From 508b4ee5180b75b88445366e1968ba49ae2e2677 Mon Sep 17 00:00:00 2001 From: Martin Thoma Date: Sun, 26 Jan 2014 22:43:30 +0100 Subject: [PATCH] Textsetzung: Figure bekommt htp; stackrel -> overset; \ref -> \cref; enumerate --- documents/GeoTopo/Arbeitszeit.md | 1 + documents/GeoTopo/GeoTopo.pdf | Bin 4941876 -> 4943948 bytes documents/GeoTopo/Kapitel1.tex | 21 ++-- documents/GeoTopo/Kapitel2-UB.tex | 2 +- documents/GeoTopo/Kapitel2.tex | 98 ++++++++-------- documents/GeoTopo/Kapitel3.tex | 122 ++++++++++---------- documents/GeoTopo/Kapitel4-UB.tex | 8 +- documents/GeoTopo/Kapitel4.tex | 50 ++++---- documents/GeoTopo/Loesungen.tex | 8 +- documents/GeoTopo/Readme.md | 2 + documents/GeoTopo/definitions/shortcuts.sty | 10 +- documents/GeoTopo/shortcuts.sty | 25 +++- 12 files changed, 185 insertions(+), 162 deletions(-) diff --git a/documents/GeoTopo/Arbeitszeit.md b/documents/GeoTopo/Arbeitszeit.md index eb18877..eb1ca7c 100644 --- a/documents/GeoTopo/Arbeitszeit.md +++ b/documents/GeoTopo/Arbeitszeit.md @@ -46,3 +46,4 @@ in dem Erstellen dieses Skripts steckt: |24.01.2014 | 23:00 - 00:15 | Digitalisieren der Vorlesung von 23.01.2014 |25.01.2014 | 09:30 - 12:45 | Digitalisieren der Vorlesung von 23.01.2014 |25.01.2014 | 13:05 - 13:35 | Aufgabe aus Tutorium hinzugefügt +|26.01.2014 | 19:00 - 22:00 | Textsetzung: Figure bekommt htp; stackrel -> overset; \ref -> \cref; enumerate diff --git a/documents/GeoTopo/GeoTopo.pdf b/documents/GeoTopo/GeoTopo.pdf index a1fa3f51b15bfe4e1253610dd3757ccfe707381d..529c83d2a31139653d0c9db9cdd2421206d20c70 100644 GIT binary patch delta 478595 zcmX8ZLwF@jw*}x>9ozO7+qRt(+qQG!q+?qhcWm3XZQJbJ{LdY{vl`dht9CWcR)J0K zR)N(*N2Y`uVA40HGL zeqXaf-r1=l$JGk;!6Zb?E$5B`LWuW=i_#Y<_gdL$*h-tUAOaOxMb6BRGDvG7l(~$N z1y%ybPqu-3hGfsV1SsT0PeEn@;@M3B2T-?h<+L6`!!JRTqu+RG7Mp$Hd{CD{qc>6t zxRX>V<*wd#%$n`%27DZQFDifEg7uJ*+(GShZ%52?DmBIrSNNVh?Ntu`{`nwYW|jc? z(+6mPx}@l)yLfjbt(T7*c~(6^#TWeTKrc+Yd7DH;!0IObYGy-R(c$i 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zLIu@BaOenSPz2`xS5m{!@whhs^T58J2!C2V9%2V~`=>tM4z_N8h{?mwh4`tDt+!8r Xw}T_Gm^4&EL<&mG$*HcVLHxe}hRwN% diff --git a/documents/GeoTopo/Kapitel1.tex b/documents/GeoTopo/Kapitel1.tex index 7cee2e1..49c118d 100644 --- a/documents/GeoTopo/Kapitel1.tex +++ b/documents/GeoTopo/Kapitel1.tex @@ -62,8 +62,8 @@ Auch gibt es Mengen, die sowohl abgeschlossen als auch offen sind. \begin{definition} 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} + \item $\displaystyle M^\circ := \Set{x \in M | M \text{ ist Umgebung von } x} = \bigcup_{\overset{U \subseteq M} {U \in \fT}} U $ heißt \textbf{Inneres} oder \textbf{ offener Kern} von $M$. \xindex{Inneres} \xindex{Kern!offener} + \item $\displaystyle \overline{M} := \bigcap_{\overset{M \subseteq A}{A \text{ abgeschlossen}}} A$ 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} @@ -303,14 +303,14 @@ Auch gibt es Mengen, die sowohl abgeschlossen als auch offen sind. \begin{definition} Seien $X, Y$ topologische Räume und $f:X \rightarrow Y$ eine Abbildung. - \begin{enumerate}[label=\alph*),ref=\theplaindefinition.\alph*] + \begin{defenum} \item $f$ heißt \textbf{stetig}\xindex{Abbildung!stetige}, wenn für jedes offene $U \subseteq Y$ auch $f^{-1} (U) \subseteq X$ offen ist. \label{def:stetigkeit} \item $f$ heißt \textbf{Homöomorphismus}\xindex{Homöomorphismus}, wenn $f$ stetig ist und es eine stetige Abbildung $g: Y \rightarrow X$ gibt, sodass $g \circ f = \id_X$ und $f \circ g = \id_Y$. - \end{enumerate} + \end{defenum} \end{definition} \begingroup @@ -333,7 +333,7 @@ Auch gibt es Mengen, die sowohl abgeschlossen als auch offen sind. \enquote{$\Rightarrow$}: Sei $x \in X, \varepsilon > 0$ gegeben und $U := \fB_\varepsilon(f(x))$.\\ Dann ist $U$ offen in $Y$.\\ - $\xRightarrow{\ref{def:stetigkeit}} f^{-1}(U)$ ist + $\xRightarrow{\text{Def. }\ref{def:stetigkeit}} f^{-1}(U)$ ist offen in $X$. Dann ist $x \in f^{-1}(U)$.\\ $\Rightarrow \exists \delta > 0$, sodass $\fB_\delta(x) \subseteq f^{-1} (U)$\\ @@ -364,7 +364,7 @@ Auch gibt es Mengen, die sowohl abgeschlossen als auch offen sind. stetig für jeden topologischen Raum $Y$ und jede Abbildung $f$. \item Sei $X = [0, 1), Y = S^1 = \Set{z \in \mdc | \|z\| = 1}$ und $f(t) = e^{2 \pi i t}$ - \begin{figure} + \begin{figure}[htp] \centering \input{figures/topology-continuous-mapping} \caption{Beispiel einer stetigen Funktion $f$, deren @@ -476,11 +476,10 @@ sodass $\pi$ stetig wird. Es gilt: $f$ ist bijektiv und die Umkehrabbildung ist ebenfalls stetig. \end{beispiel} - +\index{Stetigkeit|)} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Mitschrieb vom 31.10.2013 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\index{Stetigkeit|)} \section{Zusammenhang}\index{Zusammenhang|(} \begin{definition}\xindex{zusammenhaengend@zusammenhängend} Ein Raum $X$ heißt \textbf{zusammenhängend}, wenn es keine offenen, @@ -550,7 +549,7 @@ sodass $\pi$ stetig wird. $\Rightarrow$ Widerspruch zu $A$ ist zusammenhängend $ \qed$ \end{beweis} -\begin{bemerkung}\label{zusammenhangVereinigung} +\begin{bemerkung}\label{bem:zusammenhangVereinigung} Sei $X$ ein topologischer Raum und $A, B \subseteq X$ zusammenhängend. Ist $A \cap B \neq \emptyset$, dann ist $A \cup B$ zusammenhängend. @@ -597,8 +596,8 @@ sodass $\pi$ stetig wird. ist unerlaubte Zerlegung. \item Nach \cref{zusammenhangAbschluss} ist $\overline{Z(x)}$ zusammenhängend $\Rightarrow \overline{Z(x)} \subseteq Z(x)$ - $\Rightarrow Z(x) = \overline{Z(x}$ - \item Ist $Z(y) \cap Z(x) \neq \emptyset \xRightarrow{\ref{zusammenhangVereinigung}} Z(y) \cup Z(x)$ + $\Rightarrow Z(x) = \overline{Z(x)}$ + \item Ist $Z(y) \cap Z(x) \neq \emptyset \xRightarrow{\text{Bem. }\ref{bem:zusammenhangVereinigung}} Z(y) \cup Z(x)$ ist zusammenhängend. \\ \begin{align*} \Rightarrow Z(x) \cup Z(y) &\subseteq Z(x) \Rightarrow Z(y) \subseteq Z(x)\\ diff --git a/documents/GeoTopo/Kapitel2-UB.tex b/documents/GeoTopo/Kapitel2-UB.tex index 1afe7cc..f76f945 100644 --- a/documents/GeoTopo/Kapitel2-UB.tex +++ b/documents/GeoTopo/Kapitel2-UB.tex @@ -7,7 +7,7 @@ \item Beweisen Sie, dass eine topologische Mannigfaltigkeit genau dann wegzusammenhängend ist, wenn sie zusammenhängend ist - \item Betrachten Sie nun wie in Beispiel~\ref{bsp:mannigfaltigkeit8} + \item Betrachten Sie nun wie in \cref{bsp:mannigfaltigkeit8} den Raum $X:= (\mdr \setminus \Set{0}) \cup \Set{0_1, 0_2}$ versehen mit der dort definierten Topologie. Ist $X$ wegzusammenhängend? diff --git a/documents/GeoTopo/Kapitel2.tex b/documents/GeoTopo/Kapitel2.tex index 0f553ef..5eabb16 100644 --- a/documents/GeoTopo/Kapitel2.tex +++ b/documents/GeoTopo/Kapitel2.tex @@ -5,7 +5,7 @@ \section{Topologische Mannigfaltigkeiten} \begin{definition} Sei $X$ ein topologischer Raum und $n \in \mdn$. - \begin{enumerate}[label=\alph*)] + \begin{defenum} \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 @@ -16,11 +16,11 @@ \item $X$ heißt (topologische) $n$-dimensionale \textbf{Mannigfaltigkeit}\xindex{Mannigfaltigkeit}, wenn $X$ hausdorffsch ist, eine abzählbare Basis der Topologie hat und ein $n$-dimensionalen Atlas besitzt. - \end{enumerate} + \end{defenum} \end{definition} \begin{bemerkung} - \begin{enumerate}[label=(\alph*)] + \begin{bemenum} \item Es gibt surjektive, stetige Abbildungen $[0,1] \rightarrow [0,1] \times [0,1]$ \item Für $n \neq m$ sind $\mdr^n$ und $\mdr^m$ nicht homöomorph. Zum Beweis benutzt man den \enquote{Satz von der Gebietstreue} (Brouwer): @@ -32,11 +32,11 @@ \[f:\mdr^n \rightarrow \mdr^m \rightarrow \mdr^n, \;\;\; (x_1, \dots, x_n) \mapsto (x_1, x_2, \dots, x_n, 0, \dots, 0)\] eine stetige injektive Abbildung. Also müsste $f(\mdr^n)$ offen sein $\Rightarrow$ Widerspruch - \end{enumerate} + \end{bemenum} \end{bemerkung} \begin{beispiel} - \begin{enumerate}[label=\arabic*)] + \begin{bspenum} \item Jede offene Teilmenge $U \subseteq \mdr^n$ ist eine $n$-dimensionale Mannigfaltigkeit mit einem Atlas aus einer Karte. @@ -105,7 +105,7 @@ \item $\GL_n(\mdr)$ ist eine Mannigfaltigkeit der Dimension $n^2$, weil offene Teilmengen von $\mdr^{n^2}$ eine Mannigfaltigkeit bilden. - \end{enumerate} + \end{bspenum} \end{beispiel} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -148,7 +148,7 @@ \item oder mehr Henkel, wie z.B. der Zweifachtorus in \cref{fig:double-torus} \end{enumerate} - \begin{figure} + \begin{figure}[htp] \centering \includegraphics[width=0.2\linewidth, keepaspectratio]{figures/Double-torus-illustration.png} \caption{Zweifachtorus} @@ -161,15 +161,15 @@ und $X = V(F) := \Set{x \in \mdr^n | F(x) = 0}$ das \enquote{vanishing set}. Dann gilt: - \begin{enumerate}[label=\alph*),ref=\theplaindefinition.\alph*] + \begin{bemenum} \item $X$ ist abgeschlossen in $\mdr^n$ \item Ist $\grad(F)(X) \neq 0 \;\;\;\forall{x \in X}$, so ist - $X$ eine Mannigfaltigkeit der Dimension $n-1$. \label{Mannigfaltigkeitskriterium} - \end{enumerate} + $X$ eine Mannigfaltigkeit der Dimension $n-1$. \label{bem:Mannigfaltigkeitskriterium} + \end{bemenum} \end{bemerkung} \begin{beweis}\leavevmode - \begin{enumerate}[label=\alph*),ref=\theplaindefinition.\alph*] + \begin{enumerate}[label=\alph*),ref=\thedefinition.\alph*] \item Sei $y \in \mdr^n \setminus V(F)$. Weil $F$ stetig ist, gibt es $\delta > 0$, sodass $F(\fB_\delta(y)) \subseteq \fB_\varepsilon(F(y))$ mit $\varepsilon = \frac{1}{2} \|F(y)\|$. Folgt @@ -189,9 +189,9 @@ \end{beweis} \begin{beispiel}\xindex{Neilsche Parabel} - \begin{enumerate}[label=\alph*)] + \begin{bspenum} \item $F: \mdr^3 \rightarrow \mdr,\;\;\; (x, y, z) \mapsto x^2 + y^2 + z^2 - 1$, - $V(F) = S^2$, $\grad(F) = (2x, 2y, 2z) \xRightarrow{\ref{Mannigfaltigkeitskriterium}} S^n$ + $V(F) = S^2$, $\grad(F) = (2x, 2y, 2z) \xRightarrow{\text{Bem. }\ref{bem:Mannigfaltigkeitskriterium}} S^n$ ist $n$-dimensionale Mannigfaltigkeit in $\mdr^{n+1}$ \item $F: \mdr^2 \rightarrow \mdr, \;\;\; (x,y) \mapsto y^2 - x^3$ \begin{figure}[ht] @@ -208,10 +208,10 @@ \caption{Rechts ist die Neilsche Parabel für verschiedene Parameter $a$.} \end{figure} Es gilt: $\grad(F) = (-3x^2, 2y)$. Also: $\grad(0,0) = (0,0)$. - Daher ist \cref{Mannigfaltigkeitskriterium} + Daher ist \cref{bem:Mannigfaltigkeitskriterium} nicht anwendbar, aber $V(F)$ ist trotzdem eine 1-dimensionale topologische Mannigfaltigkeit. - \end{enumerate} + \end{bspenum} \end{beispiel} \begin{definition}\xindex{Mannigfaltigkeit!mit Rand} @@ -278,21 +278,21 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. \begin{definition} Sei $X$ eine $n$-dimensionale Mannigfaltigkeit mit Atlas $(U_i, \varphi_i)_{i \in I}$. - \begin{enumerate}[label=\alph*)] + \begin{defenum} \item $X$ heißt \textbf{differenzierbare Mannigfaltigkeit der Klasse $C^k$}\xindex{Mannigfaltigkeit!differenzierbare}, wenn jede Kartenwechselabbildung $\varphi_{ij},\;i,j \in I$ $k$-mal stetig differenzierbar ist. \item $X$ heißt \textbf{differenzierbare Mannigfaltigkeit}\xindex{Mannigfaltigkeit!glatte}, wenn $X$ eine differenzierbare Mannigfaltigkeit der Klasse $C^\infty$ ist. - \end{enumerate} + \end{defenum} \end{definition} \begin{definition} Sei $X$ eine differenzierbare Mannigfaltigkeit der Klasse $C^k$ ($k \in \mdn \cup \Set{\infty}$) mit Atlas $(U_i, \varphi_i)_{i \in I}$. - \begin{enumerate}[label=\alph*)] + \begin{defenum} \item Eine Karte $(U, \varphi)$ auf $X$ heißt \textbf{verträglich}\xindex{verträglich} mit $\atlas$, wenn alle Kartenwechsel $\varphi \circ \varphi_i^{-1}$ und $\varphi_i \circ \varphi^{-1}$ ($i \in I$ mit $U_i \cap U \neq \emptyset$) @@ -303,7 +303,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. Eine $C^\infty$-Struktur heißt auch \textbf{differenzierbare Struktur}\xindex{Struktur!differenzierbare} auf $X$. - \end{enumerate} + \end{defenum} \end{definition} \begin{bemerkung} @@ -315,7 +315,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. Seien $X, Y$ differenzierbare Mannigfaltigkeiten der Dimension $n$ bzw. $m$, $x \in X$. - \begin{enumerate}[label=\alph*),ref=\theplaindefinition.\alph*] + \begin{defenum} \item Eine stetige Abbildung $f:X \rightarrow Y$ heißt\label{def:stetigeAbbildungDiffbar} \textbf{differenzierbar}\xindex{Abbildung!differenzierbare} in $x$ (von Klasse $C^k$), @@ -331,11 +331,11 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. es eine differenzierbare Abbildung $g: Y \rightarrow X$ von Klasse $C^\infty$ gibt mit $g \circ f = \id_X$ und $f \circ g = \id_Y$. - \end{enumerate} + \end{defenum} \end{definition} \begin{bemerkung} - Die Bedingung in Definition~\ref{def:stetigeAbbildungDiffbar} hängt nicht + Die Bedingung in \cref{def:stetigeAbbildungDiffbar} hängt nicht von den gewählten Karten ab. \end{bemerkung} @@ -381,14 +381,14 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. \end{definition} \begin{beispiel} - \begin{enumerate}[label=\arabic*)] + \begin{bspenum} \item Rotationsflächen: Sei $r:\mdr \rightarrow \mdr_{> 0}$ eine differenzierbare Funktion. $F: \mdr^2 \rightarrow \mdr^3 \;\;\; (u,v) \mapsto (r(u) \cos (u), r(v) \sin(u), v)$ - \begin{figure} + \begin{figure}[htp] \centering \subfloat[Kugelkoordinaten]{ \includegraphics[width=0.45\linewidth, keepaspectratio]{figures/spherical-coordinates.pdf} @@ -433,7 +433,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. \end{pmatrix}\] hat Rang 2 für $\cos v \neq 0$. In $N$ und $S$ ist $\cos v = 0$. - \end{enumerate} + \end{bspenum} \end{beispiel} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -497,7 +497,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. eine Abbildung, $(g,h) \mapsto g \cdot h$, sodass $(G, \circ)$ eine Gruppe ist. - \begin{enumerate}[label=(\alph*)] + \begin{defenum} \item $G$ heißt \textbf{topologische Gruppe}\xindex{Gruppe!topologische}, wenn die Abbildungen $\circ: G \times G \rightarrow G$ und $\iota: G \rightarrow G$. @@ -506,11 +506,11 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. \item Ist $G$ eine differenzierbare Mannigfaltigkeit, so heißt $G$ \textbf{Lie-Gruppe}\xindex{Lie-Gruppe}, wenn $(G, \circ)$ und $(G, \iota)$ differenzierbar sind. - \end{enumerate} + \end{defenum} \end{definition} \begin{beispiel} - \begin{enumerate}[label=\arabic*)] + \begin{bspenum} \item Alle endlichen Gruppen sind 0-dimensionale Lie-Gruppen. \item $\GL_n(\mdr)$ \item $(\mdr^\times, \cdot)$ @@ -537,7 +537,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. $\frac{\partial}{\partial a_{11}} (\det -1) = 1 \cdot \det A_{11}$ Es gibt $i \in \Set{1, \dots, n}$ mit $\frac{\partial}{\partial a_{1i}} (\det -1) A \neq 0$ - \end{enumerate} + \end{bspenum} \end{beispiel} \begin{bemerkung} @@ -553,16 +553,16 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. \section{Simplizialkomplex} \begin{definition}\xindex{Lage!allgemeine} Seien $v_0, \dots, v_k \in \mdr^n$ Punkte. - \begin{enumerate}[label=\alph*),ref=\theplaindefinition.\alph*] + \begin{defenum} \item $v_0, \dots, v_k$ sind \textbf{in allgemeiner Lage} $\gdw$ es gibt keinen $(k-1)$-dimensionalen affinen Untervektorraum, der $v_0, \dots, v_k$ enthält \gdw $v_1 - v_0, \dots, v_k - v_0$ sind linear abhängig. \item $\conv(v_0, \dots, v_k) = \Set{\sum_{i=0}^k \lambda_i v_i | \lambda_i \geq 0, \sum_{i=0}^k \lambda_i = 1} $ - \end{enumerate} + \end{defenum} \end{definition} \begin{definition} - \begin{enumerate}[label=\alph*),ref=\theplaindefinition.\alph*] + \begin{defenum} \item Sei $\Delta^k = \conv(e_0, \dots, e_k) \subseteq \mdr^{n+1}$ die konvexe Hülle der Standard-Basisvektoren $e_0, \dots, e_k$. @@ -578,7 +578,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. von $\Delta$. $s_{i_0, \dots, i_r}$ ist $r$-Simplex. - \end{enumerate} + \end{defenum} \end{definition} \begin{figure}[ht] @@ -609,7 +609,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. % Mitschrieb vom 21.11.2013 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{definition} - \begin{enumerate}[label=\alph*),ref=\theplaindefinition.\alph*] + \begin{enumerate}[label=\alph*),ref=\thedefinition.\alph*] \item Eine endliche Menge $K$ von Simplizes im $\mdr^n$ heißt (endlicher) \textbf{Simplizialkomplex}\xindex{Simplizialkomplex}, wenn gilt: @@ -655,7 +655,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. \label{fig:simplizialkomplex-cube-divided} } - \subfloat[$P$ ist kein Teilsimplex, da Eigenschaft \ref{def:simplizialkomplex.ii} verletzt ist]{ + \subfloat[$P$ ist kein Teilsimplex, da Eigenschaft \cref{def:simplizialkomplex.ii} verletzt ist]{ \parbox[c][4cm]{5cm}{\centering\input{figures/topology-triangle-no-simplicial-complex.tex}} \label{fig:no-simplizialkomplex-triangles} }% @@ -672,15 +672,15 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. \[f:|K| \rightarrow |L|\] heißt \textbf{simplizial}\xindex{Abbildung!simpliziale}, wenn für jedes $\Delta \in K$ gilt: - \begin{enumerate}[label=(\roman*)] + \begin{defenum} \item $f(\Delta) \in L$ \item $f|_{\Delta} : \Delta \rightarrow f(\Delta)$ ist eine affine Abbildung. - \end{enumerate} + \end{defenum} \end{definition} \begin{beispiel} - \begin{enumerate}[label=\arabic*)] + \begin{bspenum} \item $\varphi(e_1) := b_1$, $\varphi(e_2) := b_2$\\ $\varphi$ ist eine eindeutig bestimmte lineare Abbildung @@ -693,7 +693,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. \item \todo[inline]{Wozu dient das Beispiel?} \resizebox{0.9\linewidth}{!}{\input{figures/topology-2.tex}} - \end{enumerate} + \end{bspenum} \end{beispiel} \begin{definition} @@ -707,7 +707,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. \end{definition} \begin{beispiel} - \begin{enumerate}[label=\arabic*)] + \begin{bspenum} \item $\chi(\Delta^1) = 2 - 1 = 1$\\ $\chi(\Delta^2) = 3 - 3 + 1 = 1$\\ $\chi(\Delta^3) = 4 - 6 + 4 - 1 = 1$ @@ -716,7 +716,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. $\chi(\text{Ikosaeder}) = 12 - 30 + 20 = 2$ \item $\chi(\text{Würfel}) = 8 - 12 + 6 = 2$\\ $\chi(\text{Würfel, unterteilt in Dreiecksflächen}) = 8 - (12 + 6) + (6 \cdot 2) = 2$ - \end{enumerate} + \end{bspenum} \end{beispiel} \begin{bemerkung} @@ -729,7 +729,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. definiert ein $k$-Simplex.\\ $\Rightarrow a_k(\Delta^n) = \binom{n+1}{k+1}, \;\;\; k = 0, \dots, n$\\ $\Rightarrow \chi(\Delta^n) = \sum_{k=0}^n (-1)^k \binom{n+1}{k+1}$\\ - $f(x) = (x+1)^{n+1} \stackrel{\substack{\text{\tiny{Binomischer}}\\\text{\tiny{Lehrsatz}}}}{=} \sum_{k=0}^{n+1} \binom{n+1}{k} x^k$\\ + $f(x) = (x+1)^{n+1} \overset{\substack{\text{\tiny{Binomischer}}\\\text{\tiny{Lehrsatz}}}}{=} \sum_{k=0}^{n+1} \binom{n+1}{k} x^k$\\ $\Rightarrow 0 = \sum_{k=0}^{n+1} \binom{n+1}{k} (-1)^k = \chi(\Delta^n) -1$\\ $\Rightarrow \chi(\Delta^n) = 1 \qed$ \end{beweis} @@ -738,12 +738,12 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. % Mitschrieb vom 28.11.2013 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{definition} - \begin{enumerate}[label=\alph*),ref=\theplaindefinition.\alph*] + \begin{defenum} \item Ein 1D-Simplizialkomplex heißt \textbf{Graph}\xindex{Graph}. \item Ein Graph, der homöomorph zu $S^1$ ist, heißt \textbf{Kreis}\xindex{Kreis}. \item Ein zusammenhängender Graph heißt \textbf{Baum}\xindex{Baum}, wenn er keinen Kreis enthält. - \end{enumerate} + \end{defenum} \end{definition} \begin{figure}[ht] @@ -778,16 +778,16 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. \end{beweis} \begin{bemerkung} - \begin{enumerate}[label=\alph*),ref=\theplaindefinition.\alph*] + \begin{bemenum} \item Jeder zusammenhängende Graph $\Gamma$ enthält einen Teilbaum $T$, der alle Ecken von $\Gamma$ enthält.% \footnote{$T$ wird \enquote{Spannbaum} genannt.} \item Ist $n = a_1(\Gamma) = a_1(T)$, so ist $\chi(\Gamma) = 1 - n$. - \end{enumerate} + \end{bemenum} \end{bemerkung} \begin{beweis}\leavevmode - \begin{enumerate}[label=\alph*),ref=\theplaindefinition.\alph*] + \begin{enumerate}[label=\alph*),ref=\thedefinition.