Countably compact space
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- The first uncountable ordinal (with the order topology) is an example of a countably compact space that is not compact.
- A compact space is countably compact.
- 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 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.
- For T1 spaces, countable compactness and limit point compactness are equivalent.
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