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Math Help - circle heomomorphism

  1. #1
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    circle heomomorphism

    Why isn't  f: [0, 2 \pi ) \to S^1 a homeomorphism, where  f(x) = (\cos x, \sin x) and  S^1 is unit circle on the plane?
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  2. #2
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    What is S^1?
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  3. #3
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    The book says that  S^1 is the unit circle on the plane.

    Intuitively it says that you cant unwrap the unit circle onto the interval  [0, 2 \pi) . But you can wrap the interval  [0, 2 \pi) onto the unit circle. Why is this?
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  4. #4
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    You do realize that for f to be a homeomorphic it must be bicontinuous?
    That is, both f and \overleftarrow f must be continuous.
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    Yes I do. Then  \overleftarrow{f} is not continuous?
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  6. #6
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    What would \overleftarrow f even be?
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  7. #7
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    Wait, I think  \overleftarrow{f} is not continuous. Thats why. Something about how the points dont converge to 0 but  2 \pi instead.
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