\alph*] \item Siehe \enquote{Algorithmus von Kruskal}. \item $\begin{aligned}[t]\chi(\Gamma) &= a_0(\Gamma) - a_1(\Gamma)\\ &= a_0(\Gamma) - (n+a_1(T))\\ @@ -905,12 +905,12 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$. Sei $Z_n := \text{Kern}(d_n) \subseteq C_n$ und $B_n := \text{Bild}(d_{n+1}) \subseteq C_n$. - \begin{enumerate}[label=\alph*)] + \begin{defenum} \item $H_n = H_n(K, \mdr) := Z_n / B_n$ heißt $n$-te \textbf{Homotopiegruppe}\xindex{Homotopiegruppe} von $K$. \item $b_n(K) := \dim_{\mdr} H_n$ heißt $n$-te \textbf{Belti-Zahl}\xindex{Belit-Zahl} von $K$. - \end{enumerate} + \end{defenum} \end{definition} \begin{bemerkung} diff --git a/documents/GeoTopo/Kapitel3.tex b/documents/GeoTopo/Kapitel3.tex index 6a1f696..5d18620 100644 --- a/documents/GeoTopo/Kapitel3.tex +++ b/documents/GeoTopo/Kapitel3.tex @@ -23,7 +23,7 @@ $\gamma_1, \gamma_2: [0,1] \rightarrow X$ Wege von $a$ nach $b$, d.~h. $\gamma_1(0) = \gamma_2(0) = a$, $\gamma_1(1) = \gamma_2(1) = b$ - \begin{enumerate}[label=\alph*)] + \begin{defenum} \item $\gamma_1$ und $\gamma_2$ heißen \textbf{homotop}\xindex{Weg!homotope}, wenn es eine stetige Abbildung \[H(t,0) = \gamma_1(t), H(t,1) = \gamma_2(t) \;\;\; \forall t \in [0,1] =: I \] @@ -34,7 +34,7 @@ $\gamma_1$ und $\gamma_2$. \item $\gamma_s: I \rightarrow X, \gamma_s(t) = H(t,s)$ ist Weg in $X$ von $a$ nach $b$ für jedes $s \in I$. - \end{enumerate} + \end{defenum} \end{definition} \begin{bemerkung} @@ -60,7 +60,7 @@ \end{beweis} \begin{beispiel} - \begin{enumerate}[label=\arabic*)] + \begin{bspenum} \item Sei $X = S^1$. $\gamma_1$ und $\gamma_2$ aus \cref{fig:circle-two-paths} nicht homöotop. \item Sei $X = T^2$. $\gamma_1, \gamma_2$ und $\gamma_3$ @@ -71,7 +71,7 @@ Je zwei Wege im $\mdr^2$ mit Anfangs- und Endpunkt $(0,0)$ sind homöotop. - \begin{figure} + \begin{figure}[htp] \centering \input{figures/topology-paths-in-r2.tex} \caption{Zwei Wege im $\mdr^2$ mit Anfangs- und Endpunkt $(0,0)$} @@ -85,7 +85,7 @@ $H(t,s) := (1-s) \gamma(t)$ ist stetig, $H(t,0) = \gamma(t)\; \forall t \in I$ und $H(t,1) = 0 \; \forall t \in I$ - \end{enumerate} + \end{bspenum} \begin{figure}[ht] \centering @@ -175,7 +175,7 @@ ist $\gamma_1 * \gamma_2 \sim \gamma_1 ' * \gamma_2'$. \end{bemerkung} -\begin{figure} +\begin{figure}[htp] \centering %\includegraphics[width=0.5\linewidth, keepaspectratio]{figures/todo/skizze-bemerkung-10-6.jpg} \input{figures/topology-homotop-paths-2.tex} @@ -225,7 +225,7 @@ Für einen Weg $\gamma$ sei $[\gamma]$ seine \textbf{Homotopieklasse}\xindex{Hom \end{beweis} \begin{beispiel} - \begin{enumerate}[label=\arabic*)] + \begin{bspenum} \item $S^1 = \Set{z \in \mdc | {|z|} = 1} = \Set{(\cos \varphi, \sin \varphi) \in \mdr^2 | 0 \leq \varphi \leq 2 \pi}$ $\pi_1 (S^1, 1) = \Set{[\gamma^k] | k \in \mdz} \cong \mdz$ @@ -240,7 +240,7 @@ Für einen Weg $\gamma$ sei $[\gamma]$ seine \textbf{Homotopieklasse}\xindex{Hom Für jedes sternförmige $G \subseteq \mdr^n$ ist $\pi_1(G,x) = \Set{e}$ - \begin{figure} + \begin{figure}[htp] \centering \input{figures/star-shaped-domain.tex} \caption{Sternförmiges Gebiet}. @@ -253,7 +253,7 @@ Für einen Weg $\gamma$ sei $[\gamma]$ seine \textbf{Homotopieklasse}\xindex{Hom Dieses Argument funktioniert nicht mehr bei flächendeckenden Wegen! - \end{enumerate} + \end{bspenum} \end{beispiel} \begin{bemerkung}\label{kor:gruppenisomorphismus-wege} @@ -265,7 +265,7 @@ Für einen Weg $\gamma$ sei $[\gamma]$ seine \textbf{Homotopieklasse}\xindex{Hom ein Gruppenisomorphismus. \end{bemerkung} -\begin{figure} +\begin{figure}[htp] \centering \input{figures/todo.tex} \caption{Situation aus \cref{kor:gruppenisomorphismus-wege}}. @@ -294,13 +294,13 @@ Für einen Weg $\gamma$ sei $[\gamma]$ seine \textbf{Homotopieklasse}\xindex{Hom Es seien $X, Y$ topologische Räume, $f:X \rightarrow Y$ eine stetige Abbildung, $x \in X, y := f(x) \in Y$. - \begin{enumerate}[label=\alph*)] + \begin{bemenum} \item Dann ist die Abbildung $f_* : \pi_1(X,x) \rightarrow \pi_1(Y, y), [y] \rightarrow [f \circ y]$ ein Gruppenhomomorphismus. \item Ist $Z$ ein weiterer topologischer Raum und $g: Y \rightarrow Z$ eine stetige Abbildung $z:= g(y)$. Dann ist $(g \circ f)_* = g_* \circ f_*: \pi_1(X,x) \rightarrow \pi_1(Z,z)$ - \end{enumerate} + \end{bemenum} \end{bemerkung} \begin{beweis}\leavevmode @@ -319,14 +319,14 @@ Für einen Weg $\gamma$ sei $[\gamma]$ seine \textbf{Homotopieklasse}\xindex{Hom \end{beweis} \begin{beispiel} - \begin{enumerate}[label=\arabic*)] + \begin{bspenum} \item $f:S^1 \hookrightarrow \mdr^2$ ist injektiv, aber $f_*:\pi_1(S^1, 1) \cong \mdz \rightarrow \pi_1(\mdr^2, 1) -0 \Set{e}$ ist nicht injektiv \item $f: \mdr \rightarrow S^1, t \mapsto (\cos 2 \pi t, \sin 2 \pi t)$ ist surjektiv, aber $f_*: \pi_1(\mdr, 0) = \Set{e} \rightarrow \pi_1(S^2, 1) \cong \mdz$ ist nicht surjektiv - \end{enumerate} + \end{bspenum} \end{beispiel} \begin{bemerkung}%Folgerung 11.6 @@ -406,9 +406,9 @@ Für einen Weg $\gamma$ sei $[\gamma]$ seine \textbf{Homotopieklasse}\xindex{Hom \end{beweis} \begin{beispiel} - \begin{enumerate}[label=\arabic*)] + \begin{bspenum} \item - \begin{figure} + \begin{figure}[htp] \centering \input{figures/topologischer-raum-x.tex} \caption{Topologischer Raum $X$} @@ -419,20 +419,20 @@ Für einen Weg $\gamma$ sei $[\gamma]$ seine \textbf{Homotopieklasse}\xindex{Hom $\pi_1(U,x) = \cong \mdz, \pi_1(V,x) = \cong \mdz$, insbesondere ist $a*b$ nicht homotop zu $b*a$. \item Torus: $\pi_1(T^2, X)$ wird erzeugt von $a$ und $b$. - \begin{figure} + \begin{figure}[htp] \centering \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} - \end{enumerate} + \end{bspenum} \end{beispiel} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Mitschrieb vom 12.12.2013 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Überlagerungen}\index{Ueberlagerung@""Uberlagerung|(} -\begin{figure} +\begin{figure}[htp] \centering \includegraphics[width=4cm, keepaspectratio]{figures/topology-r-spiral-covering-s.pdf} \caption{$\mdr \rightarrow S^1$,\\$t \mapsto (\cos 2 \pi t, \sin 2 \pi t)$} @@ -449,13 +449,13 @@ Für einen Weg $\gamma$ sei $[\gamma]$ seine \textbf{Homotopieklasse}\xindex{Hom \end{definition} \begin{beispiel} - \begin{enumerate}[label=\arabic*)] + \begin{bspenum} \item siehe \cref{fig:ueberlappung-r1-spirale-s1} \item siehe \cref{fig:ueberlappung-kaestchen-torus} \item $\mdr^n \rightarrow T^n = \mdr^n / \mdz^n$ \item $S^n \rightarrow \praum^n(\mdr)$\xindex{Raum!projektiver} \item $S^1 \rightarrow S^1$, $z \mapsto z^2$, siehe \cref{fig:liftung-s1-s1} - \end{enumerate} + \end{bspenum} \begin{figure}[ht] \centering @@ -506,7 +506,7 @@ Für einen Weg $\gamma$ sei $[\gamma]$ seine \textbf{Homotopieklasse}\xindex{Hom \begin{beweis} Sei $y \in V$ und $x \in p(V)$, sodass $x=p(y)$ gilt. - Sei weiter $U = U_x$ die offene Umgebung von $x$ wie in Definition~\ref{def:12.1} + Sei weiter $U = U_x$ die offene Umgebung von $x$ wie in \cref{def:12.1} und $V_j$ die Komponente von $p^{-1}(U)$, die $y$ enthält. Dann ist $V \cap V_j$ offene Umgebung von $y$. @@ -529,10 +529,10 @@ Haben wir Häufungspunkt definiert?} \begin{bemerkung} % Bemerkung 12.3 der Vorlesung Sei $p: Y \rightarrow X$ Überlagerung, $x \in X$. - \begin{enumerate}[label=\alph*)] + \begin{bemenum} \item $X$ hausdorffsch $\Rightarrow Y$ hausdorffsch \item $p^{-1}(X)$ ist diskret in $Y$ - \end{enumerate} + \end{bemenum} \end{bemerkung} \begin{beweis}\leavevmode @@ -578,7 +578,7 @@ Haben wir Häufungspunkt definiert?} \end{bemerkung} \begin{beweis} - Sei $U$ Umgebung von $x_1$ wie in Definition~\ref{def:12.1}, $x \in U$. + Sei $U$ Umgebung von $x_1$ wie in \cref{def:12.1}, $x \in U$. Dann enthält jedes $V_j, j \in I_X$ genau ein Element von $p^{-1}(x)$ @@ -595,7 +595,7 @@ Haben wir Häufungspunkt definiert?} \textbf{Liftung} von $f$, wenn $p \circ \tilde{f} = f$ ist. \end{definition} -\begin{figure} +\begin{figure}[htp] \centering \resizebox{0.95\linewidth}{!}{\input{figures/liftung-torus-r.tex}} \caption{Beim Liften eines Weges bleiben geschlossene Wege im allgemeinen nicht geschlossen} @@ -609,7 +609,7 @@ Haben wir Häufungspunkt definiert?} $\exists z_0 \in Z: f_0(z) = f_1(z) \Rightarrow f_0 = f_1$ \end{bemerkung} -\begin{figure} +\begin{figure}[htp] \centering \input{figures/commutative-diagram-2.tex} \caption{Situation aus \cref{kor:12.5}} @@ -621,7 +621,7 @@ Haben wir Häufungspunkt definiert?} \underline{Z.~Z.}: $T$ ist offen und $Z \setminus T$ ist auch offen. - Sei $z \in T, x = f(z), U$ Umgebung von $x$ wie in Definition~\ref{def:12.1}, + Sei $z \in T, x = f(z), U$ Umgebung von $x$ wie in \cref{def:12.1}, $V$ die Komponente von $p^{-1}(U)$, die $y:=f_0(z) = f_1(z)$.\todo{deutsch?} Sei $q:U \rightarrow V$ die Umkehrabbildung zu $p|_V$. @@ -648,10 +648,10 @@ Haben wir Häufungspunkt definiert?} \begin{beweis} Existenz: Siehe \Cref{fig:satz-12.6}. - \begin{figure} + \begin{figure}[htp] \centering \includegraphics[width=0.6\linewidth, keepaspectratio]{figures/todo/skizze-1.jpg} - \caption{Skizze für den Beweis von Satz~\ref{thm:ueberlagerung-weg-satz-12.6}} + \caption{Skizze für den Beweis von \cref{thm:ueberlagerung-weg-satz-12.6}} \label{fig:satz-12.6} \end{figure} \end{beweis} @@ -709,10 +709,10 @@ $p|V_j: V_j \rightarrow U$ Homöomorphismus. \begin{bemerkung}%In Vorlesung: "Folgerung 12.8" Sei $p: Y \rightarrow X$ eine Überlagerung, $x_0 \in X, y_0 \in p^{-1}(x_0)$ - \begin{enumerate}[label=\alph*),ref=\theplaindefinition.\alph*] + \begin{bemenum} \item $p_1: \pi_1(Y, y_0) \rightarrow \pi_1(X, x_0)$ ist injektiv\label{kor:12.8a} \item $[\pi_1(X, x_0): p_* (\pi_1(Y, y_0))] = \deg(p)$\label{kor:12.8b} - \end{enumerate} + \end{bemenum} \end{bemerkung} \begin{beweis}\leavevmode @@ -720,7 +720,7 @@ $p|V_j: V_j \rightarrow U$ Homöomorphismus. \item Sei $\tilde{\gamma}$ ein Weg in $Y$ um $y_0$ und $p_* ([\tilde{\gamma}]) = e$, also $p \circ \tilde{\gamma} \sim \gamma_{x_0}$ - Nach Proposition~\ref{proposition:12.7} ist dann + Nach \cref{proposition:12.7} ist dann $\tilde{\gamma}$ homotop zum Lift des konstanten Wegs $\gamma_{x_0}$ mit Anfangspunkt $y_0$, also zu $\gamma_{y_0} \Rightarrow [\tilde{\gamma}] = e$ @@ -754,8 +754,8 @@ $p|V_j: V_j \rightarrow U$ Homöomorphismus. \end{bemerkung} \begin{beweis} - Wegen Folgerung~\ref{kor:12.8a} ist auch $Y$ einfach zusammenhängend - und wegen Folgerung~\ref{kor:12.8b} ist $\deg(p)=1$, $p$ ist also + Wegen \cref{kor:12.8a} ist auch $Y$ einfach zusammenhängend + und wegen \cref{kor:12.8b} ist $\deg(p)=1$, $p$ ist also bijektiv. Nach \todo{Was ist das?}{12.2} ist $p$ offen $\Rightarrow p^{-1}$ @@ -806,7 +806,7 @@ $p|V_j: V_j \rightarrow U$ Homöomorphismus. $\xRightarrow{q \text{ offen}} q(W)$ ist offene Umgebung von $p(z) \cdot d(\tilde{p}(z))$. - Sei $U \subseteq q(W)$ offen wie in Definition~\ref{def:12.1} und + Sei $U \subseteq q(W)$ offen wie in \cref{def:12.1} und $V \subseteq q^{-1}(U)$ die \todo{Was?}{Komp.} die $\tilde{p}(z)$ enthält. @@ -836,7 +836,7 @@ $p|V_j: V_j \rightarrow U$ Homöomorphismus. $p(\tilde{x_0}) = x_0$ und $\tilde{y_0} \in q^{-1}(x_0) \subseteq \tilde{Y}$. - Nach Satz~\ref{thm:12.11} gibt es genau eine Überlagerung + Nach \cref{thm:12.11} gibt es genau eine Überlagerung \[f:\tilde{X} \rightarrow \tilde{Y} \text{ mit } f(x_0) = \tilde{Y_0} \text{ und } q \circ f = p\] und genau eine Überlagerung \[g: \tilde{Y} \rightarrow \tilde{X} \text{ mit } g(\tilde{y_0}) = \tilde{x_0} \text{ und } p \circ g = q\] @@ -887,7 +887,7 @@ der folgende Satz: \underline{Annahme}: $[\tilde{\gamma}] \neq e$ - Mit Folgerung~\ref{kor:12.8a} folgt dann: $[\gamma] \neq e$. + Mit \cref{kor:12.8a} folgt dann: $[\gamma] \neq e$. Dann ist der Lift von $\gamma$ nach $\tilde{x}$ mit Anfangspunkt $\tilde{x_0}$ ein Weg von $\tilde{x_0}$ nach $(x_0, [\gamma])$. @@ -905,7 +905,7 @@ der folgende Satz: \end{definition} \begin{bemerkung}%In Vorlesung:12.14 - \begin{enumerate}[label=\alph*),ref=\theplaindefinition.\alph*] + \begin{bemenum} \item Die Decktransformationen von $p$ bilden eine Gruppe, die sog. \textbf{Decktransformationsgruppe}\xindex{Decktransformationsgruppe} $\Deck(p) = \Deck(Y/X) = \Deck(Y \rightarrow X)$ @@ -915,7 +915,7 @@ der folgende Satz: \item Ist $p$ eine reguläre Decktransformation, dann gilt: $\forall x \in X: \Deck(Y/X)$ operiert transitiv auf der Menge der Urbilder $p^{-1}(x)$. - \end{enumerate} + \end{bemenum} \end{bemerkung} \begin{beweis}\leavevmode @@ -935,7 +935,7 @@ der folgende Satz: $\Delta \subseteq Y \times Y$ unter der stetigen Abbildung $y \mapsto (f(y),y)$. Außerdem ist $\Fix(f)$ offen, denn ist $y \in \Fix(f)$, so sei $U$ eine - Umgebung von $p(y) \in X$ wie in Definition~\ref{def:12.1} + Umgebung von $p(y) \in X$ wie in \cref{def:12.1} und $U \subseteq p^{-1}(U)$ die Komponente, die $y$ enthält; also $p:V \rightarrow U$ ein Homöomorphismus. Dann ist $W := f^{-1}(V) \cap V$ offene Umgebung von $y$. @@ -952,16 +952,16 @@ der folgende Satz: Zu $i \in \Set{0, \dots, d-1}$ gibt es höchstens ein $f \in \Deck(Y/X)$ mit $f(y_0) = y_1$, denn ist $f(y_0) = g(y_0)$, so ist \todo{Was steht hier?