How to prove formally that these two definitions of Complex projective space are same?

Question

Definition: $\mathbb CP^n$ Complex Projective Space is defined as the space of lines through origin in $\mathbb C^{n+1}$.

Definition: $\mathbb CP^n= \frac {S^{2n+1}}{S^1}$

How to prove formally that above two definitions of $\mathbb C P^n$ are same?

Intuitively i can see that both definitions are same but i am struggling in proving formally.Any ideas?

Answer 1

This is answered in the wikipedia article itself: One may also regard $\mathbb{C}P^n$ as a quotient of the unit $(2n + 1)$-sphere in $\mathbb{C}^{n+1}$ under the action of $U(1)$: $$ \mathbb{C}P^n\cong S^{2n+1}/U(1)\cong S^{2n+1}/S^1. $$ Actually, is has been shown on MSE already here. For $n=1$ see also here.