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Symbolverzeichnis erweitert; Bild hinzugefügt

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Martin Thoma 2013-10-27 11:52:38 +01:00
parent 6535e529de
commit 66c298d738
5 changed files with 111 additions and 3 deletions
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documents/GeoTopo/GeoTopo.pdf
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documents/GeoTopo/GeoTopo.tex
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@ -21,7 +21,7 @@
\usepackage{tikz}
\usepackage{tikz-3dplot}
\usepackage{tkz-fct}
\usetikzlibrary{3d,calc,intersections,er,arrows,positioning,shapes.misc}
\usetikzlibrary{3d,calc,intersections,er,arrows,positioning,shapes.misc,patterns}
\usepackage{shortcuts}
% Setze den richtigen Namen für das Glossar und das Stichwortverzeichnis

9
documents/GeoTopo/Kapitel1.tex
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@ -169,7 +169,7 @@ Auch gibt es Mengen, die sowohl abgeschlossen als auch offen sind.
\begin{definition} \xindex{Quotiententopologie}
Sei $X$ topologischer Raum, $\sim$ eine Äquivalenzrelation auf $X$,
$\overline{X} = X / \sim$ sei die Menge der Äquivalenzklassen,
$\overline{X} = X /_\sim$ sei die Menge der Äquivalenzklassen,
$\pi: x \rightarrow \overline{x}, \;\;\; x \mapsto [x]_\sim$,
$U \subseteq \overline{X}$ heißt offen, wenn $\pi^{-1} (U) \subseteq X$
offen ist. Dadurch wird eine Topologie auf $\overline{X}$ definiert.
@ -282,6 +282,11 @@ Auch gibt es Mengen, die sowohl abgeschlossen als auch offen sind.
\item Jeder Teilraum um $X$ ist Hausdorffsch.
\item $X_1 \times X_2$ ist Hausdorffsch.
\end{enumerate}
\begin{figure}[htp]
\centering
\input{figures/topology-metric-hausdorff}
\caption{Wenn $X_1, X_2$ hausdorffsch sind, dann auch $X_1 \times X_2$}
\end{figure}
\end{bemerkung}
\todo[inline]{TODO: Es fehlt eine \enquote{Beweisskizze}, die den $\mdr^2$ darstellt sowie zwei Punkte $(x_1, y_1)$ und $(x_2, y_2)$ sowie ihre (disjunkten) Umgebungen bzgl. der $X_1$-Achse.}

21
documents/GeoTopo/Symbolverzeichnis.tex
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@ -58,6 +58,13 @@
sort=MengenoperationNSubsetneq
}
\newglossaryentry{setminus}
{
name={\ensuremath{A \setminus B}},
description={$A$ ohne $B$},
sort=MengenoperationNSetminus
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Zahlenmengen %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -113,6 +120,20 @@
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 \|}},

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documents/GeoTopo/figures/topology-metric-hausdorff.tex Normal file
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@ -0,0 +1,82 @@
% defining the new dimensions and parameters
\newlength{\hatchspread}
\newlength{\hatchthickness}
\newlength{\hatchshift}
\newcommand{\hatchcolor}{}
% declaring the keys in tikz
\tikzset{hatchspread/.code={\setlength{\hatchspread}{#1}},
hatchthickness/.code={\setlength{\hatchthickness}{#1}},
hatchshift/.code={\setlength{\hatchshift}{#1}},% must be >= 0
hatchcolor/.code={\renewcommand{\hatchcolor}{#1}}}
% setting the default values
\tikzset{hatchspread=6pt,
hatchthickness=0.4pt,
hatchshift=0pt,% must be >= 0
hatchcolor=black}
% declaring the pattern
\pgfdeclarepatternformonly[\hatchspread,\hatchthickness,\hatchshift,\hatchcolor]% variables
{custom north west lines}% name
{\pgfqpoint{\dimexpr-2\hatchthickness}{\dimexpr-2\hatchthickness}}% lower left corner
{\pgfqpoint{\dimexpr\hatchspread+2\hatchthickness}{\dimexpr\hatchspread+2\hatchthickness}}% upper right corner
{\pgfqpoint{\dimexpr\hatchspread}{\dimexpr\hatchspread}}% tile size
{% shape description
\pgfsetlinewidth{\hatchthickness}
\pgfpathmoveto{\pgfqpoint{0pt}{\dimexpr\hatchspread+\hatchshift}}
\pgfpathlineto{\pgfqpoint{\dimexpr\hatchspread+0.15pt+\hatchshift}{-0.15pt}}
\ifdim \hatchshift > 0pt
\pgfpathmoveto{\pgfqpoint{0pt}{\hatchshift}}
\pgfpathlineto{\pgfqpoint{\dimexpr0.15pt+\hatchshift}{-0.15pt}}
\fi
\pgfsetstrokecolor{\hatchcolor}
% \pgfsetdash{{1pt}{1pt}}{0pt}% dashing cannot work correctly in all situation this way
\pgfusepath{stroke}
}
\begin{tikzpicture}
\begin{axis}[
legend pos=south west,
axis x line=middle,
axis y line=middle,
%grid = major,
%width=9cm,
%height=4.5cm,
grid style={dashed, gray!30},
xmin=-1, % start the diagram at this x-coordinate
xmax= 6, % end the diagram at this x-coordinate
ymin=-0.25, % start the diagram at this y-coordinate
ymax= 5, % end the diagram at this y-coordinate
axis background/.style={fill=white},
xlabel=$X_1$,
ylabel=$X_2$,
%xticklabels={,,},
%yticklabels={,,},
%xtick={-1,0,1,2,3,4,5},
%ytick={-1,0,1,2,3,4,5},
ticks=none,
%tick align=outside,
enlargelimits=true,
tension=0.08]
\addplot[hatchcolor=red,mark=none, pattern=custom north west lines, draw=none] coordinates {(0.5, 0) (0.5,5) (1.5,5) (1.5,0) };
\addplot[red,mark=none, thick] coordinates {(0.5, 0) (0.5,5)};
\addplot[red,mark=none, thick] coordinates {(1.5, 0) (1.5,5)};
\addplot[hatchcolor=red,mark=none, pattern=custom north west lines, draw=none] coordinates {(4.5, 0) (4.5,5) (5.5,5) (5.5,0) };
\addplot[red,mark=none, thick] coordinates {(4.5, 0) (4.5,5)};
\addplot[red,mark=none, thick] coordinates {(5.5, 0) (5.5,5)};
\addplot[mark=none, dashed] coordinates {(1, 0) (1,3)};
\addplot[mark=none, dashed] coordinates {(5, 0) (5,3)};
\addplot[mark=x] coordinates {(1, 3)};
\addplot[mark=x] coordinates {(5, 3)};
\node at (axis cs:1,3) [anchor=north west] {$(x_1, y_1)$};
\node at (axis cs:5,3) [anchor=north west] {$(x_2, y_2)$};
\node at (axis cs:1,0) [anchor=north] {$x_1$};
\node at (axis cs:5,0) [anchor=north] {$x_2$};
\node[red] at (axis cs:1,-0.3) [anchor=north] {$U_1 \times X_2$};
\node[red] at (axis cs:5,-0.3) [anchor=north] {$U_2 \times X_2$};
\end{axis}
\end{tikzpicture}