}{$(g^{-1} - f) y_0 = y_0$}, - also nach \ref{kor:12.14c} $g^{-1} \circ f = \id_Y$. + also nach \cref{kor:12.14c} $g^{-1} \circ f = \id_Y$. \end{enumerate} \end{beweis} \begin{beispiel} - \begin{enumerate}[label=\arabic*)] + \begin{bspenum} \item $p: \mdr \rightarrow S^1: \Deck(\mdr / S^1) = \Set{t \mapsto t + n | n \in \mdz} \cong \mdz$ \item $p: \mdr^2 \rightarrow T^2: \Deck(\mdr^2 / T^2) \cong \mdz \times \mdz = \mdz^2$ \item $p: S^n \rightarrow \praum^n(\mdr): \Deck(g^n / \praum^n(\mdr)) = \Set{x \mapsto \pm x} \cong \mdz / 2 \mdz$ - \end{enumerate} + \end{bspenum} \end{beispiel} Nun werden wir eine Verbindung zwischen der Decktransformationsgruppe @@ -985,8 +985,8 @@ und der Fundamentalgruppe herstellen: $f, g \in \Deck(\tilde{X}/ X) \Rightarrow \gamma_{g \circ f} = \gamma_g * g(\gamma_f)$ $\Rightarrow p(\gamma_{g \circ f}) = p(\gamma_g) * \underbrace{(p \circ g)}_{=p} (\gamma_f) = \rho(g) \neq \rho(f)$ \item \underline{$\rho$ ist injektiv}: $\rho(f) = e \Rightarrow p (\gamma_f) \sim \gamma_{x_0}$ - $\xRightarrow{\ref{thm:ueberlagerung-weg-satz-12.6}} \gamma_f \sim \gamma_{\tilde{x_0}}$ - $\Rightarrow f(x_0) = \tilde{x_0} \xRightarrow{\ref{kor:12.14c}} f = \id_{\tilde{x}}$. + $\xRightarrow{\cref{thm:ueberlagerung-weg-satz-12.6}} \gamma_f \sim \gamma_{\tilde{x_0}}$ + $\Rightarrow f(x_0) = \tilde{x_0} \xRightarrow{\text{Bem. }\cref{kor:12.14c}} f = \id_{\tilde{x}}$. \item \underline{$\rho$ ist surjektiv}: Sei $[\gamma] \in \pi_1(X, x_0)$, $\tilde{\gamma}$ Lift von $\gamma$ nach $\tilde{x}$ mit Anfangspunkt $\tilde{x_0}$. Der Endpunkt von $\tilde{\gamma}$ @@ -994,7 +994,7 @@ und der Fundamentalgruppe herstellen: \underline{$p$ ist reguläre Überlagerung}: Seien $\tilde{x_0}, \tilde{x_1} \in \tilde{X}$ mit - $p(\tilde{x_0}) = p(\tilde{x_1})$. Nach Satz~\ref{thm:12.11} + $p(\tilde{x_0}) = p(\tilde{x_1})$. Nach \cref{thm:12.11} gibt es genau eine Überlagerung $\tilde{p}: \tilde{X} \rightarrow X$ mit $p=p \circ \tilde{p}$ und $\tilde{p}(\tilde{x_0}) = \tilde{x_1}$. Somit ist $\tilde{p}$ eine Decktransformation und damit @@ -1023,9 +1023,9 @@ und der Fundamentalgruppe herstellen: \[(\cos(2 \pi g(0)), \sin(2 \pi g(0))) = (p \circ g)(0) = p(0) = (1,0)\] Es existiert $n \in \mdz$ mit $g(0) = n$. Da auch $f_n(0) = 0 + n = n$ - gilt, folgt mit Folgerung~\ref{kor:12.14c} $g = f_n$. Damit folgt: + gilt, folgt mit \cref{kor:12.14c} $g = f_n$. Damit folgt: \[\Deck(\mdr/S^1) = \Set{f_n | n \in \mdz} \cong \mdz\] - Nach Satz~\ref{thm:12.15} also $\pi_1(S^1) \cong \Deck(\mdr/S^1) \cong \mdz$ + Nach \cref{thm:12.15} also $\pi_1(S^1) \cong \Deck(\mdr/S^1) \cong \mdz$ \end{beispiel} \index{Ueberlagerung@""Uberlagerung|)} @@ -1042,10 +1042,10 @@ und der Fundamentalgruppe herstellen: \[ \circ: G \times X \rightarrow X,\;\;\; (g,x) \mapsto g \cdot x,\] für die gilt: - \begin{enumerate}[label=(\roman*),ref=\theplaindefinition.\roman*] + \begin{defenum} \item $1_G \circ x = x \;\;\; \forall x \in X$\label{def:gruppenoperation.1} \item $(g \cdot h) \circ x = g \circ (h \circ x) \;\;\; \forall g,h \in G \forall x \in X$\label{def:gruppenoperation.2} - \end{enumerate} + \end{defenum} \end{definition} \begin{beispiel} @@ -1068,7 +1068,7 @@ und der Fundamentalgruppe herstellen: Sei $G$ eine Gruppe, $X$ ein topologischer Raum und $\circ: G \times X \rightarrow X$ eine Gruppenoperation. - \begin{enumerate}[label=\alph*)] + \begin{defenum} \item $G$ operiert durch Homomorphismen, wenn für jedes $g \in G$ die Abbildung \[m_g: X \rightarrow X, x \mapsto g \cdot X\] @@ -1076,7 +1076,7 @@ und der Fundamentalgruppe herstellen: \item Ist $G$ eine topologische Gruppe, so heißt die Gruppenoperation $\circ$ \textbf{stetig}\xindex{Gruppenoperation!stetige}, wenn $\circ: G \times X \rightarrow X$ stetig ist. - \end{enumerate} + \end{defenum} \end{definition} \begin{bemerkung}%In Vorlesung: Bemerkung 13.2 @@ -1090,9 +1090,9 @@ und der Fundamentalgruppe herstellen: (m_{g^{-1}} \circ m_g)(x) &= m_{g^{-1}} (m_g (x))\\ &= m_{g^{-1}} (g \circ x)\\ &= g^{-1} \circ (g \circ x)\\ - &\stackrel{\ref{def:gruppenoperation.2}}{=} (g^{-1} \cdot g) \circ x\\ + &\overset{\mathclap{\cref{def:gruppenoperation.2}}}{=} (g^{-1} \cdot g) \circ x\\ &= 1_G \circ x\\ - &\stackrel{\ref{def:gruppenoperation.1}}{=} x + &\overset{\mathclap{\cref{def:gruppenoperation.1}}}{=} x \end{align*} \end{beweis} @@ -1103,13 +1103,13 @@ und der Fundamentalgruppe herstellen: \begin{bemerkung}\label{kor:13.3}%In Vorlesung: Bemerkung 13.3 Sei $G$ eine Gruppe und $X$ eine Menge. - \begin{enumerate}[label=\alph*)] + \begin{bemenum} \item Die Gruppenoperation von $G$ auf $X$ entsprechen bijektiv den Gruppenhomomorphismen $\varrho: G \rightarrow \Perm(X) = \Sym(X) = \Set{f: X \rightarrow X | f \text{ ist bijektiv}}$ \item Ist $X$ ein topologischer Raum, so entsprechen dabei die Gruppenoperationen durch Homöomorphismus den Gruppenhomomorphismen $G \rightarrow \Homoo(X)$ - \end{enumerate} + \end{bemenum} \end{bemerkung} \begin{beweis} @@ -1123,16 +1123,16 @@ und der Fundamentalgruppe herstellen: Umgekehrt: Sei $\varrho: G \rightarrow \Perm(X)$ Gruppenhomomorphismus. Definiere $\circ: G \times X \rightarrow X$ durch $g \circ x = \varrho (g)(x)$. - z.~Z. \ref{def:gruppenoperation.2}: + z.~Z. \cref{def:gruppenoperation.2}: \begin{align*} g_1 \circ (g_2 \circ x) &= \varrho (g_1) (g_2 \circ x)\\ &= \varrho(g_1) (\varrho(g_2)(x))\\ &= (\varrho(g_1) \circ \varrho(g_2))(x)\\ - &\stackrel{\varrho \text {ist Hom.}}{=} \varrho(g_1 \cdot g_2) (x)\\ + &\overset{\varrho \text {ist Hom.}}{=} \varrho(g_1 \cdot g_2) (x)\\ &= (g_1 \cdot g_2) \circ x \end{align*} - z.~Z. \ref{def:gruppenoperation.1}: + z.~Z. \cref{def:gruppenoperation.1}: $1_G \cdot x = \varrho(1_G)(x) = \id_X(x) = x$, weil $\varrho$ Homomorphismus ist. \end{beweis} @@ -1169,7 +1169,7 @@ und der Fundamentalgruppe herstellen: \textbf{Erinnerung}:% In Vorlesung: Erinnerung 13.5 Die Konstruktion aus \cref{kor:13.3} induziert zu der Gruppenoperation $\pi_1(X, x_0)$ aus \cref{bsp:13.4} einen Gruppenhomomorphismus -$\varrho: \pi_1(X, x_0) \rightarrow \Homoo(X)$. Nach Satz~\ref{thm:12.15} +$\varrho: \pi_1(X, x_0) \rightarrow \Homoo(X)$. Nach \cref{thm:12.15} ist \begin{align*}\varrho(\pi_1(X, x_0)) &= \Deck(\tilde{X} / X)\\ &= \Set{f: \tilde{X} \rightarrow \tilde{X} \text{ Homöomorphismus} | p \circ f = p} \end{align*} diff --git a/documents/GeoTopo/Kapitel4-UB.tex b/documents/GeoTopo/Kapitel4-UB.tex index 5ffc4be..9ffbb5e 100644 --- a/documents/GeoTopo/Kapitel4-UB.tex +++ b/documents/GeoTopo/Kapitel4-UB.tex @@ -11,10 +11,10 @@ Winkel. Zeigen Sie: - \begin{enumerate}[label=(\alph*)] + \begin{aufgabeenum} \item Die beiden Nebenwinkel von $\angle PQR$ sind gleich. \item Der Winkel $\angle PQR$ ist gleich seinem Scheitelwinkel. - \end{enumerate} + \end{aufgabeenum} \end{aufgabe} \begin{aufgabe}\label{ub11:aufg3} @@ -23,7 +23,7 @@ definiert durch $d(P, Y) := \inf{d(P, y) | y \in Y}$. Zeigen Sie: - \begin{enumerate}[label=(\alph*),ref=\theenumi{} (\alph*)] + \begin{aufgabeenum} \item Ist $\triangle ABC$ ein Dreieck, in dem die Seiten $\overline{AB}$ und $\overline{AC}$ kongruent sind, so sind die Winkel $\angle ABC$ und $\angle BCA$ gleich. @@ -35,7 +35,7 @@ $g$ im rechten Winkel schneidet. Diese Grade heißt \textit{Lot}\xindex{Lot} von $P$ auf $g$ und der Schnittpunkt des Lots mit $g$ heißt \textit{Lotfußpunkt}\xindex{Lotfußpunkt}. - \end{enumerate} + \end{aufgabeenum} \end{aufgabe} \begin{aufgabe}\label{ub-tut-24:a1} diff --git a/documents/GeoTopo/Kapitel4.tex b/documents/GeoTopo/Kapitel4.tex index f638b94..6ec7737 100644 --- a/documents/GeoTopo/Kapitel4.tex +++ b/documents/GeoTopo/Kapitel4.tex @@ -71,7 +71,7 @@ aufgestellt. \end{definition} \begin{definition} - \begin{enumerate}[label=\alph*)] + \begin{defenum} \item $P, Q, R$ liegen \textbf{kollinear}\xindex{kollinear}, wenn es $g \in G$ gibt mit $\Set{P, Q, R} \subseteq g$. \item $Q$ \textbf{liegt zwischen}\xindex{liegt zwischen} $P$ @@ -80,7 +80,7 @@ aufgestellt. \item \textbf{Halbgeraden}\xindex{Halbgerade}:\\ $PR^+ := \Set{Q \in X | Q \text{ liegt zwischen } P \text{ und } R \text{ oder } R \text{ liegt zwischen } P \text{ und } Q}$\\ $PR^- := \Set{Q \in X | P \text{ liegt zwischen } Q \text{ und } R}$\\ - \end{enumerate} + \end{defenum} \end{definition} \begin{figure}[htp] @@ -91,10 +91,10 @@ aufgestellt. \end{figure} \begin{bemerkung} - \begin{enumerate}[label=(\roman*)] + \begin{bemenum} \item $PR^+ \cup PR^- = PR$ \item $PR^+ \cap PR^- = \Set{P}$ - \end{enumerate} + \end{bemenum} \end{bemerkung} \begin{beweis}\leavevmode @@ -266,12 +266,12 @@ schneiden sich. \item $B$ liegt zwischen $P$ und $C$. $d(P,A) + d(A, C) > d(P,C) = d(P,B) + d(B,c) = d(P,A) + d(B,C)$ - $\Rightarrow d(A,c) > d(B,C) \Rightarrow$ Widerspruch zu \ref{enum:komischer-beweis-i} + $\Rightarrow d(A,c) > d(B,C) \Rightarrow$ Widerspruch zu \cref{enum:komischer-beweis-i} \item $C$ liegt zwischen $P$ und $B$ $d(P,C) + d(C,A) > d(P,A) = d(P,B) = d(P,C) + d(C, B)$\\ $\Rightarrow d(C, A) > d(C, B)$\\ - $\Rightarrow$ Widerspruch zu \ref{enum:komischer-beweis-i} + $\Rightarrow$ Widerspruch zu \cref{enum:komischer-beweis-i} \end{enumerate} \end{enumerate} @@ -438,7 +438,7 @@ schneiden sich. \end{figure} \begin{definition}\label{def:14.8}%In Vorlesung: 14.8 - \begin{enumerate}[label=\alph*),ref=\alph*] + \begin{defenum} \item \label{def:14.8a} Ein \textbf{Winkel}\xindex{Winkel} ist ein Punkt $P \in X$ zusammen mit $2$ Halbgeraden mit Anfangspunkt $P$.\\ Man schreibt: $\angle R_1 P R_2$ bzw. $\angle R_2 P R_1$\footnote{Für dieses Skript gilt: $\angle R_1 P R_2 = \angle R_2 P R_1$. Also sind insbesondere alle Winkel $ \leq 180^\circ$.} @@ -452,7 +452,7 @@ schneiden sich. bzgl. $PR_2$ wie $R_1$ \item \label{def:14.8d} Im Dreieck $\triangle PQR$ gibt es Innenwinkel und Außenwinkel. - \end{enumerate} + \end{defenum} \end{definition} \begin{figure}[ht] @@ -580,7 +580,7 @@ Sei im Folgenden \enquote{IWS} die \enquote{Innenwinkelsumme}. gleich $\pi$. \end{bemerkung} -\begin{figure} +\begin{figure}[htp] \centering \input{figures/triangle-2.tex} \caption{Situation aus \cref{bem:14.12}} @@ -604,10 +604,10 @@ Sei im Folgenden \enquote{IWS} die \enquote{Innenwinkelsumme}. In ähnlichen Dreiecken sind Verhältnisse entsprechender Seiten gleich. \end{satz} -\begin{figure} +\begin{figure}[htp] \centering - \input{figures/todo.tex} - \caption{Bild 2} + \input{figures/hyberbolische-geometrie-2.tex} + \caption{Strahlensatz} \label{fig:bild-2} \end{figure} @@ -615,7 +615,7 @@ Sei im Folgenden \enquote{IWS} die \enquote{Innenwinkelsumme}. TODO \end{beweis} -\begin{figure} +\begin{figure}[htp] \centering \input{figures/triangle-similar.tex} \caption{Die Dreiecke $\triangle ABC$ und $\triangle AB'C'$ sind ähnlich.} @@ -645,7 +645,7 @@ Sei im Folgenden \enquote{IWS} die \enquote{Innenwinkelsumme}. Der Flächeninhalt eines Dreiecks ist $\nicefrac{1}{2} \cdot \text{Grundseite} \cdot \text{Höhe}$. -\begin{figure} +\begin{figure}[htp] \centering \input{figures/todo.tex} \caption{Flächenberechnung im Dreiecks} @@ -654,7 +654,7 @@ Der Flächeninhalt eines Dreiecks ist $\nicefrac{1}{2} \cdot \text{Grundseite} \ \underline{Zu zeigen:} Unabhängigkeit von der gewählten Grundseite. -\begin{figure} +\begin{figure}[htp] \centering \input{figures/triangle-3.tex} \caption{$\triangle ABL_a$ und $\triangle C{L_C}B$ sind ähnlich, weil $\IWS = \pi$} @@ -774,15 +774,15 @@ $\overset{\text{Strahlensatz}}{\Rightarrow} \frac{a}{h_c} = \frac{c}{h_a} \right \begin{bemerkung}[Eigenschaften der hyperbolischen Geraden] Die hyperbolischen Geraden erfüllen\dots - \begin{enumerate}[label=\alph*)] + \begin{bemenum} \item \dots die Inzidenzaxiome \ref{axiom:1} \item \dots das Anordnungsaxiom \ref{axiom:3.2} \item \dots nicht das Parallelenaxiom \ref{axiom:5} - \end{enumerate} + \end{bemenum} \end{bemerkung} \begin{beweis}\leavevmode - \begin{enumerate}[label=\alph*)] + \begin{enumerate}[label=\alph*), ref=\theproposition (\alph*)] \item Offensichtlich sind \ref{axiom:1.3} und \ref{axiom:1.2} erfüllt. Für \ref{axiom:1.1} gilt:\\ Gegeben $z_1, z_2 \in \mdh$\\ @@ -815,7 +815,7 @@ $\overset{\text{Strahlensatz}}{\Rightarrow} \frac{a}{h_c} = \frac{c}{h_a} \right \end{beweis} \begin{proposition}%In Vorlesung: Proposition 15.2 - \begin{enumerate}[label=\alph*)] + \begin{propenum} \item Die Gruppe $\SL_2(\mdr)$ operiert auf $\mdh$ durch \[\sigma(z):= \begin{pmatrix}a & b\\c & d\end{pmatrix} \circ z := \frac{az + b}{cz + d}\] \item Es ist $\sigma(z) = (-\sigma)(z)$ für alle $\sigma \in \SL_2(\mdr)$ @@ -831,8 +831,8 @@ $\overset{\text{Strahlensatz}}{\Rightarrow} \frac{a}{h_c} = \frac{c}{h_a} \right \begin{pmatrix}1 & a\\ 0 & 1\end{pmatrix}, a \in \mdr\;\;\; \begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix}\] erzeugt - \item $\PSL_2(\mdr)$ operiert auf $G$ - \end{enumerate} + \item \label{prop:15.2e} $\PSL_2(\mdr)$ operiert auf $G$ + \end{propenum} \end{proposition} \begin{beweis}\leavevmode @@ -862,7 +862,9 @@ $\overset{\text{Strahlensatz}}{\Rightarrow} \frac{a}{h_c} = \frac{c}{h_a} \right \item Es genügt die Aussage für Matrizen aus \cref{prop:15.2d} zu zeigen. \begin{itemize} - \item $\sigma = \begin{pmatrix}\lambda & 0\\ 0 & \lambda^{-1}\end{pmatrix}$, also $\sigma(z) = \lambda^2 z$ + \item $\sigma = \begin{pmatrix}\lambda & 0\\ 0 & \lambda^{-1}\end{pmatrix}$, also $\sigma(z) = \lambda^2 z$. + Daraus ergeben sich die Situationen, die in \cref{fig:prop15.2.e.fall1.1} und + \cref{fig:prop15.2.e.fall1.2} dargestellt sind. \begin{figure}[ht] \centering \subfloat[Fall 1]{ @@ -871,10 +873,10 @@ $\overset{\text{Strahlensatz}}{\Rightarrow} \frac{a}{h_c} = \frac{c}{h_a} \right }% \subfloat[Fall 2 (Strahlensatz)]{ \resizebox{0.45\linewidth}{!