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:
“Though there were numerous vessels at this great distance in the horizon on every side, yet the vast spaces between them, like the spaces between the stars,—far as they were distant from us, so were they from one another,—nay, some were twice as far from each other as from us,—impressed us with a sense of the immensity of the ocean, the “unfruitful ocean,” as it has been called, and we could see what proportion man and his works bear to the globe.”
—Henry David Thoreau (1817–1862)