---
tag: 0QPHW703PV
kind: definition
name: Compact space
msc: [54D30]
defines: 0TVJZYB6SH
uses: [0NVPNC12JY]
sources:
  - url: https://en.wikipedia.org/wiki/Compact_space
    locator: Open cover definition
---

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is compact when each [[open cover|0NVPNC12JY]] of $X$ has a finite subcover.

## Remarks

Some authors, mainly in the French tradition, also require a compact space to be Hausdorff, and they call the general notion quasi-compact.
Each finite topological space is compact. An infinite set with the [[discrete topology|0S2TEHX4E1]] is not compact.
For metric spaces, compactness is the same as [[sequential compactness|0CARW59WD2]], by [[this theorem|0K8SGQSV7G]].
The [[continuous image of a compact space is compact|0K9AAJ8G51]].

## Explanations

### L1 human

In a compact space, each way to cover the space by open sets can be reduced to finitely many of them.
This finiteness lets you turn local facts, which hold near each point, into global facts with one bound for the whole space.