(i) Define the notion of the fundamental group $\pi_{1}\left(X, x_{0}\right)$ of a path-connected space $X$ with base point $x_{0}$.

(ii) Prove that if a group $G$ acts freely and properly discontinuously on a simply connected space $Z$, then $\pi_{1}\left(G \backslash Z, x_{0}\right)$ is isomorphic to $G$. [You may assume the homotopy lifting property, provided that you state it clearly.]

(iii) Suppose that $p, q$ are distinct points on the 2 -sphere $S^{2}$ and that $X=S^{2} /(p \sim q)$. Exhibit a simply connected space $Z$ with an action of a group $G$ as in (ii) such that $X=G \backslash Z$, and calculate $\pi_{1}\left(X, x_{0}\right)$.