inverse function theorem (topological spaces)

Let $X$ and $Y$ be topological spaces, with $X$ compact and $Y$ Hausdorff. Suppose $f:X\to Y$ is a continuous bijection. Then $f$ is a homeomorphism, i.e. $f^{-1}$ is continuous.

Note if $Y$ is a metric space, then it is Hausdorff, and the theorem holds.