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zd)@363#Z*j^8V+Z0K;7eK%!OYrS@E`aksyvSMrUH4WLK$x;_h2+jJ{JPbJE=d($ZN z7f&l46=Mm}g=<2wCFIhcgt9;xCwL~tW^kPpn|GTbv#@%84tayl9JHR!5NU-bN8;q= NLZYFO`XY_=zW|xCySo4Y diff --git a/documents/GeoTopo/GeoTopo.tex b/documents/GeoTopo/GeoTopo.tex index 8a6dfc8..25ea446 100644 --- a/documents/GeoTopo/GeoTopo.tex +++ b/documents/GeoTopo/GeoTopo.tex @@ -1,11 +1,12 @@ \documentclass[a4paper,oneside,DIV15,BCOR12mm]{scrbook} +\usepackage{etoolbox} \usepackage{amsmath,amssymb} \usepackage[utf8]{inputenc} % this is needed for umlauts \usepackage[ngerman]{babel} % this is needed for umlauts \usepackage[T1]{fontenc} % this is needed for correct output of umlauts in pdf \usepackage[framed,amsmath,thmmarks,hyperref]{ntheorem} -\usepackage[bookmarks,bookmarksnumbered,hypertexnames=false,pdfpagelayout=OneColumn,colorlinks]{hyperref} -\usepackage{makeidx} % index +\usepackage{makeidx} % for automatically generation of an index +\usepackage[bookmarks,bookmarksnumbered,hypertexnames=false,pdfpagelayout=OneColumn,colorlinks,hyperindex=false]{hyperref} % has to be after makeidx \usepackage{enumerate} \usepackage{braket} % needed for \Set \usepackage{csquotes} @@ -28,6 +29,24 @@ pdftitle = {Geometrie und Topologie} } +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% make the index link to the correct part of the page % +% http://tex.stackexchange.com/q/74493/5645 % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\newcounter{indexanchor} +\newcommand*{\xindex}[1]{% + \stepcounter{indexanchor}% make anchor unique + \def\theindexterm{#1}% + \edef\doindexentry{\noexpand\index + {\expandonce\theindexterm|indexanchor{index-\theindexanchor}}}% + \raisebox{\baselineskip}{\hypertarget{index-\theindexanchor}% + {\doindexentry}}% +} +\newcommand*{\indexanchor}[2]{\hyperlink{#1}{#2}} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\makeindex + \begin{document} \maketitle diff --git a/documents/GeoTopo/Kapitel1.tex b/documents/GeoTopo/Kapitel1.tex index 013be17..189c0f6 100644 --- a/documents/GeoTopo/Kapitel1.tex +++ b/documents/GeoTopo/Kapitel1.tex @@ -5,12 +5,12 @@ \input{figures/s2.tex} & \input{figures/cube.tex} & TODO & \input{figures/pyramid.tex} \end{tabular} - aber nicht zum $\mdr^2$ oder zu einem Rhombus + aber nicht zum $\mdr^2$ oder zu einem Torus: \input{figures/torus.tex} \section{Topologische Räume} -\begin{definition} \index{Topologischer Raum} \index{offen} \index{abgeschlossen} +\begin{definition} \xindex{Topologischer Raum} \xindex{offen} \xindex{abgeschlossen} Ein \textbf{topologischer Raum} ist ein Paar $(X, \fT)$ bestehend aus einer Menge $X$ und $\fT \subseteq \powerset{X}$ mit folgenden Eigenschaften @@ -22,7 +22,7 @@ \end{enumerate} Die Elemente von $\fT$ heißen \textbf{offene Teilmengen} von $X$. - $A \setminus X$ heißt \textbf{abgeschlossen}, wenn $X \setminus A$ offen ist. + $A \subseteq X$ heißt \textbf{abgeschlossen}, wenn $X \setminus A$ offen ist. \end{definition} @@ -30,13 +30,14 @@ Es gibt auch Mengen, die weder abgeschlossen, noch offen sind wie z.~B. $[0,1)$. \begin{beispiel} \begin{enumerate}[1)] - \item $X = \mdr^n$ mit der euklidischen Metrik.\\ + \item $X = \mdr^n$ mit der euklidischen Metrik.\\ \xindex{Topologie!euklidische} $U \subseteq \mdr^n$ offen $\gdw$ für jedes $x \in U$ - gibt es $r > 0$, sodass $B_r(x) = \Set{y \in \mdr^n | d(x,y) < r} \subseteq U$ + gibt es $r > 0$, sodass $B_r(x) = \Set{y \in \mdr^n | d(x,y) < r} \subseteq U$\\ + Also: $\fT = \Set{M \subseteq X | M \text{ ist offene Kugel}}$ \item Allgemeiner: $(X, d)$ metrischer Raum - \item $X$ Menge, $\fT = \Set{\emptyset, X}$ heißt \enquote{triviale Topologie} \index{Topologie!triviale} - \item $X$ Menge, $\fT = \powerset{X}$ heißt \enquote{diskrete Topologie} \index{Topologie!diskrete} - \item $X :=\mdr, \fT_Z := \Set{U \subseteq \mdr | \mdr \setminus U \text{ endlich}} \cup \Set{\emptyset}$ heißt \enquote{Zariski-Topologie} \index{Topologie!Zariski}\\ + \item $X$ Menge, $\fT = \Set{\emptyset, X}$ heißt \enquote{triviale