Countably compact space

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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

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. MunkresMore info, Page 181, Exercise 4 (definition introduced in exercise)