It's easy to prove that the inverse of a continuous bijection is continuous—basically, "because is monotone." Proving the same for the inverse of a continuous bijection is quite a bit harder.
What are some general situations where it's easy prove that a continuous bijection has a continuous inverse? The following is probably my favorite short result from point-set topology.
Proposition: A continuous map from a compact space to a Hausdorff space is closed.
The proof is just a collection of definitions: a closed set in a compact space is compact, the continuous image of a compact set is compact, and a compact set in a Hausdorff space is closed. However, this is enough to prove that a continuous bijection from a compact space to a Hausdorff space is a homeomorphism.
What if our domain is not compact? Can we use the compact-to-Hausdorff idea to get anything for free? I was thinking about this today and came up with the following.
Proposition. A continuous, proper (preimage of compacts are compact) bijection from anything into a locally compact Hausdorff space is a homeomorphism.
Proof. Let be our map. Proving that is continuous is the same as proving that is closed. If is compact then we're done by the above. Otherwise, lift to a map between Alexandroff extensions, defining . (We take , with the additional open sets for all open such that is compact.)
We need three facts about the Alexandroff extension.
Now the proof is easy—again a collection of definitions.
Let be a closed set in . Suppose that it was not previously compact. It extends to a compact set in . Since is proper, is continuous and is compact in . Since is Hausdorff, is closed, and hence is closed relative to . □