Jordan curve theorem for a simple polygon (elementary proof) Let $C$ be a simple (Jordan) polygon in the plane. I would like to prove the Jordan curve theorem for $C$ using an elementary method. I think that we can prove it using the winding number (via complex analysis) with respect to $C$. Am I mistaken?
Related question: Triangle inside a simply connected open subset of the complex plane.
 A: Indeed you can use the winding number defined by the contour integral to define interior and exterior of a piecewise smooth non-self-intersecting closed curve in the complex plane. However, that is not what I would call elementary, and you still need powerful theorems about winding number that probably boil down to the Jordan curve for a polygon... Basically, how do you know that the winding number is always $0$ or $1$ or $-1$? To prove that it cannot be any other integer is the intrinsic core of the Jordan curve theorem.
See this post for an elementary proof of the Jordan curve theorem for polygons. We can now easily define the winding number of a polygon around a point in the following way. If the point is outside the polygon, the winding number is $0$. If the point is inside the polygon, the winding number is the sum of the turning angles, which can be easily shown to be either $1$ or $-1$ full turns since the sum of the internal angles is $(n-2)180^\circ$.
By the way, there is an alternative definition of winding number that does not involve complex analysis and works for arbitrary continuous curves, which I'll sketch out below.
Let $C$ be the curve in question and $P$ be a point not on $C$. By compactness $C$ is contained in a finite set of open disks that do not contain $P$. Then we can partition $C$ into finitely many sections each of which is contained in one of those disks, and define the winding number as the winding number of the polygon given by the points between sections, which is just the sum of the directed angles made by the polygonal sides with respect to $P$.
We can then prove that winding number changes continuously as $P$ moves continuously without touching $C$, and so path-connected regions have the same winding number. From the polygonal case we know that there are only two possible winding numbers ignoring sign, which means that the general rectifiable curve $C$ also divides the plane into two path-connected regions.
