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Show P^1 is homemorphic to S^1

  1. Oct 2, 2012 #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!!
     
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  3. Oct 2, 2012 #2

    micromass

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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.
     
  4. Oct 2, 2012 #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.
     
  5. Oct 2, 2012 #4

    micromass

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    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.
     
  6. Oct 2, 2012 #5
    Would you say it in detail?I have no idea about it.
     
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