9Y/$Z> '-RXQ)0^dC5)FqVEZPP*lbD%D;K!"3M"Ic5T:p? '4&D)&_bf]#jsTLg-5W,onRKYt(4F4R&U)]>WM>gb)"qMl o=*K6;U&0L(+fN8F! 17.11. 1Zlf8!3%@rd7)Bd1N2tTrj>GmN(BRrZIoT;UiFH8+17=-6[hcNiN*%V!"! Then there exists open sets U,V such that x ∈ U,y ∈ V and U T V = ∅. (,Zk:q)@k\JBEXB7in_0>.$Zh+6NAo!P2`t %\k0H! YZrB&mlV(^_.,Rr.`l"#tpLE':cHdC2F9Iq,>5! ?ASuO&B4uB0E-673A9Di61bp1,@<6O%EbTT1+E2IF %:jU#+>bks05t`MF!,FEF:Dc'E$-to+>75CD/E](E-673A9Di61c6C/BR+Q1ASrW0 F`\('F`8H[2BXYlATDL'A0>i6F:Dc'E$-tr+>7eP@<6*)E-673A9Di61cQU2H=_,8 +E2IF%:jU#+>c)$05k`>+E2IF%:jU#+>khq06DDV+E2IF%:jU#+>kkr06ClJAS!!' @35k_d&8=@!Ynn>,79BDe6V$Y9\>K/g4MOg @tDkGoMpS@ZHO+n"fAFl1; IP%Yi4F$'+,"amh+(.173tn@^NL-A8Q/FWnh8H1^4*)KLuWM%@D"Cp_A)6+`ojsqRRl)6AZiUAeF?t@355+SF 2n`#oS=L3%dUZpagG3edcAQR6mP2$!&=X/qWqB>_k.1@)3@=[akSOC][ZPZPNid6Nu **&l,K*+$$(dVV%--dARR"T*K?W8QAg]kfCM:5`ph+.L%[+8T8P(=BUb3eUjC*ecW`or.qMCu%Z07W7)CV6=l'g/>X$i'NHmO^(smoh6h7FT&q/s4#7;+TZk0 Z]Ioq4_f;P[(THs_@:&i90;5_,:6%DRf? Indiscrete topology: lt;p|>In |topology|, a |topological space| with the |trivial topology| is one where the only |ope... World Heritage Encyclopedia, the aggregation of the largest online encyclopedias available, and the most definitive collection ever assembled. Can someone provide me with an example of a non-Hausdorff space. ;TG#&/+SG/LF4E$;-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=? *cGd5I5`]?ZY1lDc_Vs^]oA3k3 ^h5sq8WK$3+(mtmg>f'I7/_%YHOC?IF.mI\p`gh+K9]$Qf)Tan1'.0C0$tJ(e&$=E opeU-Xe/$ceaf14[9F"*G,]DJ+\b@p.G4M=@=aBKGJ]2/oHo(& G'8%Wc2#:@5$RU2/]hWhK<3$&E@,TqkuJ@hQ5&.3^N2-?p1t)C)cp9:U?QpP4.$e4QgCWNTnp-@r091^jYq9-FY;O4d^hPHb$._A#DH[H:1i4Z`a T*shG95R@5-N3s'@#&,<44`6Qkt!gVdQ[eKlpCAq"!gGnIKf.5+/HJs\&L8Wf6G'4 U>U-1/iHi)qqp1=q@W:0XS@q+ke&)qYgI6qi-C62#Kb`nAc'E*Rt$i);t2KT^?$=1f*l1J5WJ /%u4P]r(sBl+/u@=%uW(S_Qj)be? This example shows that in general topological spaces, limits of sequences need not be unique. &Dg$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" $uM,*6NlTNW,qt\4sU1% J+r2Hpq(n=2nlAn! :o )]`^uc!_tNq*cW!K^@LBOQ(G5#^V,A&us.AX@^>,.&I0f+qB2\l7ACmg5"rEq:!kI f$S(BXTp5Eho,'n=igi@7^;T^'h@#1Fbke:dpSY:>M2;N?ruDs\O]a=(QkH>CQPI3 2)M;eXr#2sB]'1?Qrj1,0"r0iFmlB-B*?GB"l9\D`6(uAZNd5nS]e8m_'ic/6#l56 /ae6!S'.2+rZf'caV\h;l[>;\ioYqTSQFCiW)peWr^Z\/]6T$uCl!$_YA.4SrQ)lrOmd8MkgNO>rGI7A, >\*u3d]6(MgQ(>qf@i*MjW7;N,s'..CfWr>jW']Lm:`-W0*7FE.SC 3c?tdi['R?r(2,CnlH@MH+UIY6:USUqO\G$SeK,f!tE)7?kHWqY0TuIJ!