“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.

Answer

Continuing my comment, Jordan’s Curve Theorem is, perhaps, one of the most well-known, easy-to-grasp, and very hard to prove theorems. We could write it as:

For any closed non self-intersecting smooth curve (i.e., a continuous and injective map from the circle S1 to the real plane), its complement in the plane has exactly two connectedness components: one bounded and the other one unbounded.

Why does it look easy? Because most curves we can think of fulfilling the above conditions “trivially” fulfill the claim.

Why isn’t its proof easy? Because, in the general case, it requires advanced stuff like homotopy groups, Hopf maps, covering maps, lifting properties for maps, etc. (I’m just talking of some aspects of the proof in the above mentioned book. There might be, and almost sure there are, other proofs).

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