Compact Hausdorff space
Definition
Equivalent definitions in tabular format
No. | Shorthand | A topological space is said to be compact Hausdorff if ... | A topological space is said to be compact Hausdorff if ... |
---|---|---|---|
1 | Compact and Hausdorff | it is compact (i.e., every open cover has a finite subcover) and Hausdorff (i.e., any two distinct points can be separated by disjoint open subsets) | Fill this in later |
2 | Closed iff compact on subsets | a subset is closed iff it is compact in the subspace topology | for any subset of , is closed in if and only if it is a compact space in the subspace topology |
3 | Ultrafilter formulation | every ultrafilter converges to a unique point. | for any ultrafilter of , there is a unique point such that . |
This article is about a basic definition in topology.
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This article describes a property of topological spaces obtained as a conjunction of the following two properties: compactness and Hausdorffness
Relation with other properties
Stronger properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
compact metrizable space | | | |||
compact polyhedron | | | |||
compact manifold | | |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
compact T1 space | ||||
Baire space | Any countable intersection of dense open subsets is dense | | | ||
locally compact Hausdorff space | locally compact and Hausdorff | | | ||
paracompact Hausdorff space | paracompact and Hausdorff | |||
normal space | Hausdorff and any two disjoint closed subsets can be separated by disjoint open subsets. | compact Hausdorff implies normal | Binormal space, Paracompact Hausdorff space|FULL LIST, MORE INFO | |
completely regular space | Hausdorff and there is a continuous function to separating any point and closed subset disjoint from it. | (via normal) | Normal Hausdorff space, Tychonoff space|FULL LIST, MORE INFO |
Metaproperties
Products
This property of topological spaces is closed under taking arbitrary products
View all properties of topological spaces closed under products
An arbitrary product of compact Hausdorff spaces is compact Hausdorff. This follows from independent statements to that effect for compactness, and for Hausdorffness.
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
A closed subset of a compact Hausdorff space is compact Hausdorff. In fact, a subset is compact Hausdorff iff it is closed:
- Every subspace is anyway Hausdorff
- Since the whole space is compact, any closed subset is compact
- Since the whole space is Hausdorff, any compact subset is closed
Effect of property operators
The subspace operator
Applying the subspace operator to this property gives: completely regular space
A topological space can be embedded in a compact Hausdorff space iff it is completely regular. Necessity follows from ths fact that compact Hausdorff spaces are completely regular, and any subspace of a completely regular space is completely regular. Sufficiency follows from an explicit construction, such as the Stone-Cech compactification.