Define the "inside" of a simple closed polygonal path. Define the "inside" of a simple closed polygonal path.  Show that if such a path is contained in a simply connected domain, so is its "inside."
For the first part (defining the "inside") I am reading the author's solution to this problem.  He states "We can view a simple closed curve $\Gamma$ with $k$ levels as a union of rectangles and one or more closed curves $\Gamma'$ with $k-1$ levels.  We can then define the inside of $\Gamma$ as the points in the rectangles together with the inside of the closed curves $\Gamma'$.
This seems like crap to me.  He's just pushing the definition of "inside" off to the next lower level of closed curves.  Could anybody care to explain or help me to come up with a better definition?
 A: The approach by the author is valid, after all this pushing down the number of levels must sooner or later hit the bottom. This allows a proof of the claim about simply connected domains $D$ by induction on the number of levels: Show explicitly that the claim holds for rectangles; and given a simple closed axe parallel polygonal path $\Gamma$ such that $\Gamma\subset D$ and $\Gamma$ has $k>2$ levels, partition $\Gamma$ into rectangles (i.e., polygons with $2$ levels) and less-level curves $\Gamma'$, show that the new edges you added in this process are $\subset D$, and conclude by induction that the interior of the rectangles and the $\Gamma'$s is $\subset D$ (the union of which, together with the added edges, is presumably what the author defines as interior of $\Gamma$).
I might suggest some alternative definitions of interior of a simple closed polygonal path $\Gamma$: 


*

*The point $a\notin\Gamma$ is inside $\Gamma$ iff all but finitely many of the rays originating at $a$ intersect $\Gamma$ in an odd number of points. 

*Or $a\notin \Gamma$ is inside $\Gamma$ iff the set of axe parallel polygonal paths originating at $a$ and disjoint from $\Gamma$ is bounded 


Question is if you can show the claim about simply connected domains from one of those easily ...
