Let $f : X \longrightarrow Y$ be a continuous surjective map such that for every closed $A \subseteq X$, $f(A)$ is closed in $Y$. Show that if $Y$ and all the fibres $f^{-1}(y)$, $y \in Y$ are compact, then $X$ is compact. Show that if $Y$ is Hausdorff and $X$ is compact, then $Y$ and all the fibres $f^{-1}(y)$, $y \in Y$ are compact.