Ok, I am totally stuck on this problem, I just am not really sure where to go with it.

Let X and Y be topological spaces and let

be a one-to-one, onto function with a continuous inverse. Prove that if X is Hausdorff and Y is compact, then f is a homeomorphism.

The only thing I could really get is that having a continuous inverse takes the place of the function needing to be open, so all that is left is showing that the function is continuous. So I have to show for any set V in Y,

is in X. I am not sure how compactness and Hausdorff can be used to prove this.