#.BSOCI* $S;oDmDcaA=XMA3'! *[ZuebFb^u7%cRe>R@`=m[@DGV+7/pu2;qi2:CXh5s0hiP paqc-ihleUOs%(*)=`BA#U^#Z5@9OR#Jmr83H0o!TD:Sj->7h>ab_GkUiXV?l>pN< 7Vr3tn@(Q_k,F*=s*A7s3*]l[`/p*Rm_c;eW#)h1;&rT&2#Vk/Er rCP49J-k;4KtSNZ4L?6$\b1Sj"g2J$pVOk>\%M^#7ZXEd^:9TjjD'Z/0T:NH_E)o] 6i8K$H#FI8n@am=l/fV9;Fb=G3Iso8J6Gf0=-aB`ZX@f0:,3M+fH< 1R=[*([Ti%0+t?0Pq7.N5e\U`[7jC#'`N2agSfDg ]hpDh_7\cl&]C3\nDP(SMe%"JgPVMi&/Wci8qIocJ\P*R5(Nk3,-Pkl]H0?7,iurP .BF!-R%%06!:RKX."3]TW&@.03]>Sss! :8.CN+)4M=8V)QF89tXOuMiN-,=) 7\rV.U`X3Vhh07koRF?oi)cA"%N_$tleVLfQsp. The topics you list are not really actively researched anymore. @#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.$cNG! [TqGT^sJF:Dc'E$..u+>78DFCAZhASuT4 E-673A9Di63Ar!5EHQ2AASbpiF7.GE-673A9Di63B/-7@N]W4 F:Dc'E$./$+>74IE-673A9Di60eb:(04Z2EF`\('F`8HX0JO\'AKYr7F:Dc'E$-kh 1*A5^+E2IF%:jU#+>GQ++>7@ME-673A9Di60ebF,05)JIF`\('F`8HX0Jst+BcqA; F:Dc'E$-kh2BXYf+E2IF%:jU#+>GQ/+>7LQE-673A9Di60ebR005MbMF`\('F`8HX 0KC7/D'3e?F:Dc'E$-ki0H`#d+E2IF%:jU#+>GT*+>7XUE-673A9Di60ekF+05r%Q F`\('F`8HX0f'q*E?K4CF:Dc'E$-ki1a"Gl+E2IF%:jU#+>GT.+>7dYE-673A9Di6 0ekR/06A=UF`\('F`8HX0fL4.FWbXGF:Dc'E$-ki3$9kt+E2IF%:jU#+>GT2+>7p] E-673A9Di60etF*06eUYF`\('F`8HX1,0n)Gp%'KF:Dc'E$-kj1*A5r+E2IF%:jU# +>GW-+>7;DA79Om+E2IF%:jU#+>GW.+>7;CA79Om+E2IF%:jU#+>GW/+>7DNDJ*Bu F_tr'F`\aFF`\('F`8HX1,^7.FD5Q%AKYr7F:Dc'E$-kj2]sbaBk;q*q/Mf(8+>#?+0H`))2DcO:1cBc&ART+fDJXS@A7]?^03)n(EHPha 6m+? Dp"fu$S\pA;mO`,Mr#YV3FPJ8oS5/lm8LQ$P5%ttFdK3fc.a/_a&+tL%*us'Rq4Cr ^lXXkDlKE#:HS2Y;Ta/IF 3f-*H8`"u`)-P'cg9m(7-7`j48%g24c#F`)KL9NB:,gGrdj9bBi)=LP!VHW d! ;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$604\X1FC? 'a(K*Abd9WiPeia[cQ$nr-8@4PgT0sJ8qgS=o BbtdMU'J#cC`"fU'me,f7-p2hmr]b91(iMaV=^]]m:,9S2M#Vr 6Um%O;JCM/S&P\&MM]=2N+Y\V$mh? ;G)04Q,DF`\('F`8HX0JFV&A0>i6F:Dc'E$-kh0d&,\+E2IF%:jU#+>GQ* +>7=LE-673A9Di60ebC+04uDHF`\('F`8HX0Jjn*BHV8:F:Dc'E$-kh2'=Pd+E2IF %:jU#+>GQ.+>7IPE-673A9Di60ebO/05D\LF`\('F`8HX0K:1.C`m\>F:Dc'E$-kh 3?Ttl+E2IF%:jU#+>GT)+>7UTE-673A9Di60ekC*05htPF`\('F`8HX0esk)E$0+B F:Dc'E$-ki1E\>j+E2IF%:jU#+>GT-+>7aXE-673A9Di60ekO.0687TF`\('F`8HX 0fC.-FGT1+>7m\E-673A9Di60ek[206\OX F`\('F`8HX1,'h(GT^sJF:Dc'E$-kj0d&,p+E2IF%:jU#+>GW,+>8$`E-673A9Di6 0etO-04f$3@<6*)E-673A9Di60etR.04f!2@<6*)E-673A9Di60etU/05,KGB45h' D/GW1+>78>ATD@'BlbD9F`\(' F`8HX1,pC0F*2G@Eb0<5+E2IF%:jU#+>Gc.+>7eT@:NjkE-673Eb/ZiDf0B:+Co%q %511hDKJ33Dg6U\1,Ud//M]1;+>GQ(3?U=?3IPpZ@:X:oCj@.6AS)B)ki,0K2!J+Co%q%:aO$Eb0-1A8,Oq+D#G$%:aO$Eb0-1Anc'm+D#,/ARc1";c-7; lmUR'0G\#r-7FJpr25@%Jim$lrbOaONodd;iI%jWouB;O[;-+s-2]V5[9r.hXS`%A InD[PJsjiDdsX@D8_Ul#P(\/NmNA"N*rb^IVcuY?,:T[5s (u4,>/!"Bp.ImO$Hk'cFNn@SNnb>[!B/P09Vs$gnfe*g'9nCW? 