---
tag: S3N3CHZV89
kind: concept
name: Countably compact space
msc: [54D30]
specializes: [E8XKSQY76M]
sources:
  - url: https://en.wikipedia.org/wiki/Countably_compact_space
same_as:
  wikidata: Q859275
  wikipedia: Countably compact space
---

## Summary

A countably compact space is a topological space in which every countable open cover has a finite subcover.
It is a weakening of [[compactness|0TVJZYB6SH]] that is equivalent to the statement that every sequence has a cluster point.
Compact spaces and [[sequentially compact|0FRMJH3CY7]] spaces are countably compact.
For metric spaces, countable compactness, compactness and sequential compactness all coincide.

## Explanations

### L1 human

Compactness asks that every open cover can be cut down to finitely many sets.
Countable compactness only asks this for covers with countably many sets.
An equivalent picture: every sequence in the space keeps coming back near some point.
For example, the interval $[0, 1]$ is countably compact, but $(0, 1]$ is not: the cover by the sets $(1/n, 1]$ has no finite subcover.