Seifert-van Kampen Theorem

Theorem
Let $\left(X, \tau\right)$ be a topological space.

Let $U_1,U_2 \in \tau$ such that
 * $U_1 \cup U_2 = X$
 * $U_1 \cap U_2 \ne \varnothing$ is connected

Let $\ast \in U_1 \cap U_2$ and be the maps induced by the inclusions.
 * $i_k : \pi_1\left(U_1 \cap U_2, \ast\right) \hookrightarrow \pi_1\left(U_k, \ast\right)$
 * $j_k : \pi_1\left(U_k, \ast\right) \hookrightarrow \pi_1\left(U_1 \cup U_2, \ast\right)$

Then $\pi_1\left(X, \ast\right)$ is the pushout of $i_1$ and $i_2$ in the catefory of groups.


 * $\begin{xy}\xymatrix{

\pi_1\left(U_1 \cap U_2, \ast\right) \ar[r]^*+{i_1} \ar[d]_*+{i_2} & \pi_1\left(U_1, \ast\right) \ar[d]^*+{j_1}

\\ \pi_1\left(U_2, \ast\right) \ar[r]_*+{j_2} & \pi_1\left(X, \ast\right) }\end{xy}$

In other words: The functor $\pi_1\left(-, \ast\right) : \mathbf{Top_\bullet} \to \mathbf{Grp}$ preserves pushouts of inclusions.