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Thread: circle heomomorphism

  1. #1
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    circle heomomorphism

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

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