This is a theorem proved in Munkres.
Let $f:X\to Y$ be a bijective continous function. If X is compact and Y is hausdorff, then f is a homeomorphism.
I knew Y being hausdorff which will be good to imply compact set in Y being closed which will imply continuity for the inverse. I have three questions. They may be related.
What kind of role is Y being hausdorff?
Why it is necessary to be hausdorff?
What is the intuition behind Y being hausdorff?
On
The roles of $X$ being compact and $Y$ being Hausdorff, as well as $f$ being continuous, are important for the following reasons:
These three facts allow you to show that the function $f$ is a closed mapping (images of closed sets are closed). Since a homeomorphism is nothing more than a continuous closed bijection, the conditions on $X$ and $Y$ allow you to prove the only one of these properties that is not explicitly assumed.
It occurs to me that the Hausdorffness of $Y$ can be replaced by the slightly weaker condition that "all compact subsets are closed"; such spaces are sometimes called KC spaces. (So a continuous bijection from a compact space onto a KC space is a homeomorphism.) While there are non-Hausdorff KC spaces, most of the KC spaces we run into in the wild are Hausdorff, so there's little need for the added generality, especially in an introductory topology course.
To generalize Mathmo123's example: since $f$ is bijective we may as well consider $Y$ to be the same set as $X$, but with a weaker topology (i.e. some of the open sets of $X$ are no longer open). Then $f$ is still continuous, but no longer a homeomorphism. So what the theorem is saying is that you can't weaken a compact topology and have it be Hausdorff, and you can't strengthen a Hausdorff topology and have it be compact.