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.
• The continuous image of a countably compact space is countably compact.
• Every countably compact space is pseudocompact.
• In a countably compact space, every locally finite family of nonempty subsets is finite.
• Every countably compact paracompact space is compact.[1]

See also

• Sequentially compact space
• Compact space
• Limit point compact

Notes

1. https://math.stackexchange.com/q/171182/52912

References

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