Paracompact Space - Paracompact Hausdorff Spaces

Paracompact Hausdorff Spaces

Paracompact spaces are sometimes required to also be Hausdorff to extend their properties.

  • (Theorem of Jean Dieudonné) Every paracompact Hausdorff space is normal.
  • Every paracompact Hausdorff space is a shrinking space, that is, every open cover of a paracompact Hausdorff space has a shrinking: another open cover indexed by the same set such that the closure of every set in the new cover lies inside the corresponding set in the old cover.
  • On paracompact Hausdorff spaces, the cohomology of a sheaf is equal to its Čech cohomology.

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