Topologie} \xindex{Topologie!triviale} + \item $X$ Menge, $\fT = \powerset{X}$ heißt \enquote{diskrete Topologie} \xindex{Topologie!diskrete} + \item $X :=\mdr, \fT_Z := \Set{U \subseteq \mdr | \mdr \setminus U \text{ endlich}} \cup \Set{\emptyset}$ heißt \enquote{Zariski-Topologie} \xindex{Topologie!Zariski}\\ Beobachtung: $U \in \fT_Z \gdw \exists f \in \mdr[X]$, sodass $\mdr \setminus U = V(f) = \Set{x \in \mdr | f(x) = 0}$ \item $X := \mdr^n, \fT_Z = \{U \subseteq \mdr^n | \text{Es gibt Polynome } f_1, \dots, f_r \in \mdr[X_1, \dots, X_n] \text{ sodass }\\\mdr^n \setminus U = V(f_1, \dots, f_r)\}$ \item $X = \Set{0,1}, \fT = \Set{\emptyset, \Set{0,1}, \Set{0}}$\\ @@ -44,7 +45,7 @@ Es gibt auch Mengen, die weder abgeschlossen, noch offen sind wie z.~B. $[0,1)$. \end{enumerate} \end{beispiel} -\begin{definition} \index{Umgebung} +\begin{definition} \xindex{Umgebung} Sei $(X, \fT)$ ein topologischer Raum, $x \in X$. Eine Teilmenge $U \subseteq X$ heißt \textbf{Umgebung} von $x$, @@ -54,16 +55,16 @@ Es gibt auch Mengen, die weder abgeschlossen, noch offen sind wie z.~B. $[0,1)$. \begin{definition} Sei $(X, \fT)$ ein topologischer Raum, $M \subseteq X$ eine Teilmenge. \begin{enumerate}[a)] - \item $\displaystyle M^\circ := \Set{x \in M | M \text{ ist Umgebung von } x} = \bigcup_{\stackrel{U \subseteq M} {U \in \fT}} U $ heißt \textbf{Inneres} oder \textbf{ offener Kern} von $M$. \index{Inneres} \index{Kern!offener} - \item $\displaystyle \overline{M} := \bigcap_{\stackrel{M \subseteq A}{A \text{ abgeschlossen}}} A$ heißt \textbf{abgeschlossene Hülle} oder \textbf{Abschluss} von $M$. \index{Abschluss} - \item $\partial M := \overline{M} \setminus M^\circ$ heißt \textbf{Rand} von $M$. \index{Rand} - \item $M$ heißt \textbf{dicht} in $X$, wenn $\overline{M} = X$ ist. \index{dicht} + \item $\displaystyle M^\circ := \Set{x \in M | M \text{ ist Umgebung von } x} = \bigcup_{\stackrel{U \subseteq M} {U \in \fT}} U $ heißt \textbf{Inneres} oder \textbf{ offener Kern} von $M$. \xindex{Inneres} \xindex{Kern!offener} + \item $\displaystyle \overline{M} := \bigcap_{\stackrel{M \subseteq A}{A \text{ abgeschlossen}}} A$ heißt \textbf{abgeschlossene Hülle} oder \textbf{Abschluss} von $M$. \xindex{Abschluss} + \item $\partial M := \overline{M} \setminus M^\circ$ heißt \textbf{Rand} von $M$. \xindex{Rand} + \item $M$ heißt \textbf{dicht} in $X$, wenn $\overline{M} = X$ ist. \xindex{dicht} \end{enumerate} \end{definition} \begin{beispiel} \begin{enumerate}[1)] - \item $X = \mdr$ mit endlicher Topologie\\ + \item $X = \mdr$ mit euklidischer Topologie\\ $M = \mdq \Rightarrow \overline{M} = \mdr, \;\;\; M^\circ = \emptyset$ \item $X = \mdr$, $M=(a,b) \Rightarrow \overline{M} = [a,b]$ \item $X = \mdr, \fT = \fT_Z$\\ @@ -71,7 +72,7 @@ Es gibt auch Mengen, die weder abgeschlossen, noch offen sind wie z.~B. $[0,1)$. \end{enumerate} \end{beispiel} -\begin{definition} \index{Basis} \index{Subbasis} +\begin{definition} \xindex{Basis} \xindex{Subbasis} Sei $(X, \fT)$ ein topologischer Raum. \begin{enumerate}[a)] \item $B \subseteq \fT$ heißt \textbf{Basis} der Topologie $\fT$, @@ -84,7 +85,7 @@ Es gibt auch Mengen, die weder abgeschlossen, noch offen sind wie z.~B. $[0,1)$. \end{definition} \begin{beispiel} - $X = \mdr^n$ heißt euklidische Topologie und + $X = \mdr^n$ mit euklidischer Topologie und \[B = \Set{B_r(x) | r \in \mdq_{> 0}, x \in \mdq^n}\] ist eine Basis. \end{beispiel} @@ -94,10 +95,10 @@ Es gibt auch Mengen, die weder abgeschlossen, noch offen sind wie z.~B. $[0,1)$. genau eine Topologie $\fT$ auf $X$, für die $B$ Subbasis ist. \end{bemerkung} -\begin{definition} \index{Spurtopologie} \index{Teilraum} +\begin{definition} \xindex{Spurtopologie} \xindex{Teilraum} Sei $(X, \fT)$ ein topologischer Raum, $Y \subseteq X$.\\ $\fT_Y := \Set{U \cap Y | U \in \fT}$ ist eine Topologie auf $Y$. - $\fT$ heiß \textbf{Spurtopologie} und $(Y, \fT_Y)$ heißt ein + $\fT_Y$ heißt \textbf{Spurtopologie} und $(Y, \fT_Y)$ heißt ein \textbf{Teilraum} von $(X, \fT)$ \end{definition}