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we define the projectif space [tex]P^n \mathbb{R}[/tex]

by the quotient space :[tex]\mathbb{R}^{n+1}/\sim[/tex] where:

[tex]x\sim y\Leftrightarrow x[/tex] et[tex]y[/tex] are colinaires.

my questions are :

1. How we proof that the restiction de [tex]\sim[/tex] on [tex]S^n[/tex] (where S^n is the sphere on n dimension) identify x and -x?

2. How this projectif reel space is homeomorphe to the quotient of S^n by this identification?

3.How we proof that [tex]P^{n}\mathbb{R}[\tex] is compact?

thanks

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# Algebric geometry

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