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

We say that the map $f$ is

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

*Bourbaki-proper*^{1}if the map $Z \times X\to Z \times Y$ is closed for any topological space $Z$.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.We say that $f$ is

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

*proper*if $f$ is separated and universally closed.

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