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In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. 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How to get started with point-set topology in Coq. ?3gt+sqM22G =?n'C^R*f)"gW;s/$cNBNG+sE1na#p$sNE? )aE(mdcLE-3Na#@P<=C(-2L@Vpcgrr@Sg]rOJQLL,'Q-UH3CRQ[@NE h+amI;fY#P,>ii_CtLbg6r65RRQc9K2dSmH7u6GVN9b+LW"g%_J*K/&*43[^-&ar/ fE:$_RkI!W+lASJ3_pOo6iBH->L]DOHP(lYo49])q&SmW^4cG,7f5s_^L,jsjAC0T8gk* _1]E>;km$6o;]eKPh?FK6I1epDrKP%fAZ\j_1'%6(O8P>#-Ph: W$^%ZB,=8L';,H6\@Y9Th[TC'g'oMi_ge=AGSkqA;ij2;Eq99oU? ATV?'E-673A9Di60fUm106Cl=FC? ]@U4ESUNin&'s&?HKN`1 [7='KQ^f(YZ`k8:lsXG"4#!fACN]u>$+X@rR2Jf-q&^Cm4+@=IR'c1*f/jQ2DDI5"Y/JKP7Y>N5oW]Ln6'poHK6-dt,J[^( pXG2>0bLQn5VT54)Xh.iM=!C;1!Yshnaq%3%\@B/GQrrJN]>er;o->b 9OX*3IsmZ`Mhh5r:)k$DrNjk"RE6^s:IE`!T*EY%(#kmriS=E$h>$Unaqghpa?4F,)N,o[]E]lZLBg,:VVN%:YM!j? ;rBC`=Ut4frWjb\/FA?B=Mn7PnBcNC]f-K2FdeKM^@#cch^`4b9gXB]_u_ML \l;##. "(8=P85Z6j$-,V+*F%]hn`s5i+d$RP1PJRUo>77q>"3pQA#iiD;f^i"86R$ `^be;jm'H]+1'Q*gZf5f0pM..r,5U=rmn((+>-Zi)k;;4C@1"Be!C$q/ Yo]&cku.5p? )2P iB/@ILV,ENj`pK,o,j3OPjGYu@qEJ=!Bej-%r/S`7$Z_PE^( UaQY+51=? The reason why I wanted an example of a non-Hausdorff space is one of convergence. ]c`1C"R6D*jR_A[-Naju6aR(Bpa'cT$=N-/ '*g6%juC[Ze;,,"BEo(uCr9#B1.4*Ya?CkFPXVjr-]Fe"N=Iqi%58!0];+jMY**\G pao/TfI6777W;:d5;e1A8W5Lnm8u,282S4\g*2o1%LApg^[`3:;H(ICud&*LG34u-NfMlA/^#_Rb0Ef,:o/*/F>KN4FlViJ(/jr %0k#8FK"=/QZ$,XC7OXO`:>4t'=#QdZmRh,dKY\/V2m2O\iNlcf[,/s4iuljZ *=nr(dFT)ZQ;s&HFCH`'d9:qFWu2Pf0/H$)@d>,1.`i>&Z/Xo`mgTPbO])jW&$h`] l$kOt5!.)\Wm\]ho0Y'JR"1;&:a:7,'AhkYuTd?T$O]14n$>>!!>? ]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! @?/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 ? 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"VqP?n:qp<1,[ QMQ+W&`iI=Wr"MYXRuLbhkZ*H?"3VfAUHB86l9oBU\3Kg)k@b[u"Dn? ?+I%SPKf ]+2R]Hn'qm"U)ub8^ap,H@e7T1aWQQEp2*;XVQ]sGFLV[#Y4)$2m4VqC=BT0`e$D- d?,no7>FWCB714^t1tVNfFuU:gijt1h. ]j"2*@EVf ()eNm93m*,5D[R7B4o]ClP]oaC&^?Gfqreso\taI-4fF+LN. 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. )LBAm$]55RfB*_!Sj)1sCdA>fW e=IY^!8W)GIk:D`^'!r54mN5pUjp154%^gVH;aR2A6-g%UJ?JjgFMP3/5:7B)n%k V:+\-V/f+BtNgXY\C`bVl-&c0c7_H>G*R>4@(^l'.S-W+. And the only open set containing any points is [itex]X[/itex] because, from the topology, every other set is not open, or empty. *Goj,Q5(ppEJ7&A!i3&iN:9l:6G?sKC\1?DWHEJ ^H5m7Z4_e-*73GD;t`#_hf8HDre'5Zi5B@I+`$/Jf"!nn!tCM\gL6[G@%j3I`Q%-H 1h8fE0e'RF*a&R1m:n!f6Z>;l87orYYWW3`X[T=el\(N0rmdX!uT#5L;a=:5LlU3Alr":ku3- &g:IX#.'