Jordan Curve Theorem

I find interesting the fact that Camille Jordan thought he had a proof of the simple theorem that, on a plane, a closed non-intersecting curve, divides it in inside and outside.

Later on he realized his proof was wanting and finally provided the first elementary proof of this seemingly obvious fact.

The picture shows part of the problem, how many turns are allowed?

I am fascinated by the fact that intuitive ideas, sometimes require a lot of thought to prove.