Countably compact space

Symbol-free definition
A topological space is said to be countably compact if it satisfies the following equivalent conditions:


 * 1) Every countable open cover has a finite subcover. In other words, given a countable collection of open subsets whose union is the whole space, there is a finite subcollection whose union is again the whole space.
 * 2) Every point-finite open cover has a finite subcover.

Refinement formal expression
In the refinement formalism, a refinement formal expression is:

Countable open $$\to$$ Finite open

viz, every countable open cover has a finite open refinement.

It is also an instance of the countably qualifier applied to compactness-like properties.

Another refinement formal expression is:

Point-finite open $$\to$$ Finite open

Stronger properties

 * Compact space
 * Sequentially compact space

Weaker properties

 * Limit point-compact space

Metaproperties
The proof of this follows from a version of the tube lemma.

Any closed subset of a countably compact space is countably compact, when endowed with the subspace topology.

Switching to a coarser topology preserves countable compactness.

Textbook references

 * , Page 181, Exercise 4 (definition introduced in exercise)