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:
- 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
Formalisms
Refinement formal expression
In the refinement formalism, a refinement formal expression is:
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
Stronger properties
Weaker properties
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)