Definition simply connected in $\Bbb C$ I recently saw a different definition for simply connected which I had never seen before. A connected subset $\Omega\subset\Bbb C$ is called simply connected if the boundary is the image of a simple closed curve. Is this equivalent to the usual definitions? 
Thanks
 A: 
image of a simple closed curve

Interpretation 1: the word "image" refers to the fact that a curve is really a map (parametrization), and the geometric shape that we think of as a curve is the image of that map. Some people, including me, would omit the words "image of" in this case.  
With this interpretation, we have the definition of a Jordan domain, which is a  smaller class of domains than  simply connected. For example, the slit disk $\{z:|z|<1\}\setminus [0,1]$ is a simply connected domain but is not a Jordan domain. 

image of a simple closed curve

Interpretation 2: the word "image" means continuous image. A set is a continuous image of a simple closed curve if and only if it is compact, connected, and locally connected (this follows from the Hahn-Mazurkiewicz  Theorem. Again, we have a more restrictive definition that simple-connectedness: 


*

*the boundary of a simply connected domain need not be compact (consider the halfplane)

*the boundary of a simply connected domain need not be locally connected. To construct an  example, remove the vertical line segments from $1/n$ to $1/n+i$, $n=1,2,\dots$ from the open quadrant $x>0,y>0$. 



Remarks: 


*

*I guess the first interpretation was the intended one.

*It is true that a connected subset of $\Omega\subset\mathbb C$ is simply connected if and only if the set $\overline{\mathbb C}\setminus \Omega $ is connected in the topology of the Riemann sphere $\overline{\mathbb C}$. In the special  case when $\Omega$ is bounded, this amounts to  $\partial \Omega$ being connected. 
