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Viele kleine Verbesserungen
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Martin Thoma
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9 changed files with 214 additions and 21 deletions
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@ -587,7 +587,7 @@ sodass $\pi$ stetig wird.
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\end{enumerate}
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\end{korollar}
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\begin{beweis}
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\begin{beweis}\leavevmode
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\begin{enumerate}[label=\alph*)]
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\item Sei $Z(x) = A_1 \cup A_2$ mit $A_i \neq \emptyset$ abgeschlossen,
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disjunkt.
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@ -162,8 +162,7 @@ U_i = \Set{(x_0: \dots : x_n) \in \mdp^n(\mdr) | x_i \neq 0} &\rightarrow \mdr^n
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\end{enumerate}
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\end{korollar}
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\begin{beweis}
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von a und b:
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\begin{beweis}\leavevmode
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\begin{enumerate}[label=\alph*),ref=\theplaindefinition.\alph*]
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\item Sei $y \in \mdr^n \setminus V(F)$. Weil $F$ stetig ist,
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gibt es $\delta > 0$, sodass $F(\fB_\delta(y)) \subseteq \fB_\varepsilon(F(y))$
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@ -321,7 +320,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$.
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\item $f$ heißt \textbf{differenzierbar}
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(von Klasse $C^k$), wenn $f$ in jedem $x \in X$
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differenzierbar ist.
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\item $f$ heißt \textbf{Diffieomorphismus}\xindex{Diffieomorphismus},
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\item $f$ heißt \textbf{Diffeomorphismus}\xindex{Diffeomorphismus},
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wenn $f$ differenzierbar von Klasse $C^\infty$ ist und
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es eine differenzierbare Abbildung $g: Y \rightarrow X$
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von Klasse $C^\infty$ gibt mit $g \circ f = \text{id}_X$
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@ -347,7 +346,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$.
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\begin{beispiel}
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$f: \mdr \rightarrow \mdr, \;\;\; x \mapsto x^3$ ist kein
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Diffieomorphismis, aber Homöomorphismus, da mit $g(x) := \sqrt[3]{x}$
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Diffeomorphismis, aber Homöomorphismus, da mit $g(x) := \sqrt[3]{x}$
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gilt: $f \circ g = \text{id}_\mdr, \;\;\; g \circ f = \text{id}_\text{\mdr}$
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\end{beispiel}
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@ -535,7 +534,7 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$.
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h &\mapsto g \cdot h
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\end{align*}
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ein Diffieomorphismus.
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ein Diffeomorphismus.
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\end{bemerkung}
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\section{Simplizialkomplex}
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@ -618,33 +617,33 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$.
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\begin{figure}[ht]
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\centering
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\subfloat[1D Simplizialkomplex]{
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\parbox{4cm}{\centering\input{figures/topology-1-d-simplizialkomplex}}
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\parbox[c][4cm]{4cm}{\centering\input{figures/topology-1-d-simplizialkomplex}}
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\label{fig:simplizialkomplex-1-d}
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}%
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\subfloat[2D Simplizialkomplex (ohne untere Fläche!)]{
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\parbox{4cm}{\centering\input{figures/topology-pyramid.tex}}
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\parbox[c][4cm]{4cm}{\centering\input{figures/topology-pyramid.tex}}
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\label{fig:simplizialkomplex-2-d}
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}%
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\subfloat[2D Simplizialkomplex]{
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\parbox{5cm}{\centering\input{figures/topology-oktaeder.tex}}
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\parbox[c][4cm]{5cm}{\centering\input{figures/topology-oktaeder.tex}}
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\label{fig:simplizialkomplex-2-d-okateder}
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}%
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\subfloat[1D Simplizialkomplex]{
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\parbox{5cm}{\centering\input{figures/topology-cube.tex}}
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\parbox[c][4cm]{5cm}{\centering\input{figures/topology-cube.tex}}
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\label{fig:simplizialkomplex-cube}
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}%
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\subfloat[2D Simplizialkomplex]{
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\parbox{5cm}{\centering\input{figures/topology-cube-divided.tex}}
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\parbox[c][4cm]{5cm}{\centering\input{figures/topology-cube-divided.tex}}
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\label{fig:simplizialkomplex-cube-divided}
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}
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\subfloat[$P$ ist kein Teilsimplex, da Eigenschaft (ii) verletzt ist]{
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\parbox{5cm}{\centering\input{figures/topology-triangle-no-simplicial-complex.tex}}
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\parbox[c][4cm]{5cm}{\centering\input{figures/topology-triangle-no-simplicial-complex.tex}}
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\label{fig:no-simplizialkomplex-triangles}
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}%
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\subfloat[Simplizialkomplex]{
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\parbox{5cm}{\centering\input{figures/topology-triangle-simplicial-complex.tex}}
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\parbox[c][4cm]{5cm}{\centering\input{figures/topology-triangle-simplicial-complex.tex}}
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\label{fig:simplizialkomplex-triangles}
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}%
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\label{fig:simplizialkomplexe}
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@ -733,20 +732,20 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$.
