---
tag: N6TE6WJ4EZ
kind: concept
name: Perfect map
msc: [54C10]
specializes: [00FSJTXPJM]
sources:
  - url: https://en.wikipedia.org/wiki/Perfect_map
same_as:
  wikidata: Q7168087
  wikipedia: Perfect map
---

## Summary

A perfect map is a [[continuous|0VGFHTM75Y]] [[closed map|00FSJTXPJM]] onto its target whose fibres are all [[compact|0TVJZYB6SH]].
Perfect maps carry many properties, such as compactness, local compactness and paracompactness, in both directions between the source and the target.
They are a central tool of general topology for comparing spaces.

## Explanations

### L1 human

Under a perfect map, each point of the target comes from a compact piece of the source, and the space is not torn.
Closedness means that the image of a closed set has no missing limit points.
The projection of the cylinder $S^1 \times \R$ onto the line $\R$ is perfect: each fibre is a circle, which is compact.
The projection of the plane onto a line is not, since each fibre is a whole line.