Homeomorphism that maps a closed set to an open set?

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.

Solution 1:

It is possible to have a space $X$ and subsets $A,B$ such that $A$ is closed, $A$ is homeomorphic to $B$ (when both are endowed with the subspace topology), and yet $B$ is not closed. If this is what you did, your instructor is correct. Consider the Sierpiński space, that is the set $\{0,1\}$ with open sets $\emptyset,\{0\}$ and $\{0,1\}$. Then $\{0\}$ is not closed yet $\{1\}$ is and $\{0\},\{1\}$ are homeomorphic in the subspace topology.

What you actually need is a homeomorphism $f:X\to X$ such that $f(A)=B$. Then $A$ is closed iff $B$ is.