Wanted to make sure I have all the technical details in the following argument correct

Let and be topological spaces and be a bijection. Prove that is a homeomorphism if and only if

Proof

Let be a homeomorphism. Consider and let . is open, and is continuous, so is open, so

And let , , is open and is continuous, so is open and

Now let , so if , then , so both are open so is continuous and when , then so is continuous, thus a homeomorphism.

QED

Are all the details in the proof correct?