---
tag: 15XEBGNGHC
kind: concept
name: Proper map
msc: [54C10]
specializes: [0VGFHTM75Y]
sources:
  - url: https://en.wikipedia.org/wiki/Proper_map
same_as:
  wikidata: Q1090009
  wikipedia: Proper map
---

## Summary

A proper map is a [[continuous map|0VGFHTM75Y]] between [[topological spaces|0CQHFPGJ45]] under which the preimage of every [[compact set|0R6EWFKRCF]] is compact.
It is a map that does not let points "escape to infinity".
Proper maps between locally compact Hausdorff spaces are [[closed|00FSJTXPJM]], and they are the maps that extend continuously to one-point compactifications.

## Explanations

### L1 human

Under a proper map, a bounded region of the target only receives a bounded region of the source.
So points that run off to infinity in the source must also run off to infinity in the target.
Every polynomial map $\R \to \R$ of degree at least $1$ is proper.
The map $\arctan \colon \R \to \R$ is not, since the whole line lands in the bounded interval $(-\pi/2, \pi/2)$.