From 1baaf85a1adf3a7c9f339ab622ed51eb68007020 Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Tue, 26 Nov 2013 23:00:58 +0100 Subject: [PATCH] Viele Kleinigkeiten verbessert --- documents/GeoTopo/Bildquellen.tex | 21 ++ documents/GeoTopo/GeoTopo.pdf | Bin 3749055 -> 3758760 bytes documents/GeoTopo/GeoTopo.tex | 2 + documents/GeoTopo/Kapitel1-UB.tex | 8 + documents/GeoTopo/Kapitel1.tex | 154 +++++---- documents/GeoTopo/Kapitel2.tex | 33 +- documents/GeoTopo/Loesungen.tex | 6 +- documents/GeoTopo/Symbolverzeichnis.tex | 311 +++--------------- .../figures/topology-continuous-mapping.tex | 21 +- .../figures/topology-kartenwechsel.tex | 3 + documents/GeoTopo/shortcuts.sty | 3 + 11 files changed, 199 insertions(+), 363 deletions(-) create mode 100644 documents/GeoTopo/Bildquellen.tex diff --git a/documents/GeoTopo/Bildquellen.tex b/documents/GeoTopo/Bildquellen.tex new file mode 100644 index 0000000..8336d9e --- /dev/null +++ b/documents/GeoTopo/Bildquellen.tex @@ -0,0 +1,21 @@ +\chapter*{Bildquellen} +\addcontentsline{toc}{chapter}{Bildquellen} + +Alle Bilder, die hier nicht aufgeführt sind, wurden selbst erstellt. + +Teilweise wurden die im folgenden aufgelisteten Bilder noch leicht +modifiziert. + +\begin{itemize} + \item[Abb. \ref{fig:stereographic-projection}] Stereographische Projektion: \href{http://texample.net/tikz/examples/map-projections/}{texample.net/tikz/examples/map-projections} + \item[Abb. \ref{fig:Knoten}] Knoten von Jim.belk aus der \enquote{\href{https://commons.wikimedia.org/wiki/Category:Blue_knots}{Blue knots}}-Serie: + \begin{itemize} + \item Trivialer Knoten: \href{https://commons.wikimedia.org/wiki/File:Blue_Unknot.png}{commons.wikimedia.org/wiki/File:Blue\_Unknot.png} + \item Kleeblattknoten: \href{https://commons.wikimedia.org/wiki/File:Blue_Trefoil_Knot.png}{commons.wikimedia.org/wiki/File:Blue\_Trefoil\_Knot.png} + \item Achterknoten: \href{https://commons.wikimedia.org/wiki/File:Blue_Figure-Eight_Knot.png}{commons.wikimedia.org/wiki/File:Blue\_Figure-Eight\_Knot.png} + \item $6_2$-Knoten: \href{https://commons.wikimedia.org/wiki/File:Blue_6_2_Knot.png}{commons.wikimedia.org/wiki/File:Blue\_6\_2\_Knot.png} + \end{itemize} + \item[Abb. \ref{fig:reidemeister-zuege}] Reidemeister-Züge: YAMASHITA Makoto (\href{https://commons.wikimedia.org/wiki/File:Reidemeister_move_1.png}{1}, \href{https://commons.wikimedia.org/wiki/File:Reidemeister_move_1.png}{2}, \href{https://commons.wikimedia.org/wiki/File:Reidemeister_move_1.png}{3}) + \item[Abb. \ref{fig:treefoil-knot-three-colors}] Kleeblattknoten, 3-Färbung: Jim.belk, \href{https://commons.wikimedia.org/wiki/File:Tricoloring.png}{commons.wikimedia.org/wiki/File:Tricoloring.png} + \item[Abb. \ref{fig:double-torus}] Doppeltorus: Oleg Alexandrov, 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Eine Teilmenge $U \subseteq X$ heißt \textbf{Umgebung} von $x$, wenn es ein $U_0 \in \fT$ gibt mit $x \in U_0$ und $U_0 \subseteq U$. \end{definition} \begin{definition} - Sei $(X, \fT)$ ein topologischer Raum, $M \subseteq X$ eine Teilmenge. + Sei $(X, \fT)$ ein topologischer Raum und $M \subseteq X$ eine Teilmenge. \begin{enumerate}[label=\alph*)] \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} @@ -71,11 +71,13 @@ Auch gibt es Mengen, die sowohl abgeschlossen als auch offen sind. \begin{beispiel} \begin{enumerate}[label=\arabic*)] - \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 $X = \mdr, \fT = \fT_Z$\\ - $M = (a,b) \Rightarrow \overline{M} = \mdr$ + \item Sei $X = \mdr$ mit euklidischer Topologie und + $M = \mdq$. Dann gilt: $\overline{M} = \mdr$ und + $M^\circ = \emptyset$ + \item Sei $X = \mdr$ und $M=(a,b)$. Dann gilt: + $\overline{M} = [a,b]$ + \item Sei $X = \mdr, \fT = \fT_Z$ und $M = (a,b)$. Dann gilt: + $\overline{M} = \mdr$ \end{enumerate} \end{beispiel} @@ -103,7 +105,7 @@ Auch gibt es Mengen, die sowohl abgeschlossen als auch offen sind. \end{bemerkung} \begin{definition} \xindex{Spurtopologie} \xindex{Teilraum} - Sei $(X, \fT)$ ein topologischer Raum, $Y \subseteq X$.\\ + Sei $(X, \fT)$ ein topologischer Raum und $Y \subseteq X$.