In my Real Analysis class I got a bit frisky and broke out a homeomorphism in a problem to show that a set was closed (that is, I had a closed set, and I made a homeomorphism between it and the set in question to show that the set in question was closed). My reason for doing this was simple: A homeomorphism maps closed sets to closed sets.
My instructor made a note saying that this is not true in general. He says homeomorphisms do not in general map closed sets to closed sets.
Everything I have ever read about homeomorphisms contradicts this. But maybe I'm wrong, so if someone on here could provide a counterexample (that is, a homeomorphism mapping a closed set to an open set), I would certainly appreciate it.