Let $f: X\to Y$ be a map between topological spaces. $f$ is called proper when every preimage $f^{-1}(K)$ of every compact $K \subset Y$ is also compact.
On wiki's page I found following statements:
If X is Hausdorff and Y is locally compact Hausdorff then proper is equivalent to universally closed.
Remark:A map is universally closed if for any topological space $Z$ the map $f × id_Z : X × Z → Y × Z$ is closed.
In the case that $Y$ is Hausdorff, this is equivalent to requiring that for any map $Z → Y$ the pullback $X ×_Y Z → Z$ be closed, as follows from the fact that $X ×_Y Z$ is a closed subspace of $X × Z$.
Does anybody know a recomendable source where these two statements are rigorously proved?