\\ $\fT_Y := \Set{U \cap Y | U \in \fT}$ ist eine Topologie auf $Y$. $\fT_Y$ heißt \textbf{Spurtopologie} und $(Y, \fT_Y)$ heißt ein @@ -168,10 +170,9 @@ Auch gibt es Mengen, die sowohl abgeschlossen als auch offen sind. \end{beispiel} \begin{beispiel} - \begin{align*} - X = \mdr^2, (x_1, y_1) \sim (x_2, y_2) \Leftrightarrow &x_1 - x_2 \in \mdz\\ - &y_1 - y_2 \in \mdz - \end{align*} + Sei $X = \mdr^2$ und $(x_1, y_1) \sim (x_2, y_2) \gdw x_1 - x_2 \in \mdz$ + und $y_1 - y_2 \in \mdz$. + $X /_\sim$ ist ein Torus. \end{beispiel} @@ -191,9 +192,9 @@ Auch gibt es Mengen, die sowohl abgeschlossen als auch offen sind. heißt \textbf{Metrik}, wenn gilt: \begin{enumerate}[label=(\roman*)] - \item Definitheit: \tabto{4cm} $d(x,y) = 0 \gdw x = y$ - \item Symmetrie: \tabto{4cm} $d(x,y) = d(y,x)$ - \item Dreiecksungleichung: \tabto{4cm} $d(x,z) \leq d(x,y) + d(x+z)$ + \item Definitheit: \tabto{4cm} $d(x,y) = 0 \gdw x = y \;\;\; \forall x, y \in X$ + \item Symmetrie: \tabto{4cm} $d(x,y) = d(y,x) \;\;\; \forall x, y \in X$ + \item Dreiecksungleichung: \tabto{4cm} $d(x,z) \leq d(x,y) + d(x+z) \;\;\; \forall x, y, z \in X$ \end{enumerate} Das Paar $(X, d)$ heißt ein \textbf{metrischer Raum}. @@ -257,7 +258,7 @@ Auch gibt es Mengen, die sowohl abgeschlossen als auch offen sind. \begin{bemerkung}[Trennungseigenschaft]\label{Trennungseigenschaft} Metrische Räume sind hausdorffsch, da - \[d(x,y) > 0 \Rightarrow \exists \varepsilon: \fB_\varepsilon(x) \cap \fB_\varepsilon(y) = \emptyset\] + \[d(x,y) > 0 \Rightarrow \exists \varepsilon > 0: \fB_\varepsilon(x) \cap \fB_\varepsilon(y) = \emptyset\] Ein Beispiel für einen topologischen Raum, der nicht hausdorfsch ist, ist $(\mdr, \fT_Z)$. \end{bemerkung} @@ -291,12 +292,12 @@ Auch gibt es Mengen, die sowohl abgeschlossen als auch offen sind. \end{korollar} \begin{beweis} - \underline{Annahme}: $x$ und $y$ mit $x \neq y$ sind Grenzwerte der Folge $(x_n)$. + Sei $(x_n)$ eine konvergierende Folge und $x$ und $y$ Grenzwerte der Folge. Nach Voraussetzung gibt es Umgebungen $U_x$ von $x$ und $U_y$ - von $y$ mit $U_x \cap U_y = \emptyset$. Nach Annahme gibt es + von $y$ mit $U_x \cap U_y = \emptyset$. Es existiert ein $n_0$ mit $x_n \in U_x \cap U_y$ für alle $n \geq n_0$ - $\Rightarrow$ Widerspruch $\qed$ + $\Rightarrow x = y \qed$ \end{beweis} \section{Stetigkeit}\index{Stetigkeit|(} @@ -306,22 +307,28 @@ Auch gibt es Mengen, die sowohl abgeschlossen als auch offen sind. \begin{enumerate}[label=\alph*),ref=\theplaindefinition.\alph*] \item $f$ heißt \textbf{stetig}, wenn für jedes offene $U \subseteq Y$ auch $f^{-1} (U) \subseteq X$ offen ist. \label{def:stetigkeit} - \item $f$ heißt \textbf{Homöomorphismus}, wenn es eine + \item $f$ heißt \textbf{Homöomorphismus}, wenn $f$ stetig ist + und es eine stetige Abbildung $g: Y \rightarrow X$ gibt, sodass $g \circ f = \text{id}_X$ und $f \circ g = \text{id}_Y$. \end{enumerate} \end{definition} +\begingroup +\renewcommand{\thmfoot}{\footnotemark} \begin{korollar} - % Im Grunde wird die Äquivalenz von Stetigkeit im Sinne der - % Analysis und Topologie auf metrischen Räumen gezeigt. - Seien $X, Y$ metrische Räume und $f: X \rightarrow Y$ eine Abbildung. + \footnotetext[\thefootnote]{Im Grunde wird die Äquivalenz + von Stetigkeit im Sinne der Analysis und Topologie auf metrischen + Räumen gezeigt.} + Seien $X, Y$ metrische Räume und $f\colon X \rightarrow Y$ eine + Abbildung. - Dann gilt: $f$ ist stetig $\gdw$ zu jedem $x \in X$ und jedem - $\varepsilon > 0$ gibt es $\delta(x, \varepsilon) > 0$, sodass für - alle $y \in X$ mit $d(x,y) < \delta $ gilt - $d_Y(f(x), f(y)) < \varepsilon$. + Dann gilt: $f$ ist stetig $\Leftrightarrow$ zu jedem $x \in X$ und + jedem $\varepsilon > 0$ gibt es $\delta(x, \varepsilon) > 0$, sodass + für alle $y \in X$ mit $d(x,y) < \delta $ gilt $d_Y(f(x), f(y)) < + \varepsilon$. \end{korollar} +\endgroup \begin{beweis} \enquote{$\Rightarrow$}: Sei $x \in X, \varepsilon > 0$ gegeben @@ -352,7 +359,7 @@ Auch gibt es Mengen, die sowohl abgeschlossen als auch offen sind. \begin{enumerate}[label=\arabic*)] \item Für jeden topologischen Raum $X$ gilt: $\text{Id}_X : X \rightarrow X$ ist Homöomorphismus. - \item Ist $Y$ trivialer topologischer Raum, d.h. $\fT = \fT_\text{triv}$, + \item Ist $Y$ trivialer topologischer Raum, d.~h. $\fT = \fT_\text{triv}$, so ist jede Abbildung $f:X \rightarrow Y$ stetig. \item Ist $X$ diskreter topologischer Raum, so ist $f:X \rightarrow Y$ stetig für jeden topologischen Raum $Y$ und jede Abbildung $f$. @@ -366,7 +373,7 @@ Auch gibt es Mengen, die sowohl abgeschlossen als auch offen sind. \label{fig:nicht-stetige-umkehrabbildung} \end{figure} Die Umkehrabbildung $g$ ist nicht stetig, da $g^{-1}(U)$ - nicht offen ist (vgl. Abb. \ref{fig:nicht-stetige-umkehrabbildung}) + nicht offen ist (vgl. Abb. \ref{fig:nicht-stetige-umkehrabbildung}). \end{enumerate} \end{beispiel} @@ -396,19 +403,19 @@ Auch gibt es Mengen, die sowohl abgeschlossen als auch offen sind. \begin{enumerate}[label=\alph*)] \item Für jeden topologischen Raum ist $\text{Homöo}(X) := \Set{f: X \rightarrow X | f \text{ ist Homöomorphismus}}$ - eine Gruppe. + eine Gruppe.