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.