---
tag: 30R6VND1T7
kind: definition
name: Perfect map
msc: [54C10]
defines: N6TE6WJ4EZ
uses: [0CQHFPGJ45, 0VGFHTM75Y, 00FSJTXPJM, 0TVJZYB6SH, 01AHG4G9RT]
sources:
  - url: https://en.wikipedia.org/wiki/Perfect_map
---

## Definition

Let $X$ and $Y$ be [[topological spaces|0CQHFPGJ45]].
A map $p \colon X \to Y$ is a *perfect map* if:

1. $p$ is [[continuous|0VGFHTM75Y]].
2. $p$ is [[closed|00FSJTXPJM]]: the image of every closed subset of $X$ is closed in $Y$.
3. $p$ is surjective.
4. For every $y \in Y$, the fibre $p^{-1}(y)$ is [[compact|0TVJZYB6SH]].

## Remarks

Some authors do not require clause 3, and some also require $X$ to be Hausdorff.
A perfect map is [[proper|15XEBGNGHC]]: the [[preimage|01AHG4G9RT]] of every compact set is compact.

Example: let $K$ be a compact space and $Y$ any space, and let $p \colon Y \times K \to Y$ be the projection.
It is continuous and surjective when $K \ne \emptyset$, each fibre $\set{y} \times K$ is homeomorphic to $K$, and the projection is closed because $K$ is compact, by the [[tube lemma|W8XEZR3VKQ]].

Non-example: the projection $p \colon \R^2 \to \R$, $p(x, y) = x$.
It is continuous and surjective, but the closed set $\set{(x, y) : xy = 1}$ has image $\R \setminus \set{0}$, which is not closed.
Also its fibres are lines, which are not compact.