TITLE: Why is the Jordan Curve Theorem not "obvious"?
QUESTION [40 upvotes]: I am horribly confused about Jordan's Curve Theorem (henceforth JCT). Could you give me some reason why should the validity of this theorem be in doubt? I mean for anyone who trusts the eye theorem is obvious. Therefore answers like "do not trust the eye" is not going to help me.
I am looking for answers along the following lines. If the JCT were true for the "obvious" reason, then it might lead to some contradiction someplace else. To be more concrete, I can give an analogy with another theorem for which I had similar feelings -- namely the unique factorization theorem for natural numbers which subsided when I learnt about Kummer Primes.
In case you think that I am being too demanding when I ask this question, here is another direction you could help me with. In that case, I would just like one or two quick sentences about your personal experience with Jordan's curve theorem -- kind of like when you had your aha moment with this theorem. Something like, "I see, now I know (or can guess) why proving it was such a big deal". Please reply when you get time -- I am horribly confused.
Thanks for your patience,

REPLY [16 votes]: This is an old question, but here are some visual examples that help to suggest that indeed the Jordan curve theorem is actually not so intuitively obvious.


Both are not rectifiable. We have relatively good intuition about rectifiable curves but not non-rectifiable ones. These examples may seem simple, but imagine sprouting lots of similar smaller pieces along the way like a fractal. For the blue one, imagine the 'arms' intertwining more and more the deeper you go in. It would still be continuous but would be less obvious why there might not just be a way to divide the plane into three or more parts!