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?

You have besides that $\Bbb P_2(\Bbb R)$ is homeomorphic to each of the quotient spaces deduced from (1) the sphere $S: x^2+y^2+z^2=1$ in which diametrically opposed points are identified; (2) the closed disk $\Delta: x^2+y^2\le 1$ in which the diametrically opposed points of the border are identified.You have even homeomorphism with the hemisphere (identifying what points?). — Ataulfo, Oct 21 '16 at 14:23

score 1 · Accepted Answer · edited Apr 13 '17 at 12:19

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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.

edited Apr 13 '17 at 12:19

Community

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answered Oct 21 '16 at 13:08

Dietrich Burde

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In wikipedia it is given as a fact with no proof. Thank you :) – Math Lover Oct 21 '16 at 13:11
Well, the idea with $U(1)$ action is given, but some details are skipped, this is right. But the reader is assumed to fill in the details. – Dietrich Burde Oct 21 '16 at 13:13
I was having trouble in filling the details.Thanks for the link :-) – Math Lover Oct 21 '16 at 13:14

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

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