\xindex{Homöomorphismengruppe} \item Jede Isometrie $f:X \rightarrow Y$ zwischen metrischen Räumen ist ein Homöomorphismus. \item $\text{Isom}(X) := \Set{f:X \rightarrow X | f \text{ ist Isometrie}}$ ist - Untergruppe von $\text{Homöo}(X)$ für jeden metrischen - Raum $X$. + eine Untergruppe von $\text{Homöo}(X)$ für jeden + metrischen Raum $X$. \end{enumerate} \end{bemerkung} \begin{korollar} Seien $X, Y$ topologische Räume. $\pi_X: X \times Y \rightarrow X$ und $\pi_Y: X \times Y \rightarrow Y$ die Projektionen - \[(x,y) \mapsto x \;\;\;(x,y) \mapsto y\] + \[\pi_X: (x,y) \mapsto x \text{ und } \pi_Y: (x,y) \mapsto y\] Wird $X \times Y$ mit der Produkttopologie versehen, so sind $\pi_X$ und $\pi_Y$ stetig. \end{korollar} @@ -437,14 +444,16 @@ sodass $\pi$ stetig wird. \begin{beispiel}[Stereographische Projektion] \xindex{Projektion!stereographische} $\mdr^n$ und $S^n \setminus \Set{N}$ sind homöomorph für - beliebiges $N \in S^n$ + beliebiges $N \in S^n$. Es gilt: \begin{align*} S^n &= \Set{x \in \mdr^{n+1} | \|x\| = 1}\\ &= \Set{x \in \mdr^{n+1} | \sum_{i=1}^{n+1} x_i^2} \end{align*} - \OE{} sei $N = \begin{pmatrix}0\\ \vdots\\ 1\end{pmatrix}$. + \Obda sei $N = \begin{pmatrix}0\\ \vdots\\ 1\end{pmatrix}$. Die + Gerade durch $N$ und $P$ schneidet die Ebene $H$ in genau einem + Punkt $\hat{P}$. $P$ wird auf $\hat{P}$ abgebildet. \begin{align*} f: &S^n \setminus \Set{N} \rightarrow \mdr^n\\ @@ -457,7 +466,8 @@ sodass $\pi$ stetig wird. \begin{figure}[htp] \centering \input{figures/stereographic-projection} - \caption{Visualisierung der sphärischen Projektion\\Bildquelle: \href{http://texample.net/tikz/examples/map-projections/}{texample.net/tikz/examples/map-projections}} + \caption{Visualisierung der stereographischen Projektion} + \label{fig:stereographic-projection} \end{figure} Sei $P = \begin{pmatrix}x_1\\ \vdots \\ x_{n+1}\end{pmatrix}$, so @@ -474,14 +484,15 @@ sodass $\pi$ stetig wird. \index{Stetigkeit|)} \section{Zusammenhang}\index{Zusammenhang|(} \begin{definition}\xindex{zusammenhängend} - Ein Raum $X$ heißt \textbf{zusammenhängend}, wenn es keine offenen - nichtleeren Teilmengen $U_1, U_2$ von $X$ gibt mit $U_1 \cap U_2 = \emptyset$ - und $U_1 \cup U_2 = X$. + Ein Raum $X$ heißt \textbf{zusammenhängend}, wenn es keine offenen, + nichtleeren Teilmengen $U_1, U_2$ von $X$ gibt mit + $U_1 \cap U_2 = \emptyset$ und $U_1 \cup U_2 = X$. \end{definition} \begin{bemerkung} - $X$ ist zusammenhängend $\gdw$ Es gibt keine nichtleeren abgeschlossenen - Teilmengen $A_1, A_2$ mit $A_1 \cap A_2 = \emptyset$ und $A_1 \cup A_2 = X$. + $X$ ist zusammenhängend $\gdw$ Es gibt keine abgeschlossenen, + nichtleeren Teilmengen $A_1, A_2$ mit $A_1 \cap A_2 = \emptyset$ + und $A_1 \cup A_2 = X$. \end{bemerkung} \begin{bemerkung} @@ -494,12 +505,13 @@ sodass $\pi$ stetig wird. %\end{beispiel} \begin{beispiel}[Zusammenhang von Räumen] - \begin{enumerate} + \begin{enumerate}[label=\arabic*)] \item $\mdr^n$ ist mit der euklidischen Topologie zusammenhängend, denn: - Angenommen, $\mdr^n = U_1 \cup U_2$ mit $U_i$ offen, $U_i \neq \emptyset$ - und $U_1 \cap U_2 = \emptyset$ existiert. + \underline{Annahme}: $\mdr^n = U_1 \cup U_2$ mit $U_i$ + offen, $U_i \neq \emptyset$ und $U_1 \cap U_2 = \emptyset$ + existieren. Sei $x \in U_1, y \in U_2$ und $[x,y]$ die Strecke zwischen $x$ und $y$. Dann ist $U_1 \cap [x,y]$ die Vereinigung von offenen @@ -510,33 +522,33 @@ sodass $\pi$ stetig wird. $\mdr \setminus \Set{0} = \mdr_{< 0} \cup \mdr_{> 0}$ \item $\mdr^2 \setminus \Set{0}$ ist zusammenhängend. \item $\mdq \subsetneq \mdr$ ist nicht zusammenhängend, da - \[(\mdq \cap \mdr_{< \sqrt{2}}) \cup (\mdq \cap \mdr_{> \sqrt{2}}) = \mdq\] - \item $\Set{x}$ ist zusammenhängedn für jedes $x \in X$, wobei $X$ ein - topologischer Raum ist. + $(\mdq \cap \mdr_{< \sqrt{2}}) \cup (\mdq \cap \mdr_{> \sqrt{2}}) = \mdq$ + \item $\Set{x}$ ist zusammenhängend für jedes $x \in X$, + wobei $X$ ein topologischer Raum ist. \item $\mdr$ mit Zariski-Topologie ist zusammenhängend\xindex{Topologie!Zariski} \end{enumerate} \end{beispiel} \begin{korollar}\label{zusammenhangAbschluss} - Sei $X$ ein topologischer Raum, $A \subseteq X$ zusammenhängend. + Sei $X$ ein topologischer Raum und $A \subseteq X$ zusammenhängend. Dann ist auch $\overline{A}$ zusammenhängend. \end{korollar} \begin{beweis} - Angenommen $\overline{A} = A_1 \cup A_2, A_i$ abgeschlossen, $\neq \emptyset$, - $A_1 \cap A_2 = \emptyset$ + \underline{Annahme}: $\overline{A} = A_1 \cup A_2,\; A_i$ abgeschlossen, $\neq \emptyset$, + $\;A_1 \cap A_2 = \emptyset$ \begin{align*} &\Rightarrow A = \underbrace{\underbrace{(A \cap A_1)}_\text{abgeschlossen} \cup \underbrace{(A \cap A_2)}_\text{abgeschlossen}}_\text{disjunkt}\\ \end{align*} - Wäre $A \cap A_1 = \emptyset$ - \begin{align*} - &\Rightarrow A \subseteq A_2\\ - &\Rightarrow \overline{A} \subseteq A_2\\ - &\Rightarrow A_1 = \emptyset - &\Rightarrow \text{Widerspruch} - \end{align*} - $\qed$ + Wäre $A \cap A_1 = \emptyset$\\ + $\Rightarrow A \subseteq A_2$\\ + $\Rightarrow \overline{A} \subseteq A_2$\\ + $\Rightarrow A_1 = \emptyset$\\ + $\Rightarrow$ Widerspruch zu $A_1 \neq \emptyset$\\ + $\Rightarrow A \cap A_1 \neq \emptyset$ und analog + $A \cap A_2 \neq \emptyset$\\ + $\Rightarrow$ Widerspruch zu $A$ ist zusammenhängend $ \qed$ \end{beweis} \begin{korollar}\label{zusammenhangVereinigung} @@ -548,7 +560,7 @@ sodass $\pi$ stetig wird. \begin{beweis} Sei $A \cup B = U_1 \cup U_2, U_i \neq \emptyset$ offen, disjunkt \begin{align*} - &\xRightarrow{\text{\OE{}}} A = (A \cap U_1) \cup (A \cap U_2) \text{ offen, disjunkt}\\ + &\xRightarrow{\text{\obda}} A = (A \cap U_1) \cup (A \cap U_2) \text{ offen, disjunkt}\\ &\xRightarrow{A \text{ zhgd.}} A \cap U_1 = \emptyset\\ &\xRightarrow{A \cap B \neq \emptyset} U_1 \subseteq B\\ &B = \underbrace{(B \cap U_1)}_{= U_1} \cup \underbrace{(B \cap U_2)}_{= \emptyset} \text{ ist unerlaubte Zerlegung} @@ -580,7 +592,7 @@ sodass $\pi$ stetig wird. \item Sei $Z(x) = A_1 \cup A_2$ mit $A_i \neq \emptyset$ abgeschlossen, disjunkt. - \OE{} sei $x \in A_1$ und $y \in A_2$. $y$ liegt in einer zusammehängenden + \Obda sei $x \in A_1$ und $y \in A_2$. $y$ liegt in einer zusammehängenden Teilmenge $A$, die auch $x$ enthält. $\Rightarrow A = \underbrace{(A \cap A_1)}_{\ni x} \cup \underbrace{(A \cap A_2)}_{\ni y}$ ist unerlaubte Zerlegung. @@ -612,6 +624,13 @@ sodass $\pi$ stetig wird. \end{beweis}\index{Zusammenhang|)} \section{Kompaktheit} +\begin{definition}\xindex{Ueberdeckung@""Uberdeckung} + Sei $X$ eine Menge und $T \subseteq \powerset{X}$. + + $T$ heißt eine \textbf{Überdeckung} von $X$, wenn gilt: + \[\forall x \in X: \exists M \in T: x \in M\] +\end{definition} + \begin{definition}\xindex{kompakt} Ein topologischer Raum $X$ heißt \textbf{kompakt}, wenn jede offene Überdeckung von $X$ eine endliche Teilüberdeckung besitzt. @@ -620,13 +639,6 @@ sodass $\pi$ stetig wird. \end{definition} -\begin{definition}\xindex{Ueberdeckung@""Uberdeckung} - Sei $X$ eine Menge und $T \subseteq \powerset{X}$. - - $T$ heißt eine \textbf{Überdeckung} von $X$, wenn gilt: - \[\forall x \in X: \exists M \in T: x \in M\] -\end{definition} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Mitschrieb vom 05.11.2013 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -663,7 +675,7 @@ $\qed$ \begin{beispiel} \begin{enumerate}[label=\arabic*)] - \item $\mdr$ ist nicht kompakt. + \item $\mdr$ ist nicht kompakt.\todo{mit der std-topo? Warum? Def?} \item $(0,1)$ ist nicht kompakt.\\ $U_n = (\nicefrac{1}{n}, 1-\nicefrac{1}{n}) \Rightarrow \bigcup_{n \in \mdn} U_n = (0,1)$ \item $\mdr$ mit der Zariski-Topologie ist kompakt und jede @@ -937,8 +949,8 @@ $\qed$ \label{fig:knot-6-2} } - \label{Knoten} \caption{Beispiele für verschiedene Knoten} + \label{fig:Knoten} \end{figure} \end{beispiel} @@ -983,8 +995,8 @@ $\qed$ \label{fig:reidemeister-3} } + \caption{Reidemeister-Züge} \label{fig:reidemeister-zuege} - \caption{Reidemeister-Züge\newline Urheber:YAMASHITA Makoto} \end{figure} \begin{beweis} @@ -1002,8 +1014,8 @@ $\qed$ \centering \includegraphics[height=0.3\linewidth, keepaspectratio]{figures/tricoloring.png} - \label{fig:reidemeister-zuege} \caption{Ein 3-gefärber Kleeblattknoten} + \label{fig:treefoil-knot-three-colors} \end{figure} \index{Knoten|)} diff --git a/documents/GeoTopo/Kapitel2.tex b/documents/GeoTopo/Kapitel2.tex index 4669a82..d83f053 100644 --- a/documents/GeoTopo/Kapitel2.tex +++ b/documents/GeoTopo/Kapitel2.tex @@ -5,12 +5,12 @@ \section{Topologische Mannigfaltigkeiten} \begin{definition} Sei $X$ ein topologischer Raum und $n \in \mdn$. - \begin{enumerate}[label=(\alph*)] + \begin{enumerate}[label=\alph*)] \item Eine $n$-dimensionale \textbf{Karte}\xindex{Karte} auf $X$ ist ein Paar $(U, \varphi)$, wobei $U \subseteq X$ offen und $\varphi: U \rightarrow V$ Homöomorphismus von $U$ auf eine offene Teilmenge $V \subseteq \mdr^n$. - \item Ein $n$-dimensionaler \textbf{Atlas}\xindex{Atlas} auf $X$ ist eine + \item Ein $n$-dimensionaler \textbf{Atlas}\xindex{Atlas} $\atlas$ auf $X$ ist eine Familie $(U_i, \varphi_i)_{i \in I}$ von Karten auf $X$, sodass $\bigcup_{i \in I} U_i = X$. \item $X$ heißt (topologische) $n$-dimensionale \textbf{Mannigfaltigkeit}\xindex{Mannigfaltigkeit}, @@ -21,7 +21,7 @@ \begin{bemerkung} \begin{enumerate}[label=(\alph*)] - \item Es gibt surjektive, stetige Abbildungen $[0,1] \rightarrow [0,1] \times [0,1]$ + \item Es gibt surjektive, stetige Abbildungen $[0,1] \rightarrow [0,1] \times [0,1]$\todo{Wann ist das wichtig? Ist die Hilbert-Kurve ein Beispiel?} \item Für $n \neq m$ sind $\mdr^n$ und $\mdr^m$ nicht homöomorph. Zum Beweis benutzt man den \enquote{Satz von der Gebietstreue} (Brouwer): @@ -43,7 +43,7 @@ \item $\mdc^n$ ist eine $2n$-dimensionale Mannigfaltigkeit mit einem Atlas aus einer Karte: \[(z_1, \dots, z_n) \mapsto (\operatorname{Re} z_1, \operatorname{Im}z_1, \dots, \operatorname{Re}z_n, \operatorname{Im}z_n)\] - \item $\mdp^n(\mdr) = (\mdr^{n+1} \setminus \Set{0})/_\sim = S^n /_\sim$ und $\mdp^n(\mdc)$ sind Mannigfaltigkeiten + \item $\mdp^n(\mdr) = (\mdr^{n+1} \setminus \Set{0})/_\sim = S^n /_\sim$ und $\mdp^n(\mdc)$ sind Mannigfaltigkeiten \todo{besser aufschreiben} der Dimension $n$ bzw. $2n$. $\mdp^n(\mdr) = \bigcup_{i=0}^n U_i,$ @@ -79,7 +79,7 @@ U_i = \Set{(x_0: \dots : x_n) \in \mdp^n(\mdr) | x_i \neq 0} &\rightarrow \mdr^n Es gibt keine Umgebung von $0$ in $[0,1]$, die homöomorph zu einem offenem Intervall ist. \item $V_1 = \Set{(x,y) \in \mdr^2 | x \cdot y = 0}$ ist - keine Mannigfaltigkeit. + keine Mannigfaltigkeit. \todo{Warum genau?} \item $V_2 = \Set{(x,y) \in \mdr^2 | x^3 = y^2}$ ist eine Mannigfaltigkeit. \item $X = (\mdr \setminus \Set{0}) \cup (O_1, O_2)$ @@ -155,14 +155,15 @@ U_i = \Set{(x_0: \dots : x_n) \in \mdp^n(\mdr) | x_i \neq 0} &\rightarrow \mdr^n und $X = V(F) := \Set{x \in \mdr^n | F(x) = 0}$ das \enquote{vanishing set}. Dann gilt: - \begin{enumerate}[label=\alph*)] + \begin{enumerate}[label=\alph*),ref=\theplaindefinition.\alph*] \item $X$ ist abgeschlossen in $\mdr^n$ - \item Ist $\text{grad}(F)(X) \neq 0 \forall{x \in X}$, so ist + \item Ist $\text{grad}(F)(X) \neq 0 \;\;\;\forall{x \in X}$, so ist $X$ eine Mannigfaltigkeit der Dimension $n-1$. \label{Mannigfaltigkeitskriterium} \end{enumerate} \end{korollar} \begin{beweis} + von a und b: \begin{enumerate}[label=\alph*),ref=\theplaindefinition.\alph*] \item Sei $y \in \mdr^n \setminus V(F)$. Weil $F$ stetig ist, gibt es $\delta > 0$, sodass $F(\fB_\delta(y)) \subseteq \fB_\varepsilon(F(y))$ @@ -202,7 +203,7 @@ U_i = \Set{(x_0: \dots : x_n) \in \mdp^n(\mdr) | x_i \neq 0} &\rightarrow \mdr^n \caption{Rechts ist die Neilsche Parabel für verschiedene Parameter $a$.} \end{figure} Es gilt: $\text{grad}(F) = (-3x^2, 2y)$. Also: $\text{grad}(0,0) = (0,0)$. - Daher ist Korollar \label{Mannigfaltigkeitskriterium} + Daher ist Korollar \ref{Mannigfaltigkeitskriterium} nicht anwendbar, aber $V(F)$ ist trotzdem eine 1-dimensionale topologische Mannigfaltigkeit. \end{enumerate} @@ -276,7 +277,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. \begin{enumerate}[label=\alph*)] \item $X$ heißt \textbf{differenzierbare Mannigfaltigkeit der Klasse $C^k$}\xindex{Mannigfaltigkeit!differenzierbare}, - wenn jede Kartenwechselabbildung $\varphi_{ij},\;i,j \in I$ + wenn jede Kartenwechselabbildung $\varphi_{ij},\;i,j \in I$\todo{warum Doppelindex} $k$-mal stetig differenzierbar ist. \item $X$ heißt \textbf{differenzierbare Mannigfaltigkeit}\xindex{Mannigfaltigkeit!glatte}, wenn $X$ eine differenzierbare Mannigfaltigkeit der @@ -314,12 +315,12 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. \begin{enumerate}[label=\alph*),ref=\theplaindefinition.\alph*] \item Eine stetige Abbildung $f:X \rightarrow Y$ heißt\label{def:stetigeAbbildungDiffbar} \textbf{differenzierbar}\xindex{Abbildung!differenzierbare} - in $X$ (von Klasse $C^k$), + in $x$ (von Klasse $C^k$), wenn es Karten $(U, \varphi)$ von $X$ mit $x \in U$ und $(V, \psi)$ von $Y$ mit $f(U) \subseteq V$ gibt, sodass $\psi \circ f \circ \varphi^{-1}$ stetig differenzierbar von Klasse $C^k$ in $\varphi(x)$ ist. - \item $f$ heißt \textbf{differenzierbar}\todo{stimmt das so?