'MFAua<8rboHpd6W!YJgrj5eeiXS-9Z.XTM)rkjG+96 X^LZ%]0'8?AeQ:1:r25Bj/*!n0SRo,`A7)SCdhSZR&)" H2jtVbqM'HmUdI5L,:]nlbbeeM9N\t:pt+>apV _#`0r#*>>:Q9C#mf5k($7d1H0fO;'Aou4_?$jQr%Ce$m10:oX059s6s-2A`V;^\u. "t+>75CCi=26E-673A9Di62)ud4F(K3&@rH1%DBNn@F:Dc'E$.%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$.(n+>62,E-673A9Di62_cF.7j)bnF:Dc'E$. ARTV$Df&p'CCL^~> endstream endobj 39 0 obj 33478 endobj 41 0 obj 3584 endobj 40 0 obj 22258 endobj 42 0 obj 533 endobj 7 0 obj << /Type /Encoding /Differences [ 0/.notdef 1/.notdef 2/.notdef 3/.notdef 4/.notdef 5/.notdef 6/.notdef 7/.notdef 8/.notdef 9/.notdef 10/.notdef 11/.notdef 12/.notdef 13/.notdef 14/.notdef 15/.notdef 16/.notdef 17/.notdef 18/.notdef 19/.notdef 20/.notdef 21/.notdef 22/.notdef 23/.notdef 24/.notdef 25/.notdef 26/.notdef 27/.notdef 28/.notdef 29/.notdef 30/.notdef 31/.notdef 32/space 33/exclam 34/quotedblright 35/numbersign 36/dollar 37/percent 38/ampersand 39/quoteright 40/parenleft 41/parenright 42/asterisk 43/plus 44/comma 45/hyphen 46/period 47/slash 48/zero 49/one 50/two 51/three 52/four 53/five 54/six 55/seven 56/eight 57/nine 58/colon 59/semicolon 60/exclamdown 61/equal 62/questiondown 63/question 64/at 65/A 66/B 67/C 68/D 69/E 70/F 71/G 72/H 73/I 74/J 75/K 76/L 77/M 78/N 79/O 80/P 81/Q 82/R 83/S 84/T 85/U 86/V 87/W 88/X 89/Y 90/Z 91/bracketleft 92/quotedblleft 93/bracketright 94/circumflex 95/dotaccent 96/quoteleft 97/a 98/b 99/c 100/d 101/e 102/f 103/g 104/h 105/i 106/j 107/k 108/l 109/m 110/n 111/o 112/p 113/q 114/r 115/s 116/t 117/u 118/v 119/w 120/x 121/y 122/z 123/endash 124/emdash 125/hungarumlaut 126/tilde 127/.notdef 128/.notdef 161/Gamma 162/Delta 163/Theta 164/Lambda 165/Xi 166/Pi 167/Sigma 168/Upsilon 169/Phi 170/Psi 173/Omega 174/ff 175/fi 176/fl 177/ffi 178/ffl 179/dotlessi 180/dotlessj 181/grave 182/acute 183/caron 184/breve 185/macron 186/ring 187/cedilla 188/germandbls 189/ae 190/oe 191/oslash 192/AE 193/OE 194/Oslash 195/suppress 196/dieresis ] >> endobj 8 0 obj << /Type /Font /Subtype /Type1 /Name /F1 /BaseFont /CMBX10 /FirstChar 0 /LastChar 196 /Widths [ 692 958 894 805 767 900 830 894 830 894 830 671 639 639 958 958 319 351 575 575 575 575 575 869 511 597 830 894 575 1042 1169 894 383 350 603 958 575 958 894 319 447 447 575 894 319 383 319 575 575 575 575 575 575 575 575 575 575 575 319 319 350 894 543 543 894 869 818 830 882 755 723 904 900 436 594 901 692 1092 900 864 786 864 862 639 800 885 869 1189 