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Math Help - Quotient spaces

  1. #1
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    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.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by sssitex View Post
    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?
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  3. #3
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    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.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by sssitex View Post
    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
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  5. #5
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    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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