# “It looks straightforward, but actually it isn’t”

In a previous topic, I asked about proof of statements which are simple but incorrect.

Here, I ask about statements which seems, at a first glance, straightforward, but if we try to write a proof, we can see it’s much harder than it looked. So I expect the answers to contain:

1. the statement;
2. why it looks easy to prove;
3. why actually it isn’t.

For any closed non self-intersecting smooth curve (i.e., a continuous and injective map from the circle $\,S^1\,$ to the real plane), its complement in the plane has exactly two connectedness components: one bounded and the other one unbounded.