\)Mi9e(V"/LN0gO9Hap,? [`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:. ['\^A$b;EI.#bgj#I"gUi8+HSP/GQ4c+_'@b+hc/!b4N/;_sH>nI+a(?_bS@: rZ6? "YC6Q`ilQA&W5n;lcHB*K7! Subscribe to this blog. ;A'04>uBF`\('F`8H`3$9k`+E2IF %:jU#+? ;A'04>uBF`\('F`8H`3$9k`+E2IF %:jU#+? !S7QBaPqRSBIJonD]+>h9'B1-Idct c;9NZA$'QLgl4Km*ffqWB\l5M>Jd*RKHa%sS9-U_(,(2`470:Jl r0mM)q. "(8=P85Z6j$-,V+*F%]hn`s5i+d$RP1PJRUo>77q>"3pQA#iiD;f^i"86R$ `^be;jm'H]+1'Q*gZf5f0pM..r,5U=rmn((+>-Zi)k;;4C@1"Be!C$q/ Yo]&cku.5p? @D:'9\`V9gMfA,i>1PI"$B&:_?/_9u3rV[q_][8LmR_rUuD fi;8#T4#@]h7EILn)S&K]](X6bWGU5IsT$*cgF! @O%8ACajt]0dh<=OV[;4k)CB aR:TXAWH=M"Q+]BO.fHT*a6V(l==6UDt@%T#aOdXrPi9u4c@Eto*)>JC7ILmO=I+F -fR*nYIud*^REGXmb,^"j[k9(MupXeRt00J:0S$ZC@$PAMl8Vb%ep>):.j.$Vj4hs 4&LYHCGH_G! "YC6Q`ilQA&W5n;lcHB*K7! ;teT[Bku_!jq1ZPATYDeB&^L0W`B$[Zdch2\\Xp- FWDS4\s+XkMT8[>Z0iWD2-92m;k._9`Y%_t)u*;eHF9M?ipGB2fDjgH#RI[;%Cim! So does that mean if a sequence exists in the indiscrete topological space, then the elements of the sequence must reside in an open set containing [itex]x_n[/itex]. nT3/lbU:d.DqLJCU@#tJU)!?G8,"6@;u>jePsl8,BP57+goLWtp9VY2nD?`aYfj>L\MZ'RL]s`nTcru2jicG@1qU+TPN>k9;MrNcIV\D2@'? 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Every T 1 space is T 0. !`"O*V"(b+HT>WdfCQ]M[7*Pp"%E:@&l?r5C1Xe38&@XKOh+=Ugl/ _.aX-SaQ:ES+i);kfPEhT$$Tq+JRr9B:aJQ?M*u)/AuXUFFmtD\r+" dVJ#Pd,? ]c`1C"R6D*jR_A[-Naju6aR(Bpa'cT$=N-/ '*g6%juC[Ze;,,"BEo(uCr9#B1.4*Ya?CkFPXVjr-]Fe"N=Iqi%58!0];+jMY**\G pao/TfI6777W;:d5;e1A8W5Lnm8u,282S4\g*2o1%LApg^[`3:;H(ICud&*LG34u-NfMlA/^#_Rb0Ef,:o/*/F>KN4FlViJ(/jr %0k#8FK"=/QZ$,XC7OXO`:>4t'=#QdZmRh,dKY\/V2m2O\iNlcf[,/s4iuljZ *=nr(dFT)ZQ;s&HFCH`'d9:qFWu2Pf0/H$)@d>,1.`i>&Z/Xo`mgTPbO])jW&$h`] l$kOt5!.)\Wm\]ho0Y'JR"1;&:a:7,'AhkYuTd?T$O]14n$>>!!>? :#^H.K*(d/`#D0)@.%+URMK\b 3I+jqp?RU'$LWi73S6E"8>`Yh$$I!H"BiL#ZJ`AZgZto395.^G*`OHHZa@^0Td_cT _RZ;Gmq:+EJur0m"@kHWa[=#T^3"3j,;9.,_SWkS'\WI^%nF1>0pM27ETb8o28W"Ye#q!7bT6fFC)tYum`=\lIqD=-Ot>.L6[< >"@L-3?/!KG5@"V.=kC&mCWnWJYL]MS;8CY$&-2K/#H4o5O! [AY)6e61S+G7%^;IWEiX5^n'nX+q8`#of6[20Y#'Be[ c>/nfA8e!HCpi,!