}{\input{figures/hyberbolische-geometrie-2.tex}} - \label{fig:prop15.2.e.fall1.1} + \label{fig:prop15.2.e.fall1.2} }% \label{fig:prop15.2.e.fall1.0} - \caption{TODO} + \caption{Beweis von \cref{prop:15.2e} für eine Diagonalmatrix} \end{figure} \item Offensichtlich gilt die Aussage für $\sigma = \begin{pmatrix}1 & a\\0 & 1\end{pmatrix}$ \item Sei nun $\sigma = \begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix}$, also $\sigma(z) = - \frac{1}{z}$ diff --git a/documents/GeoTopo/Loesungen.tex b/documents/GeoTopo/Loesungen.tex index a627a6c..6e2bfbf 100644 --- a/documents/GeoTopo/Loesungen.tex +++ b/documents/GeoTopo/Loesungen.tex @@ -71,7 +71,7 @@ \begin{beweis} Nach Definition der Produkttopologie bilden Mengen der Form - \[\prod_{i \in J} U_j \times \prod_{\stackrel{i \in \mdn}{i \notin J}} P_i, \text{ wobei } J \subseteq \mdn \text{ endlich und } U_j \subseteq P_j \text{offen } \forall{j \in J}\] + \[\prod_{i \in J} U_j \times \prod_{\overset{i \in \mdn}{i \notin J}} P_i, \text{ wobei } J \subseteq \mdn \text{ endlich und } U_j \subseteq P_j \text{offen } \forall{j \in J}\] eine Basis der Topologie. Damit sind die offenen Mengen von $P$ Vereinigungen von Mengen der obigen Form. $\qed$ @@ -102,12 +102,12 @@ \textbf{Bew.:} $\det: \GL_n(\mdr) \rightarrow \mdr \setminus \Set{0}$ ist stetig. Außerdem ist $\det(\GL_n(\mdr)) = \mdr \setminus \Set{0}$ nicht - kompakt. $\stackrel{\ref{kor:5.6}}{\Rightarrow}$ + kompakt. $\overset{\ref{kor:5.6}}{\Rightarrow}$ $\GL_n(\mdr)$ ist nicht kompakt. $\qed$ \item \textbf{Beh.:} $\SL_1(\mdr)$ ist nicht kompakt, für $n > 1$ ist $\SL_n(\mdr)$ kompakt.\\ \textbf{Bew.:} Für $\SL_1(\mdr)$ gilt: $\SL_1(\mdr) = \Set{A \in \mdr^{1 \times 1} | \det A = 1} = \begin{pmatrix}1\end{pmatrix} \cong \Set{1}$. - $\stackrel{\ref{kor:5.6}}{\Rightarrow} \SL_1(\mdr)$ ist + $\overset{\ref{kor:5.6}}{\Rightarrow} \SL_1(\mdr)$ ist kompakt.\\ $\SL_n(\mdr) \subseteq \GL_n(\mdr)$ lässt sich mit einer @@ -120,7 +120,7 @@ \textbf{Bew.:} $\praum(\mdr) \cong S^n/_{x \sim -x}$. Per Definition der Quotiententopologie ist die Klassenabbildung stetig. Da $S^n$ als abgeschlossene und beschränkte Teilmenge - des $\mdr^{n+1}$ kompakt ist $\stackrel{\ref{kor:5.6}}{\Rightarrow}$ + des $\mdr^{n+1}$ kompakt ist $\overset{\ref{kor:5.6}}{\Rightarrow}$ $\praum(\mdr)$ ist kompakt. $\qed$ \end{enumerate} \end{solution} diff --git a/documents/GeoTopo/Readme.md b/documents/GeoTopo/Readme.md index 71fba12..0750030 100644 --- a/documents/GeoTopo/Readme.md +++ b/documents/GeoTopo/Readme.md @@ -49,6 +49,8 @@ Was noch kommen soll ==================== 1. Alle `TODOS` auflösen + * "Punkt" suchen + * Checken, ob alle Seitenumbrüche / Bildgrößen stimmen 2. Reviews (Mathematik, LaTeX und Bilder) 3. A5-Version drucken * In `GeoTopo.tex`... diff --git a/documents/GeoTopo/definitions/shortcuts.sty b/documents/GeoTopo/definitions/shortcuts.sty index cb4f670..05a2e17 100644 --- a/documents/GeoTopo/definitions/shortcuts.sty +++ b/documents/GeoTopo/definitions/shortcuts.sty @@ -42,9 +42,7 @@ \newframedtheorem{proposition}[satz]{Proposition} \newtheorem{korollar}[satz]{Korollar} \newtheorem{folgerung}[satz]{Folgerung} -\newtheorem{plaindefinition}{Definition} -\newenvironment{definition}{\begin{plaindefinition}}{\end{plaindefinition}} -\newenvironment{definition*}{\begin{plaindefinition*}}{\end{plaindefinition*}} +\newtheorem{definition}{Definition} \newtheorem{beispiel}{Beispiel} \newtheorem{bemerkung}{Bemerkung} \theoremstyle{nonumberplain} @@ -64,6 +62,7 @@ \def\mdq{\ensuremath{\mathbb{Q}}} \def\mdz{\ensuremath{\mathbb{Z}}} \def\mdn{\ensuremath{\mathbb{N}}} +\def\mdh{\ensuremath{\mathbb{H}}} \def\gdw{\ensuremath{\Leftrightarrow}} \def\atlas{\ensuremath{\mathcal{A}}} @@ -71,6 +70,7 @@ \def\GL{\ensuremath{\mathrm{GL}}} \def\SL{\ensuremath{\mathrm{SL}}} +\def\PSL{\ensuremath{\mathrm{PSL}}} \newcommand\mapsfrom{\mathrel{\reflectbox{\ensuremath{\mapsto}}}} \newcommand\dcup{\mathbin{\dot{\cup}}} \newcommand{\id}{\textnormal{id}} @@ -84,6 +84,9 @@ \DeclareMathOperator{\Diffeo}{Diffeo} \DeclareMathOperator{\conv}{conv} \DeclareMathOperator{\IWS}{IWS} +\newcommand{\iu}{{i\mkern1mu}} % imaginary unit +%\DeclareMathOperator{\Re}{Re} +%\DeclareMathOperator{\Im}{Im} %%%Text %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand\obda{o.~B.~d.~A.\xspace} @@ -99,7 +102,6 @@ \crefname{korollar}{Korollar}{Korollare} \crefname{folgerung}{Folgerung}{Folgerungen} \crefname{definition}{Definition}{Definitionen} -\crefname{plaindefinition}{Definition}{Definitionen} \crefname{behauptung}{Behauptung}{Behauptungen} \crefname{beispiel}{Beispiel}{Beispiele} \crefname{aufgabe}{Aufgabe}{Aufgaben} diff --git a/documents/GeoTopo/shortcuts.sty b/documents/GeoTopo/shortcuts.sty index 5a2bef7..094858b 100644 --- a/documents/GeoTopo/shortcuts.sty +++ b/documents/GeoTopo/shortcuts.sty @@ -42,9 +42,7 @@ \newframedtheorem{proposition}[satz]{Proposition} \newtheorem{korollar}[satz]{Korollar} \newtheorem{folgerung}[satz]{Folgerung} -\newtheorem{plaindefinition}{Definition} -\newenvironment{definition}{\begin{plaindefinition}}{\end{plaindefinition}} -\newenvironment{definition*}{\begin{plaindefinition*}}{\end{plaindefinition*}} +\newtheorem{definition}{Definition} \newtheorem{beispiel}{Beispiel} \newtheorem{bemerkung}{Bemerkung} \theoremstyle{nonumberplain} @@ -104,10 +102,29 @@ \crefname{korollar}{Korollar}{Korollare} \crefname{folgerung}{Folgerung}{Folgerungen} \crefname{definition}{Definition}{Definitionen} -\crefname{plaindefinition}{Definition}{Definitionen} \crefname{behauptung}{Behauptung}{Behauptungen} \crefname{beispiel}{Beispiel}{Beispiele} \crefname{aufgabe}{Aufgabe}{Aufgaben} \crefname{bemerkung}{Bemerkung}{Bemerkungen} %\let\OldAngle\angle %\let\angle\sphericalangle + +\newlist{defenum}{enumerate}{1} +\setlist[defenum]{label=\alph*),ref=\textup{\thedefinition.\alph*}} +\crefalias{defenumi}{definition} + +\newlist{bemenum}{enumerate}{1} +\setlist[bemenum]{label=\alph*),ref=\textup{\thebemerkung.\alph*}} +\crefalias{bemenumi}{bemerkung} + +\newlist{bspenum}{enumerate}{1} +\setlist[bspenum]{label=\arabic*),ref=\textup{\thebeispiel.\arabic*}} +\crefalias{bspenumi}{beispiel} + +\newlist{propenum}{enumerate}{1} +\setlist[propenum]{label=\alph*), ref=\textup{\theproposition~(\alph*)}} +\crefalias{propenumi}{proposition} + +\newlist{aufgabeenum}{enumerate}{1} +\setlist[aufgabeenum]{label=(\alph*),ref=\textup{\theaufgabe~(\alph*)}} +\crefalias{aufgabeenumi}{aufgabe}