This is TeX, Version 3.14159 (Web2C 7.3.1) (format=latex 2003.2.18) 15 SEP 2003 11:13 **top_2.tex (top_2.tex LaTeX2e <2001/06/01> Babel and hyphenation patterns for american, french, german, ngerman, i talian, nohyphenation, loaded. (/usr/local/share/texmf/tex/latex/base/article.cls Document Class: article 2001/04/21 v1.4e Standard LaTeX document class (/usr/local/share/texmf/tex/latex/base/size10.clo File: size10.clo 2001/04/21 v1.4e Standard LaTeX file (size option) ) \c@part=\count79 \c@section=\count80 \c@subsection=\count81 \c@subsubsection=\count82 \c@paragraph=\count83 \c@subparagraph=\count84 \c@figure=\count85 \c@table=\count86 \abovecaptionskip=\skip41 \belowcaptionskip=\skip42 \bibindent=\dimen102 ) (/usr/local/share/texmf/tex/latex/amsmath/amsmath.sty Package: amsmath 2000/07/18 v2.13 AMS math features \@mathmargin=\skip43 For additional information on amsmath, use the `?' option. 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(/usr/local/share/texmf/tex/latex/jknappen/ursfs.fd File: ursfs.fd 1998/03/24 rsfs font definition file (jk) ) [1 ] [2] [3] [4] [5] [6] Overfull \hbox (21.25375pt too wide) in paragraph at lines 263--269 []\OT1/cmr/m/n/10 ([](0\OML/cmm/m/it/10 ; xi\OT1/cmr/m/n/10 )) \OMS/cmsy/m/n/10 ^^U []\OT1/cmr/m/n/10 (0\OML/cmm/m/it/10 ; xN\OT1/cmr/m/n/10 ) = [](\OML/cmm/m /it/10 d\OT1/cmr/m/n/10 (0\OML/cmm/m/it/10 ; xN\OT1/cmr/m/n/10 )\OML/cmm/m/it/1 0 ; \OT1/cmr/m/n/10 1) = [](\OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 xN\OMS/cmsy/m/n/ 10 j\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 1)\OML/cmm/m/it/10 :$ \OT1/cmr/m/n/10 Si nce $\OML/cmm/m/it/10 N > \OMS/cmsy/m/n/10 j\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 = x\OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 ; \OMS/cmsy/m/n/10 j\OML/cmm/m/it/10 xN\OMS /cmsy/m/n/10 j \OT1/cmr/m/n/10 = [] [7] Overfull \hbox (5.02963pt too wide) in paragraph at lines 279--286 \OT1/cmr/m/n/10 For the box topol-ogy, let $\OML/cmm/m/it/10 U$ \OT1/cmr/m/n/10 be the open set $(\OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 ; \OT1 /cmr/m/n/10 1) \OMS/cmsy/m/n/10 ^^B \OT1/cmr/m/n/10 (\OMS/cmsy/m/n/10 ^^@\OT1/c mr/m/n/10 1\OML/cmm/m/it/10 =\OT1/cmr/m/n/10 4\OML/cmm/m/it/10 ; \OT1/cmr/m/n/1 0 1\OML/cmm/m/it/10 =\OT1/cmr/m/n/10 4) \OMS/cmsy/m/n/10 ^^B \OT1/cmr/m/n/10 (\ OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 =\OT1/cmr/m/n/10 9\OML/cmm /m/it/10 ; \OT1/cmr/m/n/10 1\OML/cmm/m/it/10 =\OT1/cmr/m/n/10 9) [] )\OML/cmm/m /it/10 :$ [] Underfull \hbox (badness 10000) in paragraph at lines 279--286 [] Underfull \hbox (badness 10000) in paragraph at lines 287--294 [] [8] Overfull \hbox (1.9475pt too wide) in paragraph at lines 295--302 \OT1/cmr/m/n/10 Now ex-am-ine the open set $\OML/cmm/m/it/10 U \OT1/cmr/m/n/10 = [](\OMS/cmsy/m/n/10 ^^@\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 =i; \OT1/cmr/m/n/10 1\OML/cmm/m/it/10 =i\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 :$ \OT1/cmr/m/n/10 This i s a neigh-bor-hood of $(0\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 ; []\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 :$ [] Overfull \hbox (5.95256pt too wide) in paragraph at lines 295--302 []\OT1/cmr/m/n/10 ([][]\OMS/cmsy/m/n/10 f[]\OT1/cmr/m/n/10 (0\OML/cmm/m/it/10 ; x[]\OT1/cmr/m/n/10 )\OMS/cmsy/m/n/10 g\OML/cmm/m/it/10 ; [][]\OMS/cmsy/m/n/10 f[]\OT1/cmr/m/n/10 (0\OML/cmm/m/it/10 ; x[]\OT1/cmr/m/n/10 )\OMS/cmsy/m/n/10 g\ OT1/cmr/m/n/10 ) = []([]\OMS/cmsy/m/n/10 f[]\OT1/cmr/m/n/10 (0\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 0)\OMS/cmsy/m/n/10 g\OML/cmm/m/it/10 ; []\OMS/cmsy/m/n/10 f[] \OT1/cmr/m/n/10 (0\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 1\OML/cmm/m/it/10 =i\OT1/c mr/m/n/10 )\OMS/cmsy/m/n/10 g\OT1/cmr/m/n/10 ) = [] Underfull \hbox (badness 10000) in paragraph at lines 303--310 [] Overfull \hbox (12.41031pt too wide) in paragraph at lines 303--310 \OT1/cmr/m/n/10 (0\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 ; []\OT1 /cmr/m/n/10 )\OML/cmm/m/it/10 :$ \OT1/cmr/m/n/10 Now for any se-quence el-e-men t $\OML/cmm/m/it/10 x[]; x[] \OT1/cmr/m/n/10 = 1\OML/cmm/m/it/10 =m:$ \OT1/cmr/ m/n/10 But $1\OML/cmm/m/it/10 =m [] \OT1/cmr/m/n/10 (\OMS/cmsy/m/n/10 ^^@[]\OML /cmm/m/it/10 ; []\OT1/cmr/m/n/10 )$ [] Overfull \hbox (4.32765pt too wide) in paragraph at lines 303--310 []\OT1/cmr/m/n/10 ([][]\OMS/cmsy/m/n/10 f[]\OT1/cmr/m/n/10 (0\OML/cmm/m/it/10 ; y[]\OT1/cmr/m/n/10 )\OMS/cmsy/m/n/10 g\OML/cmm/m/it/10 ; [][]\OMS/cmsy/m/n/10 f[]\OT1/cmr/m/n/10 (0\OML/cmm/m/it/10 ; y[]\OT1/cmr/m/n/10 )\OMS/cmsy/m/n/10 g\ OT1/cmr/m/n/10 ) = []([]\OMS/cmsy/m/n/10 f[]\OT1/cmr/m/n/10 (0\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 1\OML/cmm/m/it/10 =i\OT1/cmr/m/n/10 )\OMS/cmsy/m/n/10 g\OML/c mm/m/it/10 ; []\OMS/cmsy/m/n/10 f[]\OT1/cmr/m/n/10 (0\OML/cmm/m/it/10 ; \OT1/cm r/m/n/10 0)\OMS/cmsy/m/n/10 g\OT1/cmr/m/n/10 ) = [] [9] Overfull \hbox (15.12325pt too wide) in paragraph at lines 319--325 \OT1/cmr/m/n/10 Each el-e-ment of this set sat-is-fies the tri-an-gle in-equal- ity on $\OML/cmm/m/it/10 d[]$ \OT1/cmr/m/n/10 so that $[]\OMS/cmsy/m/n/10 f\OML /cmm/m/it/10 d[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]; y[]\OT1/cmr/m/n/10 ) + [] [10] Overfull \hbox (11.56766pt too wide) in paragraph at lines 326--330 \OT1/cmr/m/n/10 Each el-e-ment of this set sat-is-fies the tri-an-gle in-equal- ity on $[]$ so that $[]\OMS/cmsy/m/n/10 f[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[ ]; y[]\OT1/cmr/m/n/10 ) + [] Overfull \hbox (16.14514pt too wide) in paragraph at lines 331--339 \OML/cmm/m/it/10 U; []\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]; y[]\OT1/cmr/m/n/10 ) \OML/cmm/m/it/10 < i^^O$ \OT1/cmr/m/n/10 for $\OML/cmm/m/it/10 i \OMS/cmsy/m /n/10 ^^T \OT1/cmr/m/n/10 1\OML/cmm/m/it/10 =^^N$ \OT1/cmr/m/n/10 which im-plie s $[](\OML/cmm/m/it/10 x[]; y[]\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 =i < ^^O:$ \OT 1/cmr/m/n/10 For $\OML/cmm/m/it/10 i > \OT1/cmr/m/n/10 1\OML/cmm/m/it/10 =^^N; []\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]; y[]\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 =i < [] Overfull \hbox (6.95964pt too wide) in paragraph at lines 331--339 \OML/cmm/m/it/10 D\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x; y\OT1/cmr/m/n/10 ) = []\ OMS/cmsy/m/n/10 f[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]; y[]\OT1/cmr/m/n/10 )\ OML/cmm/m/it/10 =i\OMS/cmsy/m/n/10 g \OT1/cmr/m/n/10 = []\OMS/cmsy/m/n/10 f[][] f[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]; y[]\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 =i\OMS/cmsy/m/n/10 g\OML/cmm/m/it/10 ; [][]\OMS/cmsy/m/n/10 f[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 x[]; y[]\OT1/cmr/m/n/10 )\OML/cmm/m/it/10 =i\OMS/cmsy/m/n/10 gg\OML/cmm/m/it/10 :$ [] [11] [12] [13] [14] (top_2.aux) ) Here is how much of TeX's memory you used: 1146 strings out of 20880 12131 string characters out of 196184 66158 words of memory out of 350001 4110 multiletter control sequences out of 10000+15000 14038 words of font info for 56 fonts, out of 400000 for 1000 14 hyphenation exceptions out of 10000 27i,14n,24p,1543b,248s stack positions out of 3000i,100n,1500p,50000b,4000s Output written on top_2.dvi (14 pages, 83756 bytes).