Added Runge's theorem.

src/topology/main/connected.tex

@@ -45,3 +45,13 @@
 \begin{proof}
     Let $C$ be the union of all connected sets that contain $A$, then $C$ is connected by \autoref{proposition:connected-union}, and is the maximum connected set containing $A$ by definition.
 \end{proof}
+
+\begin{lemma}
+\label{lemma:union-connected-components}
+    Let $X$ be a topological space and $A \subset X$ be both open and closed, then $A$ is a union of connected components of $X$. 
+\end{lemma}
+\begin{proof}
+    Let $C \subset X$ be a connected component, then $C \cap A$ and $C \setminus A$ are both open. Since $C$ is connected, either $C \cap A = \emptyset$ and $C \subset A^c$, or $C \setminus A = \emptyset$ and $C \subset A$. 
+\end{proof}
+
+