---
tag: 0NYZ2M45G9
kind: definition
name: Closed map
msc: [54C10]
defines: 00FSJTXPJM
uses: [0CNQXVBHCX, 0G5AV9Q4RR, 0Q3F20XB0A]
sources:
  - url: https://en.wikipedia.org/wiki/Open_and_closed_maps
    locator: Definitions and characterizations
---

## Definition

Let $X$ and $Y$ be [[topological spaces|0CNQXVBHCX]].
A function $f \colon X \to Y$ is a closed map when for each [[closed set|0G5AV9Q4RR]] $F$ of $X$, the [[image|0Q3F20XB0A]] $f(F)$ is closed in $Y$.

## Remarks

A continuous map from a [[compact space|0QPHW703PV]] to a [[Hausdorff space|0MXB4CKHFW]] is closed. This gives [[continuous bijections from compact to Hausdorff spaces are homeomorphisms|0430ETCMKD]].
The projection $\mathbb{R}^2 \to \mathbb{R}$, $(x, y) \mapsto x$, is open but not closed: it maps the closed set $\{(x, y) : xy = 1\}$ to $\mathbb{R} \setminus \{0\}$.
A map can be closed but not open. A constant map $\mathbb{R} \to \mathbb{R}$ is closed, but the image of $\mathbb{R}$ is a point, which is not open.

## Explanations

### L1 human

A closed map keeps closed sets closed when you push them forward.
Open maps and closed maps are different: a map can have one property without the other.