Compact Hausdorff space

Metaproperties
An arbitrary product of compact Hausdorff spaces is compact Hausdorff. This follows from independent statements to that effect for compactness, and for Hausdorffness.

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
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.