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Math Help - Torus 2-manifold

  1. #1
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    Torus 2-manifold

    The torus,
    Tcan be obtained as an identification space as
    follows: Let
    X = [0,1] x [0,1] denote the square, and define an equivalence relation on X by (0,t) ~(1,t) and (s,0) ~(s,1) for all s,t in [0,1].

    Prove carefully that
    X/~is a closed 2-manifold

    Prove that
    X is homeomorphic to S1 x S1 where S1 is the circle

    Any help would be great. Thanks
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by ejgmath View Post
    The torus,
    Tcan be obtained as an identification space as
    follows: Let
    X = [0,1] x [0,1] denote the square, and define an equivalence relation on X by (0,t) ~(1,t) and (s,0) ~(s,1) for all s,t in [0,1].

    Prove carefully that
    X/~is a closed 2-manifold

    Prove that
    X is homeomorphic to S1 x S1 where S1 is the circle

    Any help would be great. Thanks
    Let's see some work. How much knowledge do you have of quotient spaces?
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  3. #3
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    Well to be a closed 2-manifold (without boundary) it must be hausdorff and be homeomorphic to an open set in R^n. But I am at a loss as how to show prove this.

    I am see how X/~ is hausdorff, but dont know how to prove this, and I am completely stuck with the homeomorphic part.
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