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Cleaned up partition of unity and fine sheaf a bit, so I could link to them from this MO answer to the question ’Why are there so many smooth functions?’.
Thanks! That’s the way to go!
I have edited the formatting ot fine sheaf a bit (sections, floating TOC, etc) and added “Related concepts” cross-references between the entries
I have spelled out the detailed proof that smooth manifolds admit smooth partitions of unity, here
The statement of this theorem requires X to be paracompact, but in the proof it says “the smooth manifold X X is a normal topological space because it is a compact Hausdorff space”, i.e., X is compact.
A displayed formula in the proof reads:
Vi⊂Cl(U’i)⊂U’i⊂Ui
I presume this should really be
Vi⊂Cl(Vi)⊂U’i⊂Ui?
Thanks, yes, fixed now.
have added a simple example
Proposition 4.1 seems to claim that existence of partitions of unity for all open covers implies Hausdorffness. But the antidiscrete topology admits partitions of unity and is not Hausdorff.
Additionally, the article uses point-finite partitions of unity, but it seems to me that the more restrictive class of locally finite partitions occurs far more often in the literature.
Added:
Slightly more generally, a topological space (not necessarily Hausdorff) is fully normal if and only every open cover admits a subordinate partition of unity.
A T1-space is fully normal if and only if it is paracompact, in which case it is also Hausdorff.
For topological spaces that are not T1-spaces, the condition of being fully normal is strictly stronger than paracompactness.
A regular locale is fully normal if and only if it is paracompact.
The usual proof of the existence of partitions of unity goes through for such locales since it does not make any use of points.
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