Perfect map

Builds on Topological space Continuous map Closed map Compact space Preimage of a set

Definition

Let $X$ and $Y$ be topological spaces. A map $p \colon X \to Y$ is a perfect map if:

$p$ is continuous.
$p$ is closed: the image of every closed subset of $X$ is closed in $Y$ .
$p$ is surjective.
For every $y \in Y$ , the fibre $p^{-1}(y)$ is compact.

Remarks

Some authors do not require clause 3, and some also require $X$ to be Hausdorff. A perfect map is proper: the preimage 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.

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.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Closed map concept
Compact space concept
Continuous map concept
Preimage of a set concept
Topological space concept
Perfect map definition

Tag	`30R6VND1T7`
URI	`m2://30R6VND1T7`
Kind	definition
Writing	W0 stub
MSC	54C10
Defines	Perfect map
Uses	Topological space, Continuous map, Closed map, Compact space, Preimage of a set
Changed	2 November 2026

Perfect map

Definition

Remarks

Reading path

Further reading

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