(i) State the Seifert-van Kampen theorem.

(ii) Assuming any standard results about the fundamental group of a circle that you wish, calculate the fundamental group of the $n$-sphere, for every $n \geqslant 2$.

(iii) Suppose that $n \geqslant 3$ and that $X$ is a path-connected topological $n$-manifold. Show that $\pi_{1}\left(X, x_{0}\right)$ is isomorphic to $\pi_{1}\left(X-\{P\}, x_{0}\right)$ for any $P \in X-\left\{x_{0}\right\}$.