Beispiel für offene Umbebung hinzugefügt

documents/GeoTopo/Kapitel1.tex

@@ -136,9 +136,16 @@ Auch gibt es Mengen, die sowohl abgeschlossen als auch offen sind.
     $U \subseteq X_1 \times X_2$ sei offen, wenn es zu jedem $x = (x_1, x_2) \in U$
     Umgebungen $U_i$ um $x_i$  mit $i=1,2$ gibt, sodass $U_1 \times U_2 \subseteq U$
     gilt.
-    \[\fT = \Set{U \subseteq X_1 \times X_2 | U \text{ offen}}\]
+
+    \begin{figure}[htp]
+        \centering
+        \input{figures/neighbourhood-topology}
+        \caption{Zu $x=(x_1, x_2)$ gibt es Umgebungen $U_1, U_2$ mit $U_1 \times U_2 \subseteq U$}
+    \end{figure}
+
+    $\fT = \Set{U \subseteq X_1 \times X_2 | U \text{ offen}}$
     ist eine Topologie auf $X_1 \times X_2$. Sie heißt \textbf{Produkttopologie}.
-    \[\fB = \Set{U_1 \times U_2 | U_i \text{ offen in } X_i, i=1,2}\]
+    $\fB = \Set{U_1 \times U_2 | U_i \text{ offen in } X_i, i=1,2}$
     ist eine Basis von $\fT$.
 \end{definition}
 

documents/GeoTopo/figures/neighbourhood-topology.tex

@@ -0,0 +1,47 @@
+\begin{tikzpicture}
+    \begin{axis}[
+        axis x line=middle,
+        axis y line=middle,
+        %width=9cm,
+        %height=4.5cm,
+        xmin=-1,     % start the diagram at this x-coordinate
+        xmax= 5,    % end   the diagram at this x-coordinate
+        ymin=-1,     % start the diagram at this y-coordinate
+        ymax= 5,   % end   the diagram at this y-coordinate
+        xlabel=$X_1$,
+        ylabel=$X_2$,
+        ticks=none,
+        enlargelimits=true,
+        after end axis/.code={
+            \draw [decorate,decoration={brace,mirror,raise=15pt}] (axis cs:0,3.6) -- (axis cs:0,2.5) node [midway,left=20pt] {$U_2$};
+            \draw [decorate,decoration={brace,mirror,raise=12pt}] (axis cs:1.5,0) -- (axis cs:2.5,0) node [midway,below=16pt] {$U_1$};
+        }]
+
+        \addplot[mark=none, orange, smooth, thick, fill=orange!30] coordinates {(1,1) (2,0.5) (3,1.5) (3,2) (3.5,3) (3.2, 5) (2.2, 4.7) (1.5, 4.2) (1.1, 3.9) (0.9, 2.5) (1,1)};
+        \node[orange] at (axis cs:4,4) [anchor=south] {$U$};
+
+        % Draw help lines
+        \addplot[dashed] coordinates {(1.5,0) (1.5,3.6)};
+        \addplot[dashed] coordinates {(2.5,0) (2.5,3.6)};
+        \addplot[dashed] coordinates {(0,2.5) (2.5,2.5)};
+        \addplot[dashed] coordinates {(0,3.6) (2.5,3.6)};
+
+        % Draw solid square
+        \addplot[mark=none, red, thick, fill=red!30] coordinates {(2.5,2.5) (2.5,3.6) (1.5,3.6) (1.5,2.5) (2.5,2.5)};
+
+        % Draw x and annotation
+        \node[blue] at (axis cs:2,3) [anchor=south west] {$x$};
+        \addplot[mark=*, blue] coordinates {(2,3)};
+
+        % Draw ticks of help lines
+        \addplot[mark=none, red, thick] coordinates {(1.5, -0.1) (1.5,0.1)};
+        \addplot[mark=none, red, thick] coordinates {(2.5, -0.1) (2.5,0.1)};
+        \addplot[mark=none, red, thick] coordinates {(-0.1, 2.5) (0.1,2.5)};
+        \addplot[mark=none, red, thick] coordinates {(-0.1, 3.6) (0.1,3.6)};
+
+        % Draw axis text
+        \node[blue] at (axis cs:0,3) [anchor=east] {$x_2$};
+        \node[blue] at (axis cs:2,0) [anchor=north] {$x_1$};
+
+    \end{axis} 
+\end{tikzpicture}