diff --git a/documents/GeoTopo/Arbeitszeit.txt b/documents/GeoTopo/Arbeitszeit.txt index bfcb25b..8f8f2cc 100644 --- a/documents/GeoTopo/Arbeitszeit.txt +++ b/documents/GeoTopo/Arbeitszeit.txt @@ -10,4 +10,4 @@ Datum | Uhrzeit 14.12.2013 | 13:00 - 14:45 15.12.2013 | 20:30 - 21:20 16.12.2013 | 15:00 - 15:30 -17.12.2013 | 07:30 - 07:45, 14:30 - 15:40, 16:30 - 18:00 +17.12.2013 | 07:30 - 07:45, 14:30 - 15:40, 16:30 - 18:00, 22:00 - 23:00 diff --git a/documents/GeoTopo/GeoTopo.pdf b/documents/GeoTopo/GeoTopo.pdf index a3b4773..bb7b161 100644 Binary files a/documents/GeoTopo/GeoTopo.pdf and b/documents/GeoTopo/GeoTopo.pdf differ diff --git a/documents/GeoTopo/Kapitel1.tex b/documents/GeoTopo/Kapitel1.tex index 93a9990..635d9f3 100644 --- a/documents/GeoTopo/Kapitel1.tex +++ b/documents/GeoTopo/Kapitel1.tex @@ -152,7 +152,7 @@ Auch gibt es Mengen, die sowohl abgeschlossen als auch offen sind. \end{beispiel} \begin{definition} \xindex{Quotiententopologie} - Sei $X$ topologischer Raum, $\sim$ eine Äquivalenzrelation auf $X$, + Sei $X$ ein topologischer Raum, $\sim$ eine Äquivalenzrelation auf $X$, $\overline{X} = X /_\sim$ sei die Menge der Äquivalenzklassen, $\pi: x \rightarrow \overline{x}, \;\;\; x \mapsto [x]_\sim$. @@ -552,7 +552,7 @@ sodass $\pi$ stetig wird. \end{beweis} \begin{korollar}\label{zusammenhangVereinigung} - Sei $X$ topologischer Raum, $A, B \subseteq X$ zusammenhängend. + Sei $X$ ein topologischer Raum und $A, B \subseteq X$ zusammenhängend. Ist $A \cap B \neq \emptyset$, dann ist $A \cup B$ zusammenhängend. \end{korollar} diff --git a/documents/GeoTopo/Kapitel3.tex b/documents/GeoTopo/Kapitel3.tex index 11cecfc..f4228d2 100644 --- a/documents/GeoTopo/Kapitel3.tex +++ b/documents/GeoTopo/Kapitel3.tex @@ -217,17 +217,9 @@ Für einen Weg $\gamma$ sei $[\gamma]$ seine \textbf{Homotopieklasse}\xindex{Hom \item Abgeschlossenheit folgt direkt aus der Definition von $*_G$ \item Assoziativität folgt aus Korollar~\ref{kor:assoziativitaet-von-zusammensetzen-von-wegen} \item Neutrales Element $e = [\gamma_0], \gamma_0(t) = x \;\;\; \forall t \in I$. - - $e * [\gamma] = [\gamma] = [\gamma] * e$, da $\gamma_0 * \gamma \sim \gamma$ - - \begin{figure} - \centering - \input{figures/todo.tex} - \caption{Bis auf Parametrisierung sind $\gamma_0 * \gamma$ und $\gamma$ das selbe}. - \label{fig:weg-zusammengesetzt-mit-neutralem-weg} - \end{figure} + $e * [\gamma] = [\gamma] = [\gamma] * e$, da $\gamma_0 * \gamma \sim \gamma$ \item Inverses Element $[\gamma]^{-1} = [\overline{\gamma}] = [\gamma(1-t)]$, - denn $\overline{\gamma} * \gamma \sim \gamma_0 \sim \gamma * \overline{\gamma}$ + denn $\overline{\gamma} * \gamma \sim \gamma_0 \sim \gamma * \overline{\gamma}$ \end{enumerate} \end{beweis} @@ -310,13 +302,6 @@ Für einen Weg $\gamma$ sei $[\gamma]$ seine \textbf{Homotopieklasse}\xindex{Hom \end{enumerate} \end{korollar} -\begin{figure} - \centering - \input{figures/todo.tex} - \caption{Situation aus Korollar~\ref{korr:11.5}} - \label{fig:kor-bem-11.5} -\end{figure} - \begin{beweis}\leavevmode \begin{enumerate}[label=\alph*)] \item $f_*$ ist wohldefiniert: Seien $\gamma_1, \gamma_2$ homotope @@ -406,25 +391,11 @@ Für einen Weg $\gamma$ sei $[\gamma]$ seine \textbf{Homotopieklasse}\xindex{Hom Wegen um $x$, die ganz in $U$ oder ganz in $V$ verlaufen. \end{satz} -\begin{figure} - \centering - \input{figures/todo.tex} - \caption{Situation aus Satz~\ref{thm:seifert-van-kampen}} - \label{fig:satz-seifert-van-kampen} -\end{figure} - \begin{beweis} Sei $\gamma: I \rightarrow X$ ein geschlossener Weg von $x$. Überdecke $I$ mit endlich vielen offenen Intervallen, die ganz in $\gamma^{-1}(U)$ oder ganz in $\gamma^{-1}(V)$ liegen. - \begin{figure} - \centering - \input{figures/todo.tex} - \caption{Situationsskizze} - \label{fig:intervalle-auf-01} - \end{figure} - \Obda sei $\gamma(I_1) \subseteq U, \gamma(I_2) \subseteq V$, etc. Wähle $t_i \in I_i \cap I_{i+1}$, also $\gamma(t_i) \in U \cap V$. @@ -438,18 +409,18 @@ Für einen Weg $\gamma$ sei $[\gamma]$ seine \textbf{Homotopieklasse}\xindex{Hom \item \begin{figure} \centering - \input{figures/todo.tex} + \includegraphics[width=0.2\linewidth, keepaspectratio]{figures/todo/topologischer-raum-x.png} \caption{Topologischer Raum $X$} \label{fig:top-raum-kreise} \end{figure} - $\pi_1(X,x)$ wird \enquote{frei} erzeugt von $a$ und $b$, weil + Sei $X$ wie in Abb.~\ref{fig:top-raum-kreise}. $\pi_1(X,x)$ wird \enquote{frei} erzeugt von $a$ und $b$, weil $\pi_1(U,x) = \cong \mdz, \pi_1(V,x) = \cong \mdz$, insbesondere ist $a*b$ nicht homotop zu $b*a$. \item Torus: $\pi_1(T^2, X)$ wird erzeugt von $a$ und $b$. \begin{figure} \centering - \input{figures/todo.tex} + \input{figures/topology-4.tex} \caption{$a*b = b*a \Leftrightarrow a * b * \overline{a} * \overline{b} \sim e$} \label{fig:torous-a-b} \end{figure} @@ -500,11 +471,11 @@ Für einen Weg $\gamma$ sei $[\gamma]$ seine \textbf{Homotopieklasse}\xindex{Hom \end{figure} \end{beispiel} -\begin{definition} +\begin{definition}\xindex{Abbildung!offene} Seien $X, Y$ topologische Räume und $f:X \rightarrow Y$ eine Abbildung. - $f$ heißt offen $:\gdw \forall V \subseteq X$ offen: $f(V)$ ist offen in $Y$. + $f$ heißt \textbf{offen} $:\gdw \forall V \subseteq X$ offen: $f(V)$ ist offen in $Y$. \end{definition} \begin{korollar} % Bemerkung 12.2 der Vorlesung @@ -542,7 +513,7 @@ Haben wir Häufungspunkt definiert?} \end{enumerate} \end{korollar} -\begin{beweis} +\begin{beweis}\leavevmode \begin{enumerate}[label=\alph*)] \item Seien $y_1, y_2 \in Y$. diff --git a/documents/GeoTopo/figures/todo/topologischer-raum-x.png b/documents/GeoTopo/figures/todo/topologischer-raum-x.png new file mode 100644 index 0000000..70f9e15 Binary files /dev/null and b/documents/GeoTopo/figures/todo/topologischer-raum-x.png differ diff --git a/documents/GeoTopo/figures/topology-4.tex b/documents/GeoTopo/figures/topology-4.tex new file mode 100644 index 0000000..4b6af5a --- /dev/null +++ b/documents/GeoTopo/figures/topology-4.tex @@ -0,0 +1,60 @@ +\begin{tikzpicture}[thick] + \draw[pattern=north east lines] (-3,3) -- (3,3) -- (3,-3) -- (-3,-3) -- cycle; + \begin{scope} + \draw[red, pattern color=red, pattern=north west lines] (-3,-1) -- (-1,-1) -- (-1,-3) -- (-3,-3) -- cycle; + \draw[->,blue] (-3,-1.5) -- (-2,-1.5); + \draw[blue] (-2,-1.5) -- (-1.5,-1.5); + \draw[->,blue] (-1.5,-1.5) -- (-1.5,-2.3); + \draw[blue] (-1.5, -2.3) -- (-1.5, -3); + \draw[green] (-3,-2) -- (-2,-2) -- (-2, -3); + \end{scope} + \begin{scope}[rotate=90] + \draw[red, pattern color=red, pattern=north west lines] (-3,-1) -- (-1,-1) -- (-1,-3) -- (-3,-3) -- cycle; + \draw[->,blue] (-3,-1.5) -- (-2,-1.5); + \draw[blue] (-2,-1.5) -- (-1.5,-1.5); + \draw[->,blue] (-1.5,-1.5) -- (-1.5,-2.3); + \draw[blue] (-1.5, -2.3) -- (-1.5, -3); + \draw[green] (-3,-2) -- (-2,-2) -- (-2, -3); + \end{scope} + \begin{scope}[rotate=180] + \draw[red, pattern color=red, pattern=north west lines] (-3,-1) -- (-1,-1) -- (-1,-3) -- (-3,-3) -- cycle; + \draw[->,blue] (-3,-1.5) -- (-2,-1.5); + \draw[blue] (-2,-1.5) -- (-1.5,-1.5); + \draw[->,blue] (-1.5,-1.5) -- (-1.5,-2.3); + \draw[blue] (-1.5, -2.3) -- (-1.5, -3); + \draw[green] (-3,-2) -- (-2,-2) -- (-2, -3); + \end{scope} + \begin{scope}[rotate=270] + \draw[red, pattern color=red, pattern=north west lines] (-3,-1) -- (-1,-1) -- (-1,-3) -- (-3,-3) -- cycle; + \draw[->,blue] (-3,-1.5) -- (-2,-1.5); + \draw[blue] (-2,-1.5) -- (-1.5,-1.5); + \draw[->,blue] (-1.5,-1.5) -- (-1.5,-2.3); + \draw[blue] (-1.5, -2.3) -- (-1.5, -3); + \draw[green] (-3,-2) -- (-2,-2) -- (-2, -3); + \end{scope} + + \node[red] at (3.3,-1) {$V$}; + \node[green] at (3.3,2) {$U$}; + \node[blue] at (1.5, 3.2) {$a$}; + \node[purple] at (-0.4, 1.8) {$b$}; + \draw[purple] (-1.5,1.5) -- (1.5,1.5); + + \begin{scope}[xshift=6cm, yshift=-2cm] + \node[red] at (-0.5,0.5) {$V$}; + \draw[red,pattern color=red, pattern=north west lines] (-1,1) -- (1,1) -- (1,-1) -- (-1,-1) -- cycle; + \draw[red] (-1,0) -- (1,0); + \draw[red] (0,-1) -- (0,1); + \end{scope} + + \node[blue] at (4.5, 1.7) {$a$}; + \node[purple] at (7.5, 1.7) {$b$}; + \begin{scope}[xshift=5cm, yshift=1cm] + \draw[black,fill=white] (0,0) circle(1.2cm); + \draw[black, fill=white] (2,0) circle(1.2cm); + \path[fill=white] (0,0) circle(1.19cm); + \draw[black] (0,0) circle(0.5cm); + \draw[black] (2,0) circle(0.5cm); + \draw[blue,->] (0,0)+(30:0.7cm) arc (30:390:0.7cm); + \draw[purple,<-] (2,0)+(150:0.7cm) arc (150:510:0.7cm); + \end{scope} +\end{tikzpicture} diff --git a/tikz/topology-4/topology-4.png b/tikz/topology-4/topology-4.png index 83d8320..14683a4 100644 Binary files a/tikz/topology-4/topology-4.png and b/tikz/topology-4/topology-4.png differ diff --git a/tikz/topology-4/topology-4.tex b/tikz/topology-4/topology-4.tex index 4320cfc..5b55204 100644 --- a/tikz/topology-4/topology-4.tex +++ b/tikz/topology-4/topology-4.tex @@ -5,14 +5,62 @@ \begin{document} \begin{tikzpicture}[thick] \draw[pattern=north east lines] (-3,3) -- (3,3) -- (3,-3) -- (-3,-3) -- cycle; - \draw[red, pattern