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Homework Help: Simply connected domains

  1. Nov 11, 2011 #1
    1. The problem statement, all variables and given/known data
    http://imageshack.us/photo/my-images/15/unledflsq.png/


    2. Relevant equations
    A simply connected domain D in the complex plane is an open and path
    connected set such that every simple closed path in D encloses only points of D.


    3. The attempt at a solution

    The answers are a,c and d.I don't understand why they all aren't simply connected. They are all path connected and open. Am I misunderstanding the definition of open?
     
  2. jcsd
  3. Nov 11, 2011 #2

    Deveno

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    consider the following path in (b):

    p(t) = (3/2)(cos(2πt) + i sin(2πt)). does that enclose "only points of D"?
     
  4. Nov 11, 2011 #3

    Dick

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    There's a path in the annulus in b that encloses the origin z=0. What is it? Is z=0 in the annulus?
     
  5. Nov 11, 2011 #4

    Deveno

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    there's a lot of such paths. even discounting homotopic ones, there's still an infinite number.
     
  6. Nov 11, 2011 #5

    Dick

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    I know. I was just asking Applejacks to give me one.
     
  7. Nov 11, 2011 #6

    Deveno

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    and a good one, at that! :)
     
  8. Nov 11, 2011 #7

    Dick

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    Well, you gave Applejacks the path, and I gave him a point it encloses that isn't in D. So that should pretty much settle this I would hope. We make a good team.
     
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