Countably compact space

Examples

• The first uncountable ordinal 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.