Definition
A topological space is called compact if every open cover 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.
- 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
- Some authors, like Bourbaki, require compact spaces to be Hausdorff. cf. quasicompact
Reference
Kelley, General Topology, Springer (1975)