# Seifert-van Kampen Theorem

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Although this article appears correct, it's inelegant. There has to be a better way of doing it.In particular: The categorical formulation is nice and all, but this does not need category theory at all. Basic fundamental group topology suffices.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Improve}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## 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 the sake of 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:

- $\map {\pi_1} {U_1} *_{\map {\pi_1} {U_1 \cap U_2} } \map {\pi_1} {U_2}$

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Source of Name

This entry was named for Karl Johannes Herbert Seifert and Egbert Rudolf van Kampen.