Countably compact space

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

This is a variation of compactness. View other variations of compactness

Definition

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.

Equivalence of definitions

Further information: equivalence of definitions of countably compact space

Formalisms

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

Relation with other properties

Stronger properties

• Compact space
• Sequentially compact space

Weaker properties

• Limit point-compact space

Metaproperties

Products

This property of topological spaces is closed under taking arbitrary products
View all properties of topological spaces closed under products

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

Weak hereditariness

This property of topological spaces is weakly hereditary or closed subspace-closed; in other words, any closed subset (equipped with the subspace topology) of a space with the property, also has the property.
View all weakly hereditary properties of topological spaces | View all subspace-hereditary properties of topological spaces

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

Coarsening

This property of topological spaces is preserved under coarsening, viz, if a set with a given topology has the property, the same set with a coarser topology also has the property

Switching to a coarser topology preserves countable compactness.

References

Textbook references

• Topology (2nd edition) by James R. Munkres^{More info}, Page 181, Exercise 4 (definition introduced in exercise)