The complex projective space $\mathbb{C}P^n$ can be represented as $\mathbb{C}P^n\cong S^{2n+1}/S^1$ where the elements of $(z_1,\dots,z_{2n+2})\in S^{2n+1}\subseteq \mathbb{C}^{2n+2}$ is quotiented via: $$ \pi:(z_1,\dots,z_{2n+2})\mapsto \{e^{i\theta}\cdot (z_1,\dots,z_{2n+2}):\theta \in [0,2\pi)\} . $$ I've managed to show this but I can't figure if it's true that this action is properly discontinuous. But is it even true?
In other words, is: the map $\pi_\circ \iota$ where $\iota:S^{2n+1}\hookrightarrow \mathbb{C}^{2n+2}$ a covering map?