Show P^1 is homemorphic to S^1

[jack8572357]

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!

 

[micromass]

What is your definition of $P^1$?

Knowing about stereographic projection might be helpful as well, so look that up.

 

[jack8572357]

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.

 

[micromass]

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

So, that gives a function between $P^1$ except one point and $\mathbb{R}$. Then apply stereographic projection.

 

[jack8572357]

Would you say it in detail?I have no idea about it.