The Stacks project

Definition 5.17.2. Let $f : X\to Y$ be a continuous map between topological spaces.

1. We say that the map $f$ is closed if the image of every closed subset is closed.

2. We say that the map $f$ is Bourbaki-proper¹ if the map $Z \times X\to Z \times Y$ is closed for any topological space $Z$.

3. We say that the map $f$ is quasi-proper if the inverse image $f^{-1}(V)$ of every quasi-compact subset $V \subset Y$ is quasi-compact.

4. We say that $f$ is universally closed if the map $f': Z \times _ Y X \to Z$ is closed for any continuous map $g: Z \to Y$.

5. We say that $f$ is proper if $f$ is separated and universally closed.