# Countably compact space

In mathematics a topological space is **countably compact** if every countable open cover has a finite subcover.

## Examples

- The first uncountable ordinal (with the order topology) is an example of a countably compact space that is not compact.

## Properties

- Every compact space is countably compact.
- A countably compact space is compact if and only if it is Lindelöf.
- A countably compact space is always limit point compact.
- For T1 spaces, countable compactness and limit point compactness are equivalent.
- For metrizable spaces, countable compactness, sequential compactness, limit point compactness and compactness are all equivalent.
- The example of the set of all real numbers with the standard topology shows that neither local compactness nor σ-compactness nor paracompactness imply countable compactness.

## References

- James Munkres (1999).
*Topology*(2nd ed.). Prentice Hall. ISBN 0-13-181629-2.

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