I want to prove the following:
Let $f:X\rightarrow Y$ be a bijection. $f$ is an homeomorphism iff ($f(U)$ is open iff $U$ is open in $X$)
Starting from the usual definition for an homeomorphism: bijective map which is continuous and has continuous inverse.
The "only if" part of the proof is fairly easy: I let $f$ be an homeomorphism and then prove the "iff" in the parentheses.
The reciprocal is what boggles me: I let the proposition "$f(U)$ is open iff $U$ is open in $X$" be true. I do not know how to use it to prove that $f$ is an homeomorphism.