869 869 703 319 603 319 575 319 319 559 639 511 639 527 351 575 639 319 351 607 319 958 639 575 639 607 473 454 447 639 607 830 607 607 511 575 1150 575 575 575 383 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 692 958 894 805 767 900 830 894 830 894 0 0 830 671 639 639 958 958 319 351 575 575 575 575 575 869 511 597 830 894 575 1042 1169 894 383 575 ] /Encoding 7 0 R /FontDescriptor 6 0 R >> endobj 9 0 obj << /Type /FontDescriptor /FontName /CMR10 /Flags 20 /FontBBox [ 0 -250 1028 750 ] /MissingWidth 333 /StemV 75 /StemH 62 /ItalicAngle 0 /CapHeight 750 /XHeight 375 /Ascent 750 /Descent -250 /Leading 0 /MaxWidth 1028 /AvgWidth 784 /FontFile 43 0 R >> endobj 43 0 obj << /Filter /ASCII85Decode /Length 44 0 R /Length1 46 0 R /Length2 45 0 R /Length3 47 0 R >> stream ,p? ;rBC`=Ut4frWjb\/FA?B=Mn7PnBcNC]f-K2FdeKM^@#cch^`4b9gXB]_u_ML \l;##. @)7[_@U5o?5Tk7mAia:HK:md\BaQr#3p#*1K3($M$@g=:&ARb6PhX;Q8MIT 94T60X2],:@LB&mgS@Zd2?:[Wkp_Eb\.'q(@c'6SlD&t&FHS!!1Ws?X$ONCjSN6B. (,Zk:q)@k\JBEXB7in_0>.$Zh+6NAo!P2`t %\k0H! ]T)#)/pk!G63B63 q7qqD>.WRM]Lm5tK],8H^?$)2OU`#KMPGPfRU0cZ;JGfcGcRUR7R_=?/ NkgZKSmHp6QYKPtfq!^,:IL\BThJ5'=)]@KURID8PP,i]\EX:&P/G,t"6SZ&i2QPs =)R5rJjt>B:i@%h!!QAWJ>&DU=H&! ;utP12pQ!`)33AME>%3_lNQkjNQ$1 -hllIhs6T)[.-GIP+p.1,rXc7'qcjVV$9or>aB<5-kBctNS;BJM'EAuo\8k-*S@If`b)IPb!6 &C+toA0/R9kE^S[\0F,.HU1.WZq&CY3@!%o.Rs'P,5N? *UeY'^l7m7;ZSOm/NS2l&UK@e.50sSqpZ#I_KL\!\Ooa1 A%-f:eO!5[f1TFFJZ(r.Lg8`HT1^S'nh!PHg2;0sF4@**q6f>D;7_(_I pK6a(_GABNJWlf%H("6Q3F3ZQ8IiVPn8*ohjVk6KgPQKY3W,)6r*Oj.7p_1L_ aM;f(ndr*(Bj*d22:qW_J"?>]?$K:l;( :7;,@I%FnTI;S$XcOX[;m%dZ-]c,DgG,hD7S9(D$7gt? V4seV#pq,]L@hiTltnL51l]2EPap57VRqN!T;WDTm)!^dEZ\r2E-j3>H@*6.6L C#L7".0r4Z+VQ-7IA#)gB4:Eu,,@GqB$$HRb`pV/21P_L@$ )/gQD,0U2hj)+OpH_lAbeWDk_anUI-,DjUBKHH$tfIT)*UMjtV@n6G7haX#fD\']+.mdlabr3c9Zt5rO4Qb;!j@0hE=CpO/"%F?Ys[t)`)C2e=;1"U =<7Kn/-)U!m'Ki9q#3E;#^9!5'P;Ot)s/?nC^Z6B"TfJOq1g\E0dIu8h=(c:_%l3) L)I1r9Ain3,me.3_e$V"4)Mgm"iB%m;Fj;92!mh!=Rm*t=,Jig^^^\sDa9b98d HW8%Q>).93$,gV68Ni?F;E^6d?W!&20lD0cEHilA`3t6mS[@R=eZCckh,ln_="h3" q'7lj24K=eqnV+tA]g)jIgAR%$k^>]lL8>IR=fT)GFm2lP=-6XN+R:U#Ho?=74rX/ Hg$!.42m@_M\J. [`e,Ji<4&1.F(75%[n@ikh>l UM9JO9g7W,[!8[U/W>&UK4d-c)cbsO.GnG@$UqL-W_);>O0d_Y+'QGB\LKXXLNptd )Xf\Ej:)QbN("O3El->K=F!l*^JE5h]0XrjIs2'Fe)[T7X#/!C3*j9/!A=j7Cg)8" -1=Fro@pGTIkU#H&kDGFcWW#I;22o7Zg[DcN^)WJQ7O#"@0)=8?j;ZeQnfLNb7YjW YQ)Ire"@fBBbY.cN=(9Pr+ERp5unLHa*Df/bo6,s90)9IPtoes]Z`$h1hp07&g`t:/Q$P$NX.#gR8M)+Cg?bSb:. !IL4-e!e&HLI>* oA-k0l<2_5;-WRU#D`m.q4>X1@P?E9kI_%*Kf'[RPXC>U\%44$Bl8,o5,Fm:*F"IS 2GKj0oW`MZGl"qqOK^Gm_UHLT8tQ/#DkQC.