---
tag: DV8GH6HJJ8
kind: definition
name: Countably compact space
msc: [54D30]
defines: S3N3CHZV89
uses: [0CQHFPGJ45, 049PXYHW1T, 01TQB88H4C, 0TVJZYB6SH, 0FRMJH3CY7, E8XKSQY76M, 0NW6CH38YR]
sources:
  - url: https://en.wikipedia.org/wiki/Countably_compact_space
---

## Definition

A [[topological space|0CQHFPGJ45]] $X$ is *countably compact* when every [[countable|01TQB88H4C]] [[open cover|049PXYHW1T]] of $X$ has a finite subcover.

## Remarks

The following are equivalent for any topological space $X$:

1. $X$ is countably compact;
2. every sequence in $X$ has a cluster point, that is, a point $x$ such that each neighbourhood of $x$ contains $x_n$ for infinitely many $n$;
3. every countable family of closed sets with the finite intersection property has nonempty intersection.

Every [[compact|0TVJZYB6SH]] space and every [[sequentially compact|0FRMJH3CY7]] space is countably compact.
Every countably compact space is [[limit point compact|E8XKSQY76M]], and the converse holds for [[T1 spaces|0NW6CH38YR]].
A countably compact space that is also [[Lindelöf|SW0SN1NEJE]] is compact.

Example: $[0, 1]$ is countably compact, since it is compact by the Heine–Borel theorem.
Example: the space $[0, \omega_1)$ of countable ordinals with the order topology is countably compact but not compact.

Non-example: $\R$ is not countably compact.
The countable open cover by the intervals $(-n, n)$ for $n \ge 1$ has no finite subcover.