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.
Here is what I came up with:
Let A be a closed subset of Y. Since Y is compact, A is compact. Then is compact because is continuous. Since X is Hausdorff, is closed, hence f is continuous and therefore a homeomorphism.