a topological space
is countably compact if every countable open cover has a finite subcover.
Examples and Properties
A compact space is countably compact. Indeed, directly from the definitions, a space is compact if and only if it is both countably compact and Lindelöf.
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.
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 first uncountable ordinal (with the order topology) is an example of a countably compact space that is not compact.