---
tag: TPH39D9W02
kind: definition
name: Proper map
msc: [54C10]
defines: 15XEBGNGHC
uses: [0CQHFPGJ45, 0VGFHTM75Y, 0R6EWFKRCF, 01AHG4G9RT, 00FSJTXPJM]
sources:
  - url: https://en.wikipedia.org/wiki/Proper_map
---

## Definition

Let $X$ and $Y$ be [[topological spaces|0CQHFPGJ45]].
A map $f \colon X \to Y$ is *proper* if $f$ is [[continuous|0VGFHTM75Y]] and, for every [[compact|0R6EWFKRCF]] subset $K \subseteq Y$, the [[preimage|01AHG4G9RT]] $f^{-1}(K)$ is compact.

## 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|00FSJTXPJM]].
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.