compact

Definiton

• A topological space is called compact if each of its open covers has a finite subcover.

Property

• 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.
  1. Every net has a convergent subnet(the Borzano-Weierstrass theorem).
  2. Every closed family with finite intersection property has nonempty intersection.
  3. There are no maximal open covers.
• Every compact Hausdorff space is normal.
• For a Hausdorff space X the following conditions are equivalent:
  1. The space X is compact.
  2. For every topological space Y the projection p:X×Y→Y is closed.
  3. For every normal space Y the projection p:X×Y→Y is closed.

Remark

• Bourbakiなど、compactにHausdorff性を含めることもある。

Reference

• Kelley, General Topology, Springer (1975)