Simply connected domains

  • Thread starter Applejacks
  • Start date
  • #1
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Homework Statement


http://imageshack.us/photo/my-images/15/unledflsq.png/


Homework 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.


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?
 

Answers and Replies

  • #2
Deveno
Science Advisor
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6

Homework Statement


http://imageshack.us/photo/my-images/15/unledflsq.png/


Homework 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.


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?

consider the following path in (b):

p(t) = (3/2)(cos(2πt) + i sin(2πt)). does that enclose "only points of D"?
 
  • #3
Dick
Science Advisor
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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?
 
  • #4
Deveno
Science Advisor
906
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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?

there's a lot of such paths. even discounting homotopic ones, there's still an infinite number.
 
  • #5
Dick
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there's a lot of such paths. even discounting homotopic ones, there's still an infinite number.

I know. I was just asking Applejacks to give me one.
 
  • #6
Deveno
Science Advisor
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and a good one, at that! :)
 
  • #7
Dick
Science Advisor
Homework Helper
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and a good one, at that! :)

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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