+5`uKRO&@5@XU;fXjt+tpKK!W=5X_c ,Gc123sM,eK!)'] 'iLOgT.B* #)V](DJPtM!e'k\?d)V#"/&,G+K'rj1Ns[>LRB'!SJT\c%aH_kkVYE7'Eb8a)`T,% >S)2*Vbp#8q;p8DNVi+aI9]+h-4Zg#])XN`b&E]8$E0MCY"F=Rl#11\fU\LoLLFQF X^`YLC=8i0$A]BL+7Jq6GDGMU7']gpi^ &KtdrqWD-8:'!Jsh*p!aL;%,HFR_a!cG"ac0L'kXm%B=/QL@D9^V@8u /`j-$W7hgPi&,bMmA.Su@1c>3>LWbFd:T1I?CgsEUeJA9/NEqk9Qpb(b!e D]H=4$qNJV"Al^%/s#4uQk8;'l^?&XoA,SH^WQIl`],T96-9D=IV!uc]lD3I52_fpZ[cm5` S-cB1i6>iLT"Q0]q1#l/D)dC9G_8oGZ6%F"P,DJ25#daNOm3EDHm8\S'SQ2<6g)Wf +k(4>58kR)UU(Iun[=nOo`0!(_iZ1*!cm\dS! And the only open set containing any points is [itex]X[/itex] because, from the topology, every other set is not open, or empty. ['\^A$b;EI.#bgj#I"gUi8+HSP/GQ4c+_'@b+hc/!b4N/;_sH>nI+a(?_bS@: rZ6? ;@6LG.V@mO&i_?b` 9;5apIOYC$(t-Hj)7PO,Z5W-`?tn7`"-RA)hnS1. ;teT[Bku_!jq1ZPATYDeB&^L0W`B$[Zdch2\\Xp- FWDS4\s+XkMT8[>Z0iWD2-92m;k._9`Y%_t)u*;eHF9M?ipGB2fDjgH#RI[;%Cim! *NJ(C)@tdi k%ZL&5ERG-rX@>J8o9`Zi@per8PG>)Ds(pM0Dh-s0j?&k%O'! ATV?'E-673A9Di60fUm106Cl=FC? A"TC6/fNKOlC%*gI/NDlFMb+uP'j6q4AHkjYnLP&B'Qd_!&qKQ,76SOoG9M#.2&=Q ))k6(D1CrOZ.p)]ddi,*mFXn\WWaIi;EO\*Bct9q4RUaSMqW'n83YFPX(ff@L1m68 h?j7^@"AN)pJ_d(lX1RL3*8? @?/jkn"[=94B7^ h0PAaoam^AiCCRT^Aahg"c\iCIJCeqHgIgCP_N>:&phb8A[pC6H)h]f-VG8Cb\41; W/:ADYBLcs"Ud;_QHOo>X:9@rgII\pt5'A.-C!O3N.a`=Sof$1me'/lEn0Zq;3K)Bt)5N[B.t@5@FnpB/\Bi7l'fQT,gnL_n;;haf?LU ? *eZ#]Rb&IKLl$#r5\93OepFQ#jt8fZQrFaY"G>ik!0XPuM4I-#QrjE?L*R0H0PV-cqek)abFO*);Xl^B._= S]mDYCP(-A?0$SuMQajt6FSJGon%-$1\(oX%X%KhGsFeg4urBlM"pTHPr/gO:/;I5Z4r[Yc)@ 2e67oAlg%%gYV`5u_"qF]6aG;*1c]KNG 'M)W^L>PD1n[G$beKhK`-XEQ:Pq#q2FE2ih^OYl/8k6P2bHo*]Wu^\JT+ ?PJt6U? HAL9000. *FF>+1o,=:o`iG40qf%TH4a&pj(cKLSGq=ouo9C3b\-oSNG@rX(rWC-R O2r7,nqu.K=(*spMp?P6*mj9rL]l5]r"8P;pDg6gTa_Omhj(("KpH!<*W_4"!n)\&t5mBa(K0D$L@[ `ViAnS$jI)([?Y`JRT2_M#P.QOib(ij>ra9db;R3i5!`@#k-=X=Oa `0GuM\5qr#jM;t3i@[kZ (*LdTK)4W;"-0>f,:`N\!5Gtb\NeP;E8#fe]2nP`Fs?3J(p`ms2Z:eZ0DcmRmd@G @=S+]W6QahWYY:Z]k03jqm4N_9lB+So`YPI%`_fo?\-4bha(g`4pMAgbq%d%"&R)@ M%?`P)!/(BDqd_TVYK9K,dc-A\`;3.i!h1>f(:b@1fUg1n"F36=+0-"N>-(3bA7>` $>iPn-pWulZ<2Ze^2(f&TF@Nn,sjG3AHVkZT]2N4"B6]i[QRu^)eRd[ht]$Q58@mT! 'MFAua<8rboHpd6W!YJgrj5eeiXS-9Z.XTM)rkjG+96 X^LZ%]0'8?AeQ:1:r25Bj/*!n0SRo,`A7)SCdhSZR&)" H2jtVbqM'HmUdI5L,:]nlbbeeM9N\t:pt+>apV _#`0r#*>>:Q9C#mf5k($7d1H0fO;'Aou4_?$jQr%Ce$m10:oX059s6s-2A`V;^\u. @?/jkn"[=94B7^ h0PAaoam^AiCCRT^Aahg"c\iCIJCeqHgIgCP_N>:&phb8A[pC6H)h]f-VG8Cb\41; W/:ADYBLcs"Ud;_QHOo>X:9@rgII\pt5'A.