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 $\ast \in U_1 \cap U_2$.

Let:
 * $i_k : U_1 \cap U_2 \hookrightarrow U_k$
 * $j_k : U_k \hookrightarrow U_1 \cup U_2$

be inclusions.

For the sake of simplicity let:
 * $\pi_1 \left({X}\right) = \pi_1 \left({X, \ast}\right)$

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