1. ## Quotient spaces

Hi, I have a problem solving the following exercise:

Let ~ be the equivalence relation x~y iff x and y are diametrically opposite, on $\displaystyle S^1$. Then $\displaystyle S^1/\sim$ is homeomorphic to $\displaystyle S^1$. Is the corresponding result for $\displaystyle S^2$ true?

Thanks for help.

2. Originally Posted by sssitex
Hi, I have a problem solving the following exercise:

Let ~ be the equivalence relation x~y iff x and y are diametrically opposite, on $\displaystyle S^1$. Then $\displaystyle S^1/\sim$ is homeomorphic to $\displaystyle S^1$. Is the corresponding result for $\displaystyle S^2$ true?

Thanks for help.
No, it isn't. Why?

3. maybe it's a mistake in the book (General Topology, Stephen Willard).
Thanks for your answer, now i know why I couldn't find any bijection.

4. Originally Posted by sssitex
maybe it's a mistake in the book (General Topology, Stephen Willard).
Thanks for your answer, now i know why I couldn't find any bijection.
I'm fairly certain of this. The quotient space you described is called the real projective plane and is denoted $\displaystyle \mathbb{RP}^2$. If you know algebraic topology you can calculate that $\displaystyle \pi_1\left(\mathbb{RP}^2\right)=H_1\left(\mathbb{R P}^2\right)=\mathbb{Z}/2\mathbb{Z}$ where as $\displaystyle \pi_1\left(\mathbb{S}^2\right)=H_1\left(\mathbb{S} ^2\right)=0$

5. I don't know anything about algebraic topology, but it helps a lot I know the statement is not true. Thank you!