1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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!!
  2. jcsd
  3. Oct 2, 2012 #2
    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
    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook