How does one conclude that a function maps closed sets to closed sets?

Solution 1:

Let $f :X \to Y$. Here are three exercises that answer your question:

1.) Prove that $f$ is continuous if and only if, if $O$ is open in $Y$, then $f^{-1}(O)$ is open in $X$.

2.) Use properties of the preimage to conclude that the above is equivalent to preimages of closed sets being closed.

3.) Conclude homeomorphisms send closed sets to closed sets, both ways.

Solution 2:

The preimage of a closed set under a continuous function is a closed set. If $f$ is a homeomorphism then in particular it is continuous and its inverse is continuous. Therefore $K\subseteq X$ is closed if and only if $f(K)\subseteq Y$ is closed.