Quotient spaces

**sssitex** · December 19th 2010, 12:04 PM

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.

**Drexel28** · December 19th 2010, 12:10 PM

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?

**sssitex** · December 19th 2010, 12:15 PM

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.

**Drexel28** · December 19th 2010, 12:21 PM

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 $\pi_1\left(\mathbb{S}^2\right)=H_1\left(\mathbb{S} ^2\right)=0$

**sssitex** · December 19th 2010, 12:28 PM

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

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