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

This question already has an answer here:

How to prove that S2n+1/S1S^{2n+1}/S^1 is homeomorphic to CPn\mathbb CP^n under a given identification

2 answers

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

Definition: CPn=S2n+1S1\mathbb CP^n= \frac {S^{2n+1}}{S^1}

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

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

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1 Answer
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This is answered in the wikipedia article itself: One may also regard CPn\mathbb{C}P^n as a quotient of the unit (2n+1)(2n + 1)-sphere in Cn+1\mathbb{C}^{n+1} under the action of U(1)U(1):
CPn≅S2n+1/U(1)≅S2n+1/S1.
\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=1n=1 see also here.

  

 

In wikipedia it is given as a fact with no proof. Thank you 🙂
– Victor Barg
2 days ago

  

 

Well, the idea with U(1)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
2 days ago

  

 

I was having trouble in filling the details.Thanks for the link 🙂
– Victor Barg
2 days ago