} + \item $f$ heißt \textbf{differenzierbar} (von Klasse $C^k$), wenn $f$ in jedem $x \in X$ differenzierbar ist. \item $f$ heißt \textbf{Diffieomorphismus}\xindex{Diffieomorphismus}, @@ -338,11 +339,9 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. \begin{beweis} Seien $(U', \varphi')$ und $(V', \psi')$ Karten von $X$ bzw. $Y$ um $x$ bzw. $f(x)$ mit $f(U') \subseteq V'$. - - \begin{align*} - \Rightarrow &\psi' \circ f \circ (\varphi')^{-1}\\ - = &\psi' \circ ( \psi^{-1} \circ \psi) \circ f \circ (\varphi^{-1} \circ \varphi ) \circ (\varphi')^{-1} - \end{align*} + + $\Rightarrow \psi' \circ f \circ (\varphi')^{-1}$\\ + $= \psi' \circ ( \psi^{-1} \circ \psi) \circ f \circ (\varphi^{-1} \circ \varphi ) \circ (\varphi')^{-1}$ ist genau dann differenzierbar, wenn $\psi \circ f \circ \varphi^{-1}$ differenzierbar ist. @@ -686,7 +685,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. ist simplizial: \input{figures/topology-triangle-to-line.tex} - \item \todo[inline]{wozu dient das Beispiel?} + \item \todo[inline]{Wozu dient das Beispiel?} \input{figures/topology-2.tex} \end{enumerate} diff --git a/documents/GeoTopo/Loesungen.tex b/documents/GeoTopo/Loesungen.tex index 4b3a639..8e663ef 100644 --- a/documents/GeoTopo/Loesungen.tex +++ b/documents/GeoTopo/Loesungen.tex @@ -47,6 +47,8 @@ in dieser Topologie offen sein $\Rightarrow \Set{-1,1}$ ist nicht offen. $\qed$ + \textbf{Teilaufgabe c)} + \textbf{Beh.:} Es gibt unendlich viele Primzahlen. \textbf{Bew.:} durch Widerspruch @@ -61,7 +63,7 @@ \begin{solution}[\ref{ub2:aufg4}] \begin{enumerate}[label=(\alph*)] - \item \underline{Beh.:} Die offenen Mengen von $P$ sind + \item \textbf{Beh.:} Die offenen Mengen von $P$ sind Vereinigungen von Mengen der Form \[\prod_{j \in J} U_j \times \prod_{i \in \mdn, i \neq j} P_i\] wobei $J \subseteq \mdn$ endlich und $U_j \subseteq P_j$ @@ -74,7 +76,7 @@ Mengen von $P$ Vereinigungen von Mengen der obigen Form. $\qed$ \end{beweis} - \item \underline{Beh.:} Die Zusammenhangskomponenten von $P$ + \item \textbf{Beh.:} Die Zusammenhangskomponenten von $P$ sind alle einpunktig.\xindex{Total Unzusammenhängend} \begin{beweis} Es seinen $x,y \in P$ und $x$ sowie $y$ liegen in der diff --git a/documents/GeoTopo/Symbolverzeichnis.tex b/documents/GeoTopo/Symbolverzeichnis.tex index ef7a236..324a499 100644 --- a/documents/GeoTopo/Symbolverzeichnis.tex +++ b/documents/GeoTopo/Symbolverzeichnis.tex @@ -1,277 +1,68 @@ -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -% Begriffslexikon zur Beschreibung des Produkts % -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%\newglossaryentry{sortierschluessel} -%{ -% name=Sortierschlüssel, -% description={ein Schlüssel, anhand dessen diese Einträge sortiert werden} -%} -%\newacronym{abc}{Blub}{Bananarama} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -% Mengenoperationen % -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\newglossaryentry{Potenzmenge} -{ - name={\ensuremath{\mathcal{P}(M)}}, - description={Potenzmenge von $M$}, - sort=MengenoperationNPotenzmenge -} - -\newglossaryentry{Abschluss} -{ - name={\ensuremath{\overline{M}}}, - description={Abschluss der Menge $M$}, - sort=MengenoperationFAbschluss -} - -\newglossaryentry{Rand} -{ - name={\ensuremath{\partial M}}, - description={Rand der Menge $M$}, - sort=MengenoperationFRand -} - -\newglossaryentry{Inneres} -{ - name={\ensuremath{M^\circ}}, - description={Inneres der Menge $M$}, - sort=MengenoperationFInneres -} - -\newglossaryentry{Kreuzprodukt} -{ - name={\ensuremath{A \times B}}, - description={Kreuzprodukt zweier Mengen}, - sort=MengenoperationNKreuzprodukt -} -\newglossaryentry{subseteq} -{ - name={\ensuremath{A \subseteq B}}, - description={Teilmengenbeziehung}, - sort=MengenoperationNSubseteq -} -\newglossaryentry{subsetneq} -{ - name={\ensuremath{A \subsetneq B}}, - description={echte Teilmengenbeziehung}, - sort=MengenoperationNSubsetneq -} - -\newglossaryentry{setminus} -{ - name={\ensuremath{A \setminus B}}, - description={$A$ ohne $B$}, - sort=MengenoperationNSetminus -} - -\newglossaryentry{cup} -{ - name={\ensuremath{A \cup B}}, - description={Vereinigung}, - sort=MengenoperationOCup -} - -\newglossaryentry{dcup} -{ - name={\ensuremath{A \dcup B}}, - description={Disjunkte Vereinigung}, - sort=MengenoperationOCupD -} - -\newglossaryentry{cap} -{ - name={\ensuremath{A \cap B}}, - description={Schnitt}, - sort=MengenoperationOCap -} - +\chapter*{Symbolverzeichnis} +\addcontentsline{toc}{chapter}{Symbolverzeichnis} +\begin{minipage}[t]{0.45\textwidth} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Zahlenmengen % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\newglossaryentry{N} -{ - name={\ensuremath{\mdn}}, - description={Natürliche Zahlen}, - sort=KoerperAN -} - -\newglossaryentry{Z} -{ - name={\ensuremath{\mdz}}, - description={Ganze Zahlen}, - sort=KoerperAZ -} - -\newglossaryentry{Q} -{ - name={\ensuremath{\mdq}}, - description={Rationale Zahlen}, - sort=KoerperBQ -} - -\newglossaryentry{R} -{ - name={\ensuremath{\mdr}}, - description={Reele Zahlen}, - sort=KoerperR -} - -\newglossaryentry{Rplus} -{ - name={\ensuremath{\mdr^+}}, - description={Echt positive reele Zahlen}, - sort=KoerperRplus -} - -\newglossaryentry{Einheitengruppe} -{ - name={\ensuremath{\mdr^\times}}, - description={Multiplikative Einheitengruppe von $\mdr$}, - sort=KoerperREinheiten -} - -\newglossaryentry{Komplexe Zahlen} -{ - name={\ensuremath{\mdc}}, - description={Komplexe Zahlen}, - sort=KoerperSComplexeZahlen -} - -\newglossaryentry{Projektiver Raum} -{ - name={\ensuremath{\mdp}}, - description={Projektiver Raum}, - sort=KoerperXProjektion -} +\section*{Zahlenmengen} +$\mdn\;\;\;$ Natürliche Zahlen\\ +$\mdz\;\;\;$ Ganze Zahlen\\ +$\mdq\;\;\;$ Rationale Zahlen\\ +$\mdr\;\;\;$ Reele Zahlen\\ +$\mdr^+\;\;\;$ Echt positive reele Zahlen\\ +$\mdr^\times\;\;\;$ Einheitengruppe von $\mdr$\\ +$\mdc\;\;\;$ Komplexe Zahlen\\ +\section*{Weiteres} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Fraktale Symbole % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\newglossaryentry{fB} -{ - name={\ensuremath{\fB}}, - description={Basis einer Topologie}, - sort=fB -} +$\fB\;\;\;$ Basis einer Topologie\\ +$\fB_\delta(x)\;\;\;$ $\delta$-Kugel um $x$\\ +$\fT\;\;\;$ Topologie\\ -\newglossaryentry{Epsilonumgebung} -{ - name={\ensuremath{\fB_\delta(x)}}, - description={$\delta$-Kugel um $x$}, - sort=fBr -} +$\mdp\;\;\;$ Projektiver Raum\\ +$\langle \cdot , \cdot \rangle\;\;\;$ Skalarprodukt\\ +$X /_\sim\;\;\;$ $X$ modulo $\sim$\\ +$[x]_\sim\;\;\;$ Äquivalenzklassen von $x$ bzgl. $\sim$\\ +$\| x \|\;\;\;$ Norm von $x$\\ +$| x |\;\;\;$ Betrag von $x$\\ +$S^n\;\;\;$ Sphäre\\ +$T^n\;\;\;$ Torus\\ +$\pi_X\;\;\;$ Projektion auf $X$\\ +$f^{-1}(M)\;\;\;$ Urbild von $M$\\ +$\GL_n(K)\;\;\;$ Allgemeine lineare Gruppe (general linear group)\\ +$\text{Rg}(M)\;\;\;$ Rang von $M$\\ +$f|_U\;\;\;$ $f$ eingeschränkt auf $U$\\ -\newglossaryentry{fT} -{ - name={\ensuremath{\fT}}, - description={Topologie}, - sort=fT -} +\end{minipage} +\begin{minipage}[t]{0.45\textwidth} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% Mengenoperationen % +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section*{Mengenoperationen} +$A^C\;\;\;$ Komplement der Menge $A$\\ +$\mathcal{P}(M)\;\;\;$ Potenzmenge von $M$\\ +$\overline{M}\;\;\;$ Abschluss der Menge $M$\\ +$\partial M\;\;\;$ Rand der Menge $M$\\ +$M^\circ\;\;\;$ Inneres der Menge $M$\\ +$A \times B\;\;\;$ Kreuzprodukt zweier Mengen\\ +$A \subseteq B\;\;\;$ Teilmengenbeziehung\\ +$A \subsetneq B\;\;\;$ echte Teilmengenbeziehung\\ +$A \setminus B\;\;\;$ $A$ ohne $B$\\ +$A \cup B\;\;\;$ Vereinigung\\ +$A \dcup B\;\;\;$ Disjunkte Vereinigung\\ +$A \cap B\;\;\;$ Schnitt\\ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -% Sonstiges % +% Gruppen % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\newglossaryentry{Skalarprodukt} -{ - name={\ensuremath{\langle \cdot , \cdot \rangle}}, - description={Skalarprodukt}, - sort=ZZZSkalarprodukt -} - -\newglossaryentry{Modulo} -{ - name={\ensuremath{X /_\sim}}, - description={$X$ modulo $\sim$}, - sort=ZZZAuequivalenzModulo -} - -\newglossaryentry{Modulo-Aequivalenzklasse} -{ - name={\ensuremath{[x]_\sim}}, - description={Äquivalenzklassen von $x$ bzgl. $\sim$}, - sort=ZZZAuequivalenzKlassen -} - -\newglossaryentry{Norm} -{ - name={\ensuremath{\| x \|}}, - description={Norm von $x$}, - sort=ZZZNorm -} - -\newglossaryentry{Betrag} -{ - name={\ensuremath{| x |}}, - description={Betrag von $x$}, - sort=ZZZNormBetrag -} - -\newglossaryentry{Sphaere} -{ - name={\ensuremath{S^n}}, - description={Sphäre}, - sort=ZZZSphaere -} - -\newglossaryentry{Torus} -{ - name={\ensuremath{T^n}}, - description={Torus}, - sort=ZZZSphaereTorus -} - -\newglossaryentry{Projektion} -{ - name={\ensuremath{\pi_X}}, - description={Projektion auf X}, - sort=ZZZProjektion -} - -\newglossaryentry{Urbild} -{ - name={\ensuremath{f^{-1}(M)}}, - description={Urbild von $M$}, - sort=ZZZUrbild -} - -\newglossaryentry{Ohne Einschraekung} -{ - name={$\text{\OE}$}, - description={Ohne Einschränkung}, - sort=ZZZOE -} - -\newglossaryentry{Allgemeine lineare Gruppe} -{ - name={$\GL_n(K)$}, - description={Allgemeine lineare Gruppe (general linear group)}, - sort=ZZZGL -} +\section*{Gruppen} +$\text{Homöo}(X)\;\;\;$ Homöomorphismengruppe\\ +$\text{Iso}(X)\;\;\;$ Isometriengruppe\\ +\end{minipage} -\newglossaryentry{Rang} -{ - name={$\text{Rg}(M)$}, - description={Rang von $M$}, - sort=ZZZRang -} -\newglossaryentry{Einschraeunkung} -{ - name={$f|_U$}, - description={$f$ eingeschränkt auf U}, - sort=ZZZEinschraenkung -} -% Setze den richtigen Namen für das Glossar -\renewcommand*{\glossaryname}{\glossarName} -\deftranslation{Glossary}{\glossarName} -% Drucke das gesamte Glossar -\glsaddall -\printglossaries - -% Trage das Glossar in das Inhaltsverzeichnis ein -%\stepcounter{section} -%\addcontentsline{toc}{section}{\numberline {\thesection} \glossarName} diff --git a/documents/GeoTopo/figures/topology-continuous-mapping.tex b/documents/GeoTopo/figures/topology-continuous-mapping.tex index ecae0bb..dad813d 100644 --- a/documents/GeoTopo/figures/topology-continuous-mapping.tex +++ b/documents/GeoTopo/figures/topology-continuous-mapping.tex @@ -10,21 +10,22 @@ } \begin{tikzpicture} - \draw[->] (-0.5,0) -- (1.5,0) node [below] {$\mathbb{R}$}; + \draw[->] (-0.5,0) -- (1.5,0) node [below] {$\mathbb{R}$}; - \foreach \x in {0,...,1} - \draw (\x,0.1) -- (\x,-0.1) node [below] {\x}; + \foreach \x in {0,...,1} + \draw (\x,0.1) -- (\x,-0.1) node [below] {\x}; - \draw[red] (0.07,0.1) -- (0,0.1) -- (0,-0.1) -- (0.07,-0.1) node [below] {}; - \draw[red] plot [smooth] coordinates{(0.47,0.1) (0.5,0) (0.47,-0.1)}; + \draw[thick,red] (0.07,0.1) -- (0,0.1) -- (0,-0.1) -- (0.07,-0.1) node [below] {}; + \draw[thick,red] plot [smooth] coordinates{(0.47,0.1) (0.5,0) (0.47,-0.1)}; + \draw[thick,red] (0,0) -- (0.5,0); + \draw[dotted,red] (0,-0.03) -- (0.5,-0.03); \begin{scope}[shift={(4,0)}] \draw[thick] (0cm,0cm) circle(1cm); \draw[thick, red] ([shift={(180:1cm)}]-0.0,0) arc (180:0:1cm); + \draw[red, dotted] ([shift={(180:0.97cm)}]-0.0,0) arc (180:0:0.97cm); \draw (0:1cm) node[point, label=right:{$0$}] {}; - %\path node[point, blue, label={[blue,above]{$\overline{a}$}}] (posU) at (-252:1cm) {}; - %\path node[label={[red,left]{$U$}}] at (30:1cm) {}; \end{scope} \coordinate (circleUp) at (2.6, 0.1); @@ -35,10 +36,4 @@ \path[->] (numberlineUp) edge [bend left] node[label=$f$] {} (circleUp); \path[<-] (numberlineDown) edge [bend right] node[label=below:$g$] {} (circleDown); - %\draw (3.7cm,0cm) node[point, blue, label={[blue,above]{$a$}}] (posA) {}; - %\draw (0.7cm,0cm) node[point, blue, label={[blue,above]{$\pi^{-1}(u)$}}] {}; - %\draw[dashed, blue, thick] plot [smooth] coordinates{(posU) (0.2,-0.8) (2.5,-1) (posA)}; - - %\draw[blue, dashed, thick] (3.7cm,0cm) arc (0:180:1.5 and 0.5); - \end{tikzpicture} diff --git a/documents/GeoTopo/figures/topology-kartenwechsel.tex b/documents/GeoTopo/figures/topology-kartenwechsel.tex index 6310dbc..298acdc 100644 --- a/documents/GeoTopo/figures/topology-kartenwechsel.tex +++ b/documents/GeoTopo/figures/topology-kartenwechsel.tex @@ -1,5 +1,6 @@ \begin{tikzpicture}[tqft/flow=east] \draw (0,0) ellipse (2cm and 1cm); + \draw (0,-2) ellipse (3cm and 0.8cm); \def\ringa{(-0.3,0) circle (0.5cm)} \def\ringb{(+0.3,0) circle (0.5cm)} @@ -24,6 +25,8 @@ \node at (+1,0.3) {$U_j$}; \node at (-1.9,-2) {$V_i$}; \node at (+1.9,-2) {$V_j$}; + \node at (+2.0,0.7) {$X$}; + \node at (+2.4,-1.3) {$\mathbb{R}^n$}; \path[->] (-0.35,0) edge [bend angle=10,bend right] node[label={[label distance=0.1cm]210:$\varphi_i$}] {} (-1,-1.5); diff --git a/documents/GeoTopo/shortcuts.sty b/documents/GeoTopo/shortcuts.sty index 95946cf..7c16d09 100644 --- a/documents/GeoTopo/shortcuts.sty +++ b/documents/GeoTopo/shortcuts.sty @@ -30,10 +30,12 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Define theorems % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\newcommand{\thmfoot}{} \theoremstyle{break} \setlength\theoremindent{0.7cm} \theoremheaderfont{\kern-0.7cm\normalfont\bfseries} \theorembodyfont{\normalfont} % nicht mehr kursiv +\theoremseparator{\thmfoot} \newframedtheorem{satz}{Satz}[chapter] \newframedtheorem{lemma}[satz]{Lemma} @@ -67,5 +69,6 @@ \newcommand\mapsfrom{\mathrel{\reflectbox{\ensuremath{\mapsto}}}} \newcommand\dcup{\mathbin{\dot{\cup}}} \newcommand\obda{o.~B.~d.~A.} +\newcommand\Obda{o.~B.~d.~A.}