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I'm trying to prove that the projective n-space is homeomorphic to identification space [tex]B^n / [/tex] ~ where for [tex]x, x' \in B^n[/tex]: [tex]x[/tex]~[tex]x'~\Leftrightarrow~x=x'[/tex] or [tex]x'=\pm x \in S^{n-1}[/tex],

The way I have tried to solve this is, I introduced:

[tex]{H_{+}}^{n}=\{x\in S^n | x_n \geq 0\}[/tex]

Then [tex]{H_{+}}^{n}\cong B^n[/tex] by the function [tex]F(x)=(\frac{x}{|x|}sin\frac{\pi}{2}|x|,~cos\frac{\pi}{2}|x|)[/tex] [here [tex] \frac{x}{|x|}sin\frac{\pi}{2}|x|\in \mathbb{R}^n [/tex] so [tex]cos\frac{\pi}{2}|x|[/tex] is the [tex](n+1)[/tex]th component of [tex]F(x)[/tex]]

Now I need to show that [tex]{H_{+}}^{n}/[/tex]~ [tex]\cong P^n[/tex] but I'm not sure how to do this rigorously without getting into a terrible mess.

Anyone has any ideas?

Thanks.

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# Homeomorphism of the projective n-space

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