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: Why is P1 homeomorphic to S1?

  1. Sep 26, 2008 #1
    1. The problem statement, all variables and given/known data

    Prove P1 is homeomorphic to S1

    3. The attempt at a solution

    P1 is the set of all lines through the origin. I have shown for myself that this is homeomorphic to S1/R, where R is given by: (x,y) in R iff x=y or x=-y.
    This also shows there is a continuous surjective map f from S1 to S1/R. To prove S1 and S1/R are homeomorphic I need to show that f is injective, and that it has continuous inverse.

    However, I think there is something fundamentally wrong with my understanding of the problem, because I cannot imagine the map f being injective. To me it feels as though S1/R is the same as the 'northern hemisphere' of S1, because it sends -x and x to the same line through the origin (where x is an element of the real plane on the circle S1). So wouldn't that imply that f(x)=f(-x) while x is not equal to -x, so f is not injective, so P1 is not homeomorphic to S1?

    Anyone who can tell me why I'm fundamentally off here, and give me a push in the right direction, thank you.

  2. jcsd
  3. Sep 26, 2008 #2
    Think of the origin as a point at the top (north pole?) of the circle. As the lines pass through the origin they project on to the circle like a stereographic projection.
  4. Sep 26, 2008 #3
    Ah I was taking the centre as origin... it confused me deeply.

    But then again, isn't S1={x in the real plane | ||x||=1}... so why do we get to say the origin is on the top of the circle instead of in the middle?

    This new view will probably help me take a better shot at the problem later this evening.
    Last edited: Sep 26, 2008
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook