Proper map

Remarks

Bourbaki uses another definition: $f$ is continuous and, for every topological space $Z$, the map $f \times \id_Z \colon X \times Z \to Y \times Z$ is closed. This holds if and only if $f$ is continuous, closed, and every fibre $f^{-1}(y)$ is compact. A map that is proper in this sense is proper in the sense above. When $Y$ is locally compact and Hausdorff, the two definitions agree.

Example: $f \colon \R \to \R$, $f(x) = x^2$. A compact $K \subseteq \R$ lies in some $[-M, M]$ with $M \ge 0$, and $f^{-1}(K)$ is closed and lies in $[-\sqrt{M}, \sqrt{M}]$, so it is compact.

Non-example: $f = \arctan \colon \R \to \R$. It is continuous, but $f^{-1}([-\pi/2, \pi/2]) = \R$ is not compact.