Show P^1 is homemorphic to S^1

  • #1
Show P^1 is homemorphic to S^1
I know I need to prove there is a function satisfying it's 1-1 ,onto,continuous,and the inverse of function is continuous.However, I can't find it.Please help!!
 

Answers and Replies

  • #2
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What is your definition of [itex]P^1[/itex]?

Knowing about stereographic projection might be helpful as well, so look that up.
 
  • #3
P^1 is the set of all line in R^2(or R^2\(0,0), I forget which one is right) through the origion.
 
  • #4
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3,297
P^1 is the set of all line in R^2(or R^2\(0,0), I forget which one is right) through the origion.

Can you attach to a line in [itex]\mathbb{R}^2[/itex] a real number?? For example, given a line through the origin [itex]ax+by=0[/itex], I can look at the slope [itex]-b/a[/itex] (works if a is nonzero).

So, that gives a function between [itex]P^1[/itex] except one point and [itex]\mathbb{R}[/itex]. Then apply stereographic projection.
 
  • #5
Would you say it in detail?I have no idea about it.
 

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