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
- 1 Definition
- 2 Formalisms
- 3 Relation with other properties
- 4 Metaproperties
- 5 References
A topological space is said to be countably compact if it satisfies the following equivalent conditions:
- 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.
- Every point-finite open cover has a finite subcover.
Equivalence of definitions
Further information: equivalence of definitions of countably compact space
Refinement formal expression
Countable open 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 Finite open
Relation with other properties
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.
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
Switching to a coarser topology preserves countable compactness.
- Topology (2nd edition) by James R. MunkresMore info, Page 181, Exercise 4 (definition introduced in exercise)