Countably compact space

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.
• Every countably compact space is limit point compact.
• For T1 spaces, countable compactness and limit point compactness are equivalent.
• Every sequentially compact space is countably compact. The converse does not hold. For example, the product of continuum-many closed intervals ${\displaystyle [0,1]}$ with the product topology is compact and hence countably compact; but it is not sequentially compact.
• For first-countable spaces, countable compactness and sequential compactness are equivalent. More generally, the same holds for sequential spaces.
• For metrizable spaces, countable compactness, sequential compactness, limit point compactness and compactness are all equivalent. The same holds for second countable Hausdorff spaces.
• 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.
• Closed subspaces of a countably compact space are countably compact.
• 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. More generally, every countably compact metacompact space is compact.
• Every countably compact Hausdorff first-countable space is regular.
• Every normal countably compact space is collectionwise normal.
• The product of a compact space and a countably compact space is countably compact.
• The product of two countably compact spaces need not be countably compact.