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
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“Surely, we are provided with senses as well fitted to penetrate the spaces of the real, the substantial, the eternal, as these outward are to penetrate the material universe. Veias, Menu, Zoroaster, Socrates, Christ, Shakespeare, Swedenborg,—these are some of our astronomers.”
—Henry David Thoreau (1817–1862)