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What does S^1 X S^1 really mean?

  1. Nov 4, 2007 #1
    Torus = S^1 X S^1 but what if someone was only shown S^1 X S^1, what would it mean? Would they intepret it as a torous? I know I wouldn't.

    So what is S^1 X S^1 defined as? If you say a torous then what is a torus? Don't just draw a picture.
     
  2. jcsd
  3. Nov 4, 2007 #2

    HallsofIvy

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    S1 is a circle. S1 X S1 is a Cartesian product. I would interpret it as the set of all ordered pairs, each pair being two points on a circle. If I were then to interpret the two points, (p1, p2), as coordinates for a point on a surface- go around a circle to the point given by p1, then at right angles around a circle to the point given by p2- yes, that would look like a torus to me.

    What is a torus without drawing a picture? A torus is a doughnut, of course- a glazed doughnut just out of the frier! Yum! (One good bite and it is homeomorphic to a ball.)
     
  4. Nov 4, 2007 #3
    I think I see. RXR is the plane. S^1XS^1 does map out a torus which clearly is a surface with an empty inside. Take a bite and it is homeomorphic to a cylinder, not a ball!?
     
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