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In mathematics a topological space is countably compact if every countable open cover has a finite subcover.
## Examples and Properties

## See also

## References

A compact space is countably compact. Indeed, directly from the definitions, a space is compact if and only if it is both countably compact and Lindelöf.

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.

A countably compact space is always limit point compact. For metrizable spaces, countable compactness, sequential compactness, limit point compactness and compactness are all equivalent. The first uncountable ordinal (with the order topology) is an example of a countably compact space that is not compact.

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

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Last updated on Thursday October 09, 2008 at 08:41:59 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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