Many books and papers mentioned that it is easier to prove Jordan Curve Theorem in the case of polygon and hard in the case of badly behaving curves. One example that most give is Koch snowflake. My question is specifically about Koch snowflake. Why is it hard to see and prove that the Koch snowflake divides the plane into 2 parts? Why can't we treat it as polygon and show Jordan Curve Theorem in a similar way?
[Math] Why is it hard to prove Jordan Curve Theorem in the case of Koch snowflake
general-topology
Best Answer
There are simpler proofs of the Jordan Curve Theorem when the curve is smooth or polygonal. In the smooth case, a proof can be done as follows. Fix a point $p$ in one of the parts. Let's say that a point is in the same part as $p$ if it can be joined to a $p$ by a (equivalently: any) path that hits the smooth curve an even number of times. Intuitively, the path crosses over to the other side as many times as it cross back, so the endpoint should be on the same side as $p$. The points in the other part will be joined by some (equivalently: any) path that hits my smooth curve an odd number of times.
The reason this fails in general is that to make the above argument work you need to assume that the intersection of a path and my smooth curve is a nice collection of points, and they don't overlap on arcs or in more complicated ways. If my curve is nice and smooth, or a polygon, this can always be arranged with a little wiggling. If my curve is some hideous fractal (as with the Koch snowflake), this cannot be guaranteed. Moreover, even if we get a nice intersection, we cannot prove that the parity (odd or even) is independent of the path we took. Instead, one needs to exploit more subtle techniques that do not rely on this intuitive geometric idea.