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.
Read more about this topic: Paracompact Space
Famous quotes containing the word spaces:
“Le silence éternel de ces espaces infinis m’effraie. The eternal silence of these infinite spaces frightens me.”
—Blaise Pascal (1623–1662)