Compact Hausdorff space

This article describes a property of topological spaces obtained as a conjunction of the following two properties: compactness and Hausdorffness

Definition

A topological space is termed compact Hausdorff if it satisfies the following equivalent conditions:

• It is compact and Hausdorff
• A subset is closed iff it is compact in the subspace topology

Relation with other properties

Stronger properties

• Compact metrizable space

Weaker properties

• Baire space
• Locally compact Hausdorff space
• Paracompact Hausdorff space
• Normal space

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.