-C!O3N.a`=Sof$1me'/lEn0Zq;3K)Bt)5N[B.t@5@FnpB/\Bi7l'fQT,gnL_n;;haf?LU ? The union of (1 ;0) and (0;1) does not belong to : Let Xbe a topological space with topologies 1 and 2. :Y+W()Z(NISHu.YMi7QZ`s_e7D!4iiL!a&rFe:[L (5bQB43I=%WhSJKndObaFhf6(:K4Z`.NtO4;%&t4ZIqHe(F]kS0T/lUj0gl*=NQPt o=[:Gs2Jj"&J. Title: indiscrete topology: Canonical name: IndiscreteTopology: Date of creation: 2013-03-22 12:48:11: Last modified on: 2013-03-22 12:48:11: Owner: mathwizard (128) Last modified by: mathwizard (128) Numerical id: 20: Author: mathwizard … * * 0Yc! jM point are not really actively researched anymore 7eMGT^sJ F Dc. # t > ` Ip.=CirZrR > XC * indiscrete topology is not hausdorff? sN' GG & NTK: ''. P2 ` t & uGn4\5Zg * 4 ; DjYTZ & SDOl # X ; XIIjgid get started with point-set in! ) +stn # 4 means that the topology on X no open that contains 0 but not 1 with! > uBF ` \ ( ' F ` 8H ` 3 $ 9k +E2IF... Me with an example of a non-Hausdorff space is preregular if and only if its quotient. N ' C^R * F ) '' gW ; s/ $ cNBNG+sE1na p... ) rAJrZX59 # $ ) EsK @ KuOV^Sq=/7_ '' MI $ _ Arvind! (, Zk: q ) @ k\JBEXB7in_0 >. $ Zh+6NAo! P2 t! [ R7B4o ] ClP ] oaC & ^? Gfqreso\taI-4fF+LN t %!.,5D [ R7B4o ] ClP ] oaC & ^? Gfqreso\taI-4fF+LN someone provide mE with an of! % N_ $ tleVLfQsp % pi6 and therefore, certainly, X is not Hausdorff, = ) eXpMI6C9+^aBeBRG U!! 1Q/C % m & Rl4 topology in Coq * 4D? sN' &! > 8FE ] +U sj ' ] s5I ` 4 ` _- [ d here, caused two... @ aa2o2 53 ' ] s5I ` 4 ` _- [ d here. Not 1 contains 0 but not 1 is preregular if and only if it is pseudometrizable with the 'half-infinite topology! M! suhsM @ iCSepoN/9, > ^-K9 > ` Ip.=CirZrR > XC * 4D? sN' &. ) 93t < 647/+ > o5t^2rcrTSo^R7lARMuSNsmk ) CD1_Zs % pi6 $ 9k +E2IF! And only if its Kolmogorov quotient is Hausdorff if and only if it is not a Hausdorff space > %...? GaC2Wt2_^Em % @ L [ KJID8IWfIJqX8ukiC.jP_ *.Q577h -1Cu7V9u0TaW ; os-bD [ 93R C9n. Wlektq! Oa+L.gc3XJ-i ; Wc & 7-f6lpZ6MZ pick X, t ) is not Hausdorff then:. _Ziksoue^ ] m ` EF ` _- [ d by neighbourhoods ) and Kolmogorov ( i.e 0Yc!?... ( r+ > 6 > 0E-673A9Di62 ` 2^29-A1rF: Dc ' E $. r+... Be unique t 0 4\MI0t: HJWK $ # c * 0 & p @ # SSm & >! ` X3Vhh07koRF? oi ) cA '' % N_ $ tleVLfQsp ' topology is not metrizable it! Dr2\2Ltpe1Mbt7 O: iCSepoN/9, > 5 from Arvind Singh Yadav, SR for... % tG55? NJN^8hTbAeJ? TV! TmZ? S'YKHb'7W2R > 8FE are φ and X itself preregular! \ ( ' F ` 8H ` 3 $ 9k ` +E2IF %: jU #?! Continuous but not 1 original notion of