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\begin{figure}[ht]
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\centering
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\subfloat[Dies wird häufig auch als Multigraph bezeichnet.]{
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\parbox{4cm}{\centering\input{figures/topology-graph-simple.tex}}
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\parbox[c][3cm]{4cm}{\centering\input{figures/topology-graph-simple.tex}}
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\label{fig:topology-graph-simple}
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}%
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\subfloat[Planare Einbettung des Tetraeders]{
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\parbox{4cm}{\centering\input{figures/topology-graph-tetraeder.tex}}
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\parbox[c][3cm]{4cm}{\centering\input{figures/topology-graph-tetraeder.tex}}
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\label{fig:topology-graph-tetraeder}
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}
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\subfloat[$K_5$]{
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\parbox{4cm}{\centering\input{figures/topology-graph-k-5.tex}}
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\parbox[c][3cm]{4cm}{\centering\input{figures/topology-graph-k-5.tex}}
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\label{fig:k-5}
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}%
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\subfloat[$K_{3,3}$]{
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\parbox{4cm}{\centering\input{figures/topology-graph-k-3-3.tex}}
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\parbox[c][3cm]{4cm}{\centering\input{figures/topology-graph-k-3-3.tex}}
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\label{fig:k-3-3}
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}%
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\label{fig:graphen-beispiele}
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@ -827,11 +826,13 @@ $\partial X$ ist eine Mannigfaltigkeit der Dimension $n-1$.
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eine eine Triangulierungen $T$, die sowohl um $T_1$ als
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auch um $T_2$ Verfeinerung ist.
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\todo[inline]{Komische Zeichung}
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\begin{center}
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\input{figures/topology-3.tex}\todo{Was bedeutet diese Zeichnung?}
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\end{center}
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Nach Korollar~\ref{kor:simplex-unterteilung} ist
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$\chi(\partial P_1) = \chi(T_1) = \chi(T) = \chi(T_2) = \chi(\partial P_2) = 2$.
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Weil \obda{} $P_2$ ein Tetraeder ist.
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$\chi(\partial P_1) = \chi(T_1) = \chi(T) = \chi(T_2) = \chi(\partial P_2) = 2$,
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weil \obda{} $P_2$ ein Tetraeder ist.
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\end{enumerate}
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\end{beweis}
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75
documents/GeoTopo/figures/topology-3.tex
Normal file
75
documents/GeoTopo/figures/topology-3.tex
Normal file
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@ -0,0 +1,75 @@
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\newenvironment{customlegend}[1][]{%
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\begingroup
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% inits/clears the lists (which might be populated from previous
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% axes):
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\csname pgfplots@init@cleared@structures\endcsname
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\pgfplotsset{#1}%
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}{%
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% draws the legend:
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\csname pgfplots@createlegend\endcsname
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\endgroup
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}%
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% makes \addlegendimage available (typically only available within an
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% axis environment):
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\def\addlegendimage{\csname pgfplots@addlegendimage\endcsname}
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%%--------------------------------
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% definition to insert numbers
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\pgfkeys{/pgfplots/number in legend/.style={%
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/pgfplots/legend image code/.code={%
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\node at (0.295,-0.0225){#1};
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},%
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},
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}
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\pgfdeclarelayer{background}
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\pgfdeclarelayer{foreground}
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\pgfsetlayers{background,main,foreground}
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\begin{tikzpicture}
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\tikzstyle{point}=[circle,thick,draw=black,fill=black,inner sep=0pt,minimum width=4pt,minimum height=4pt]
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\tikzstyle{smallpoint}=[circle,thick,draw=red,fill=red,inner sep=0pt,minimum width=3pt,minimum height=3pt]
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\begin{pgfonlayer}{foreground}
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\node (a)[point] at (0,0) {};
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\node (b)[point] at (2,0) {};
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\node (c)[point] at (3,0.5) {};
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\node (d)[point] at (0,3) {};
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\node (e)[point] at (2,3) {};
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\node (f)[point] at (3,1.5) {};
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\node (g)[point] at (2,2.5) {};
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\node (h)[point] at (1,1.5) {};
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\end{pgfonlayer}
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\draw (h.center) -- (e.center) -- (f.center) -- (b.center) -- cycle;
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\draw (h.center) -- (f.center);
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\draw[green, densely dashed] (a.center) -- (d.center) -- (g.center) -- (c.center) -- cycle;
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\draw[green, densely dashed] (a.center) -- (g.center);
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\begin{pgfonlayer}{foreground}
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\node (x)[point, red,fill=red] at (1.79,0.31) {};
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\node (x1)[smallpoint] at (1.2,1.5) {};
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\node (x2)[smallpoint] at (1.71,2.56) {};
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\node (x3)[smallpoint] at (2.5,1.5) {};
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\node (x4)[smallpoint] at (2.72,1.06) {};
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\end{pgfonlayer}
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\draw[blue, densely dotted] (x.center) -- (x1.center);
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\draw[blue, densely dotted] (x.center) -- (x2.center);
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\draw[blue, densely dotted] (x.center) -- (x3.center);
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\draw[blue, densely dotted] (x.center) -- (x4.center);
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\begin{customlegend}[
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legend entries={
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$T_1$,
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$T_2$,
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$?$
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},
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legend style={at={(4.5,3.5)},font=\footnotesize}] % <= to define position and font legend
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% the following are the "images" and numbers in the legend
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\addlegendimage{black}
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\addlegendimage{green,densely dashed}
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\addlegendimage{blue, densely dotted}
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\end{customlegend}
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\end{tikzpicture}
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@ -5,7 +5,7 @@
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\node (c)[point] at (3,1) {};
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\node (d)[point] at (1,1) {};
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\node (e)[point] at (1.5,3) {};
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\node (f)[point] at (1.5,-2) {};
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\node (f)[point] at (1.5,-1) {};
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\draw (a.center) -- (b.center) -- (c.center) -- (e.center) -- (b.center);
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\draw (a.center) -- (e.center);
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\draw[dashed] (a.center) -- (d.center) -- (c.center);
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