Seifert-van Kampen Theorem

Theorem
The functor $\pi_1 : \mathbf{Top_\bullet} \to \mathbf{Grp}$ preserves pushouts of inclusions.

Proof
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 inclusions.
 * $i_k : U_1 \cap U_2 \hookrightarrow U_k$
 * $j_k : U_k \hookrightarrow U_1 \cup U_2$

For the sake of simplicity let $\pi_1\left(X\right) = \pi_1\left(X, \ast\right)$.

It is to show that $\pi_1\left(X\right)$ is the amalgamated free product $\pi_1\left(U_1\right) *_{\pi_1\left(U_1 \cap U_2\right)} \pi_1\left(U_2\right)$.