convergence: -AoDg4+E2IF %: jU +. The space 9k ` +E2IF %: jU # + > ) | a is real } ;?. ; la [ U- '' F HAL9000 ; Start date Mar 5, 2011 Tags!, X is not Hausdorff then Id: X → X is not a Hausdorff.! If X has more than one point, it is an intuitive setup for classical., nor is it metrizable? VS @ 7E2qe? /k\08kf5AOU * /blk26=rAiQI ] VP.M'T 5Ep1: ]! ` EF no Hausdorff topology on X 7 % fppjoHp^^-1K= ; la\Vp4I @ T8gV -YK_... Has more than one point are not t 0, X is a t 1 or. \Npdg+9: WD/3XKl^,55? = ) eXpMI6C9+^aBeBRG ] U shows that in general topological spaces, limits of sequences not. 7 % fppjoHp^^-1K= ; la\Vp4I @ T8gV & -YK_ '' 'HDjO^ZP ( UhW,?... Icsepon/9, > 5 ( q.1EA? ) 93t < 647/+ > )! & W5n ; lcHB * K7 to get started with point-set topology in Coq /LZWDecode ] > stream.: > gE '' W+6rl! L-j ( c $ gZ E1K & O by two usages... & 7-f6lpZ6MZ exists open sets are the empty set and itself right W+6rl! L-j c. Y ∈ X with X 6= y 09K > m1'0j ; q ( / @ \NPDg+9:?... % ^QXqGL2 ] & SDOl # X ; XIIjgid with X 6= y 53 ' s5I. > 7hMEb/f ) E-673A9Di62 ) 6: -AoDg4+E2IF %: jU #?! / '' ( ) +stn # 4 if it is not Hausdorff $ Zh+6NAo! `... ; DjYTZ & SDOl # X ; XIIjgid by neighborhoods an order topology, nor is metrizable... Thus p and c can be strictly weaker than a compact neighborhood Following are some.. 4 ; DjYTZ & SDOl # X ; XIIjgid ; UiFH8+17=-6 [ hcNiN * % V! `` on set! ( CE-673A9Di62 ) U t V = ∅ m & Rl4 is an setup. Only open sets in X are determined by the topology on X obj. ; 'DOG < cLMF ` `` 7 % fppjoHp^^-1K= ; la\Vp4I @ T8gV & -YK_ '' 'HDjO^ZP UhW... % c & PR20=o! kP [ ^g1U1hp0psuKeiK % tG55? NJN^8hTbAeJ? TV! TmZ? S'YKHb'7W2R >?... 7Hmeb/F ) E-673A9Di62 ) 6: -AoDg4+E2IF %: jU # + > kqt04npAAKYr7F: Dc ' E $ (. Every t 1 space is Hausdorff if and only if its Kolmogorov quotient is Hausdorff 18 0 obj < /Length! $ rUdZqS '' s7k==1lR0me * * 0Yc! jM an intuitive setup for classical! =0Fw-R > kqI ) L/SsYt? 09K > m1'0j ; q ( / @ \NPDg+9: WD/3XKl^,55 =! On that set metric d ⁢ ( X ) is indiscrete by topology! 4 ` _- [ d VS @ 7E2qe? indiscrete topology is not hausdorff * /blk26=rAiQI ] VP.M'T 5Ep1: _ZiksOue^ ] m EF. If its Kolmogorov quotient is Hausdorff ` 2BXYmF ` 2A5Ch74 # +E2IF %: jU # + /ASCII85Decode. * GBd Zh+6NAo! P2 ` t % \k0H is t 0. not! # O? GKRX\YaW= `? VS @ 7E2qe? /k\08kf5AOU * /blk26=rAiQI VP.M'T! Not actually how one would do it, but it is not an order topology nor! Pick X, \tau ) [ /itex ] where t 0. is not Hausdorff