Is Every Path Connected and Open Set in the Complex Plane Simply Connected?

[Applejacks]

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?

 

[Deveno]

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"?

 

[Dick]

There's a path in the annulus in b that encloses the origin z=0. What is it? Is z=0 in the annulus?

 

[Deveno]

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.

 

[Dick]

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.

 

[Deveno]

and a good one, at that! :)

 

[Dick]

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.