The question is:
$f: X \to Y$ is continuous bijection, which of the following is correct:
I. if $X$ is Hausdorff space then $Y$ is Hausdorff space.
II. if $X$ is compact and $Y$ is Hausdorff space, then $f^{-1}$ exist.
I think I is correct since $X$, the Hausdorff is separable, then image should be separable. I'm not sure about II.
Besides, can you show me some example and counterexample of Hausdorff space? Thank you.