color=red, pattern=north west lines] (-3,-1) -- (-1,-1) -- (-1,-3) -- (-3,-3) -- cycle; - \draw[red, pattern color=red, pattern=north west lines] (-3,3) -- (-1,3) -- (-1,1) -- (-3,1) -- cycle; - \draw[red, pattern color=red, pattern=north west lines] (1,3) -- (3,3) -- (3,1) -- (1,1) -- cycle; - \draw[red, pattern color=red, pattern=north west lines] (1,-1) -- (3,-1) -- (3,-3) -- (1,-3) -- cycle; - \draw[->,blue] (-3,-1.5) -- (-2,-1.5); - \draw[blue] (-2,-1.5) -- (-1.5,-1.5); - \draw[->,blue] (-1.5,-1.5) -- (-1.5,-2.3); - \draw[blue] (-1.5, -2.3) -- (-1.5, -3); - \draw[green] (-3,-2) -- (-2,-2) -- (-2, -3); + \begin{scope} + \draw[red, pattern color=red, pattern=north west lines] (-3,-1) -- (-1,-1) -- (-1,-3) -- (-3,-3) -- cycle; + \draw[->,blue] (-3,-1.5) -- (-2,-1.5); + \draw[blue] (-2,-1.5) -- (-1.5,-1.5); + \draw[->,blue] (-1.5,-1.5) -- (-1.5,-2.3); + \draw[blue] (-1.5, -2.3) -- (-1.5, -3); + \draw[green] (-3,-2) -- (-2,-2) -- (-2, -3); + \end{scope} + \begin{scope}[rotate=90] + \draw[red, pattern color=red, pattern=north west lines] (-3,-1) -- (-1,-1) -- (-1,-3) -- (-3,-3) -- cycle; + \draw[->,blue] (-3,-1.5) -- (-2,-1.5); + \draw[blue] (-2,-1.5) -- (-1.5,-1.5); + \draw[->,blue] (-1.5,-1.5) -- (-1.5,-2.3); + \draw[blue] (-1.5, -2.3) -- (-1.5, -3); + \draw[green] (-3,-2) -- (-2,-2) -- (-2, -3); + \end{scope} + \begin{scope}[rotate=180] + \draw[red, pattern color=red, pattern=north west lines] (-3,-1) -- (-1,-1) -- (-1,-3) -- (-3,-3) -- cycle; + \draw[->,blue] (-3,-1.5) -- (-2,-1.5); + \draw[blue] (-2,-1.5) -- (-1.5,-1.5); + \draw[->,blue] (-1.5,-1.5) -- (-1.5,-2.3); + \draw[blue] (-1.5, -2.3) -- (-1.5, -3); + \draw[green] (-3,-2) -- (-2,-2) -- (-2, -3); + \end{scope} + \begin{scope}[rotate=270] + \draw[red, pattern color=red, pattern=north west lines] (-3,-1) -- (-1,-1) -- (-1,-3) -- (-3,-3) -- cycle; + \draw[->,blue] (-3,-1.5) -- (-2,-1.5); + \draw[blue] (-2,-1.5) -- (-1.5,-1.5); + \draw[->,blue] (-1.5,-1.5) -- (-1.5,-2.3); + \draw[blue] (-1.5, -2.3) -- (-1.5, -3); + \draw[green] (-3,-2) -- (-2,-2) -- (-2, -3); + \end{scope} + + \node[red] at (3.3,-1) {$V$}; + \node[green] at (3.3,2) {$U$}; + \node[blue] at (1.5, 3.2) {$a$}; + \node[purple] at (-0.4, 1.8) {$b$}; + \draw[purple] (-1.5,1.5) -- (1.5,1.5); + + \begin{scope}[xshift=6cm, yshift=-2cm] + \node[red] at (-0.5,0.5) {$V$}; + \draw[red,pattern color=red, pattern=north west lines] (-1,1) -- (1,1) -- (1,-1) -- (-1,-1) -- cycle; + \draw[red] (-1,0) -- (1,0); + \draw[red] (0,-1) -- (0,1); + \end{scope} + + \node[blue] at (4.5, 1.7) {$a$}; + \node[purple] at (7.5, 1.7) {$b$}; + \begin{scope}[xshift=5cm, yshift=1cm] + \draw[black,fill=white] (0,0) circle(1.2cm); + \draw[black, fill=white] (2,0) circle(1.2cm); + \path[fill=white] (0,0) circle(1.19cm); + \draw[black] (0,0) circle(0.5cm); + \draw[black] (2,0) circle(0.5cm); + \draw[blue,->] (0,0)+(30:0.7cm) arc (30:390:0.7cm); + \draw[purple,<-] (2,0)+(150:0.7cm) arc (150:510:0.7cm); + \end{scope} \end{tikzpicture} \end{document}