Let $f : X → Y$ be a continuous closed surjection such that $f^{–1}(y)$ is compact for all $y ∈ Y .$ Let $f : X → Y$ be a continuous closed surjection such that $f^{–1}(y)$ is compact for all $y ∈ Y .$ Suppose that $X$ is Hausdorff. Prove that $Y$ is Hausdorff.
I have that $f$ is a qoutient map, but I can not think of anythink useful to do with that.  Any help is appreciated.

Just take $y, y' ∈ Y$. Their preimages are compact disjoint subsets of $X$ and so can be separated by disjoint open sets. Complements of images of these sets are open disjoint and separating $y, y'$ in $Y$.
