Suppose that is a homeomorphism, and that is a subset of , assume is open in , we'll prove that is open in , since is continuous, then we know that is an open set of , since is a bijection, this means it's injective and thus as needed.
Now suppose that is open in , since we know that is continuous this means that is an open set of , we've shown that , thus, we've just shown that is an open set of , this concludes
At this point we've proven the main rightward implication. The other direction is quite a lot easier. We have to show that both and are continuous.
Suppose that is an open set of , we want to show that is an open set of , which is true iff is an open set of , since is surjective, then making it an open set of , which shows that is an open set of
Now suppose that is an open set of , our goal will be to show that is an open set of , as seen previously , and since was open in , by our assmption we know that is open in , this concludes the proof.