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Thread: complex numbers - winding number

  1. #1
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    complex numbers - winding number

    Hi all,
    I need to prove the following lemma in attachment . Our instructor wants us to do it in the following way:
    Let A be the set of all $\displaystyle x \in [0,1] $ with the property that there exists a continuous function
    $\displaystyle \theta : [0,x] -> R $ such that $\displaystyle \theta (0) = \theta_{0} $
    and

    $\displaystyle \gamma (t) = | \gamma (t) | e^{i \theta (t) } t \in [0,x] $

    Prove that A is nonempty , closed and relatively open in [0,1] and conclude that A = [0,1] since [0,1] is connected.

    Can anyone help?? I don't even have a slight idea how to proceed.
    Attached Thumbnails Attached Thumbnails complex numbers - winding number-screenshot-complex-made-simple-google-books-mozilla-firefox-1.png  
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  2. #2
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    Notice you can bound $\displaystyle \gamma$ away from zero and this together with the unifrom continuity of said function allows you to pick a neighbourhood (in $\displaystyle [0,1]$ ) of any point in A on which there is a branch of logarithm defined. There may be some ambiguity over multiples of $\displaystyle 2\pi $ but the exponential doesn't see them, so you can ignore them and take the "true" extension. This proves A is both open and closed, and A is never empty because 0 is always in your set. I leave the formal proof to you.
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  3. #3
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    OK, I saw that the set A is open. How do we see it's closed? I can't see why the complement should be open.
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