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