Proof:

The first assertion is proved in courses on point set topology. We remark that the Hausdorff assumption is not necessary for (i). The second follows from the first and we shall prove the third which will be of immense use in the sequel. Let $g$ be the inverse of $f$ and $A$ be closed in $X$ then $g^{-1}(A) = f(A)$ is closed in $Y$ from which continuity of $g$ follows.