Compact space
This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces
Definition
A topological space is said to be compact if it satisfies the following equivalent conditions:
- Every open cover has a finite subcover
- Every family of closed sets with the finite intersection property has a nonempty overall intersection
Formalisms
Refinement formal expression
In the refinement formalism, the property of compactness has the following refinement formal expression:
Open Finite open
Relation with other 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
Any product of compact spaces is compact. This result is true only in theproduct topology, not in the box topology. The result is known as the Tychonoff theorem. For the case of finite direct products, there is a much simpler proof that makes use 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 compact space is compact. In fact, given any Hausdorff space, every compact subset is closed, so we cannot in general hope for too many compact sets under than the closed ones.
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
Removing open sets reduces the number of possibilities for an open cover, and thus does not damage compactness. In other words, shifting to a coarser topology preserves compactness.