Last-modified: 2011-08-08 (月) 01:37:09
A topological space is called compact if every open cover has a finite subcover.
- 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 fip 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.
- Some authors, like Bourbaki, require compact spaces to be Hausdorff. cf. quasicompact
Kelley, General Topology, Springer (1975)