$X$ homeomorphic to $f(X)$ 
Let $X$, and $Y$ be topological spaces, and let $f:X\rightarrow Y$ be a continuous and one-to-one map. 

When is $X$ homeomorphic to $f(X)$?

Well... when $f$ is open (or closed). A nice criterion is: $X$ compact, $Y$ Hausdorff, then $f$ is a closed map. Indeed let $C\subset X$ be closed, then $C$ is compact. The continuous image of a compact set is compact, so $f(C)\subset Y$ is compact, and thus closed.
Note: I interpreted your question as: "When is $f$ an homeomorphism with its image?" Obviously as Daniel Fisher stated in a comment, $X$ and $f(X)$ can be homeomorphic without $f$ being a homeomorphism.
