Seifert-van Kampen Theorem

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

Proof
Let $\struct {X, \tau}$ be a topological space.

Let $U_1, U_2 \in \tau$ such that:
 * $U_1 \cup U_2 = X$
 * $U_1 \cap U_2 \ne \O$ 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 simplicity, let:
 * $\map {\pi_1} X = \map {\pi_1} {X, \ast}$

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