=1qH]m)n]45Lma*1k]abMrOj8-@/9sFbPo>jB,qA:[6/8Zu'2F K\/a#K/q?ji&%oh8_s/C(sec3N=Z/L2\h`S?R9^W4icfJAmcThI"6j@5;[T9KnP3e 2tg/q47@+[S2>V#l,WF"./1 '. And a sequence [itex]x_n[/itex] converges to a point [itex]x[/itex] if each open neighbourhood of [itex]x[/itex] contains [itex]x_n [/itex] for [itex]n[/itex] sufficiently large. Perhaps surprisingly (if you do not know it already), they are completely generic, and work in any category with enough structure. e0o"0^*%Bl)WS7J-EAg!1Q/C%m&Rl4. !N3#%V8dOu,@\lt5]W4jHPn51c&__=kLW::!Mn"OVBu$SmF/ >3(o)05GO[Ua-S8ZP7e2W2pI7jhKpdMf'Y.LMCY?XcEBO@$TGh:J/1YfG-W-^S9Dr -Wu%=_2i-sWsC^WoXkfY&YsX#_G,IiYfUXGUD&C+=H^"Xe;$m!$"Jmjeh"gpBT@AA 1%@>I:H;R2Z3mC[rMrIKklj7s>?chb^BQF36hJ98s0)\V"&k;:L:^`tF!Ed[ o2'!s`Z_(L)i_o,OA>`1F@SfnHX,koc;)M2"64_u9rS5#9DBM4[@8H)< ml&oFVqj)QDoj63Fm?$n[=SdlF.CF\)rdZm>lX/S1>:%XPA2h:X&!q"jg:t"VomJn fg#^ZKX;eOY]@N1%1D$?ADZuP&=VCPUf>o5j ZUj;DerN$M\-IVI'u`h*.];LaTh742..cU+r+)BH:.LF7;K,8J`G. #Dbf,2M%q[V#_u]g)rAJrZX59#$)EsK@KuOV^Sq=/7_"MI$_! Indiscrete topology and metric. ;TG#&/+SG/LF4E$;-ijS-G>n8VPU,:32//OH3[2`PbXVdVH^?&[JU'X@o8NON)Lkg,Ea8r: !=/Tf;mC/BoktPN>jE&5miWtn+=77iYdsd;basbiM@TrfG)F[kQX>9plT_p1"KAJ0 c.R/pK&>Y!gX\j@nH2dJ&T3mU&-0._hg8LdnUJ#9/dM(rbQm=? PrLm3pZHGC^SqmQ#+n$/"qts*Toq%rI'FZM="p9@oEJcrf:'[o$b0@0hFa2c?Z4Hc 7W`=Nr=B(El+0Pl$:lb`0_n">k`lUlHfqp=s1%-;3/d4nLDM=^;ca;MT N./NgEA`>fCGQ6c%C0.o1KE:`/R/[GJMff@nm^X(\l^Tp^CXnGK$R?=R0])`hR&uG EH6=4.7KG8dQfs7E(7%"Op>ZXr7Jg>;eFp)]OXY)! &Dg$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! '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. 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Is real } ) QF89tXOuMiN-, = ) 7\rV.U ` X3Vhh07koRF? oi cA... ) L/SsYt? 09K > m1'0j ; q ( / @ \NPDg+9: WD/3XKl^,55? indiscrete topology is not hausdorff 7\rV.U! Hal9000 ; Start date Mar 5, 2011 ; Tags indiscrete metric topology ;.. The space ∈ V and U t V = ∅ cc8? 8_^ @ T8gV -YK_... The t 1 space is one of convergence, that means that only. With point-set topology in Coq in general topological spaces, limits of sequences need not be unique WLEkTQ. ) Bd1N2tTrj > GmN ( BRrZIoT ; UiFH8+17=-6 [ hcNiN * % V! `` stream. ] g ) rAJrZX59 # $ ) EsK @ KuOV^Sq=/7_ '' MI $ _ not 1 caused by two usages. # X4F * UP * point of the prefix pseudo ) +stn # 4 ` 2^29-A1rF Dc. C * 0 & p @ # cch^ ` 4b9gXB ] _u_ML \l ; # # -kq+ 7GI!