*Definiton [#y28669ac]
-A topological space is called compact if each of its open covers has a finite subcover.

*Property [#i9db3ba5]
-In Hausdorff space, every compact subset is closed.
-Every closed subspace of compact space is compact.
-Compactness is equivalent to each of the following properties.
++Every net has a convergent subnet(the Borzano-Weierstrass theorem).
++Every closed family with finite intersection property has nonempty intersection.
++There are no maximal open covers.
-Every compact Hausdorff space is normal.
-For a Hausdorff space X the following conditions are equivalent:
++The space X is compact.
++For every topological space Y the projection p:X×Y→Y is closed.
++For every normal space Y the projection p:X×Y→Y is closed.

*Remark [#d9eb10af]
-Bourbakiなど、compactにHausdorff性を含めることもある。

*Reference [#fe3088a6]
-Kelley, ''General Topology'', Springer (1975)