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compact

Last-modified: 2011-08-08 (月) 01:37:09

Definition Edit

A topological space is called compact if every open cover has a finite subcover.

Property Edit

  • 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 fip 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 Edit

  • Some authors, like Bourbaki, require compact spaces to be Hausdorff. cf. quasicompact

Reference Edit

Kelley, General Topology, Springer (1975)