Countably compact space
- 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.