Let $m > 1$ be an integer and consider the following equivalence relation on $\mathbb{C} \setminus \{0\}$: $z_1 \sim z_2$ if $z_1 = z_2 e^{\frac{2\pi \imath a}{m}}$ for some $a \in \mathbb{Z}$. Write $X$ for the set of equivalence classes and $\pi : \mathbb{C} \setminus \{0\} \longrightarrow X$ for the map that takes $z$ to its equivalence class. Define a topology on $X$ by setting $U \subseteq X$ to be open if and only if $\pi^{-1}(U)$ is open in the euclidean topology of $\mathbb{C} \setminus \{0\}$. Determine (with appropriate justification) whether $X$ is compact.
Let $m > 1$ be an integer and consider the following equivalence relation on $\mathbb{C} \setminus \{0\}$: $z_1 \sim z_2$ if $z_1 = z_2 e^{\frac{2\pi \imath a}{m}}$ for some $a \in \mathbb{Z}$. Write $X$ for the set of equivalence classes and $\pi : \mathbb{C} \setminus \{0\} \longrightarrow X$ for the map that takes $z$ to its equivalence class. Define a topology on $X$ by setting $U \subseteq X$ to be open if and only if $\pi^{-1}(U)$ is open in the euclidean topology of $\mathbb{C} \setminus \{0\}$. Determine (with appropriate justification) whether $X$ is compact.