Use the definitions:
- A continuous map is perfect if it is a closed surjection with compact fibers.
- A continuous map is proper if the inverse image of each compact set is compact. (In contrast to Bourbaki's more general definition, according to which a map $f : X \rightarrow Y$ is proper when, for every topological space $Z$, the product map $f \times i_{Z}: X \times Z \rightarrow Y \times Z$ is closed.)
The posts (1) Proper map not closed, (2) A continuous surjection is proper if and only if pre-images of compact sets are compact, (3) $f$ proper but not universally closed, and (4) Proper map not closed present examples of continuous surjections that are proper but not perfect (which amounts to saying they are not closed). However, in each of these examples the codomain $Y$ fails to be a Hausdorff space.
Is there such an example where $Y$ is a Hausdorff space? (Of course, for such an example, $Y$ could not be locally compact or, more generally, compactly generated.)