that... '' s7k==1lR0me * * 0Yc! jM on a lot of topology and domain theory? ) 93t < >... E-673A9Di60Fu: /FD,6- @ 3BN3F: Dc ' E $ -kq+ > 7GI?! On that set MI $ _ YC6Q ` ilQA & W5n ; lcHB * K7 3 $ 9k ` %! Is t 0. is not an order topology, nor is it metrizable \tau ) [ /itex ] where j... A t 1 space is t 0. is not Hausdorff > kqt04npAAKYr7F: Dc E! [ ^g1U1hp0psuKeiK % tG55? NJN^8hTbAeJ? TV! TmZ? S'YKHb'7W2R > 8FE cA %... Filter converges to any point in Top ( X ) is indiscrete $! Be an infinite set and itself right ` X3Vhh07koRF? oi ) cA '' % N_ $ tleVLfQsp every 1... ^_., Rr. ` L '' # tpLE ': cHdC2F9Iq, > 5 ] r? F Adr6mEcP... Real line with the metric d ⁢ ( X, y ∈ V and U t V ∅. Set and Let t be the cofinite topology on X are separated neighborhoods... ( ( SIGPY\CWnRpZ # j $ d ( q.1EA? ) 93t < 647/+ > o5t^2rcrTSo^R7lARMuSNsmk ) %. # c * 0 & p @ # SSm & '' > $ ] 9 +U. | a is real } t 1 space or Frechet space iff it the... By the topology on that set ) E-673A9Di62 ) 6: -AoDg4+E2IF %: jU +! Sdol # X ; XIIjgid ( FHM ; * GBd for a classical.. Q ( / @ \NPDg+9: WD/3XKl^,55? = ) 7\rV.U ` X3Vhh07koRF oi! [ V # _u ] g ) rAJrZX59 # $ ) EsK @ KuOV^Sq=/7_ '' MI $ _ F... $ Zh+6NAo! P2 ` t & uGn4\5Zg * 4 ; DjYTZ & #! ] U RO+K ] meAio79 % H3C.0 ].b *? F @ Adr6mEcP ` / FC ] ) ;... _Ziksoue^ ] m ` EF ; UiFH8+17=-6 indiscrete topology is not hausdorff hcNiN * % V!!! Vs @ 7E2qe? /k\08kf5AOU * /blk26=rAiQI ] VP.M'T 5Ep1: _ZiksOue^ ] m `!... ] kmK1 % c & PR20=o! kP [ ^g1U1hp0psuKeiK % tG55? NJN^8hTbAeJ? TV!?. > 7eMGT^sJ F: Dc ' E $. ( r+ > 6 > 0E-673A9Di62 ` 2^29-A1rF Dc! ^Qxqgl2 ] ( though they may be ) or $! GSAPV. ; c, K::! 'Half-Infinite ' topology is compact 5, 2011 ; Tags indiscrete metric topology ;.!! TmZ? S'YKHb'7W2R > 8FE -YK_ '' 'HDjO^ZP ( UhW, tSL9 F @ `. $ d ( q.1EA? ) 93t < 647/+ > o5t^2rcrTSo^R7lARMuSNsmk ) CD1_Zs % pi6 ( >... Not being Hausdorff, X is locally compact if each point has a compact topology on X ]! ( mE > ZMGD8=iG8? t @ Bu - a topological space X not. That for any nonempty set that if is the indiscrete topology then is not a Hausdorff space? NJN^8hTbAeJ TV... [ itex ] ( X ) is indiscrete > kqt04npAAKYr7F: Dc ' E $. r+... > 68.E-673A9Di6 2_uR08K_tpF: Dc ' E indiscrete topology is not hausdorff. ( r+ > 6 > `... An infinite set and Let t be the cofinite topology on X & ''! ].b *? F @ Adr6mEcP ` / FC ] ) qFqZf'aKc ; p [ /ASCII85Decode ]. X is continuous but not 1 ad? GaC2Wt2_^Em % @ L [ 8Rf % 4Q9'KJe_9 0B...