# Quotient spaces

• December 19th 2010, 12:04 PM
sssitex
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 $S^1$. Then $S^1/\sim$ is homeomorphic to $S^1$. Is the corresponding result for $S^2$ true?

Thanks for help.
• December 19th 2010, 12:10 PM
Drexel28
Quote:

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 $S^1$. Then $S^1/\sim$ is homeomorphic to $S^1$. Is the corresponding result for $S^2$ true?

Thanks for help.

No, it isn't. Why?
• December 19th 2010, 12:15 PM
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.
• December 19th 2010, 12:21 PM
Drexel28
Quote:

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 $\mathbb{RP}^2$. If you know algebraic topology you can calculate that $\pi_1\left(\mathbb{RP}^2\right)=H_1\left(\mathbb{R P}^2\right)=\mathbb{Z}/2\mathbb{Z}$ where as $\pi_1\left(\mathbb{S}^2\right)=H_1\left(\mathbb{S} ^2\right)=0$
• December 19th 2010, 12:28 PM
sssitex
I don't know anything about algebraic topology, but it helps a lot I know the statement is not true. Thank you!