A question on the purpose of the condition on hausdorff to prove homeomorphism 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? 
 A: 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.  
A: The roles of $X$ being compact and $Y$ being Hausdorff, as well as $f$ being continuous, are important for the following reasons:


*

*Every closed subset of a compact space is comapct.

*The continuous image of a compact set is compact.

*Compact subsets of Hausdorff spaces are closed.


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.
