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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.
Revised:
A partition of unity is point finite if for every $x\in X$ there is only a finite number of $j\in J$ such that $u_j(x) \neq 0$.
A partition of unity is locally finite if for every $x\in X$ there is an open neighborhood $U$ of $x$ such that for only a finite number of $j\in J$ there is $x\in U$ such that $u_j(x) \neq 0$.
Often, the property of local finiteness is included in the definition of a partition of unity. This is harmless, since a result due to Michael R. Mather says that for any partition of unity we can find a locally finite partition of unity with the same indexing set and whose induced cover refines the original induced cover.
That fact is Prop. 4.5 in the entry, no?
And the definition of locally finite is Def 4.4.
(One sees here in the page history that this was a concern to the early authors :-)
I have added (here) pointer from the new paragraph to the old Definition/Proposition.
(Hm, on the other hand Prop. 4.5 starts out with “Let … be non-point finite” where it probably means “Let … be non-locally finite”.)
Also Mather’s text (here) had been cited – but not linked to, I have added the link now.
Also I have added the bibdata to the references for the proof given, and hyperlinked Mather’s name in his reference.
Well, I’m glad someone finally sorted it out ;-) Thanks all.
Sort out what? It seems we are still talking about the same statement by Mather which you had added already back in May 2010.
From memory, I was more worried about technical content and accuracy (and the relation to things like normal spaces, which I see is still there!), rather than exposition. But I agree the page still needs more work.
We are all concerned about technical content and accuracy: The question is whether #11 raised a technical point that was not yet accurately addressed (besides possibly being easy to miss) and whether it remains open.
I still think it was and is addressed in Prop. 4.5 – except that the assumption clause in Prop. 4.5 needs to be re-stated from “be a non-point finite partition” (which doesn’t even make sense) to “be a non-locally finite partition”.
Re #13: Sorry, did not see this section when I made the changes.
I myself ran into this recently when I wrote this little note https://arxiv.org/abs/2203.03120 on numerable open covers.
Mather’s argument is reproduced in Proposition 3.8 there, whereas Proposition 3.10 cites many equivalent characterization of numerable open covers.
Looks good, thanks!
Corrected references:
{#Mather65} Michael R. Mather, Paracompactness and partitions of unity, Mimeographed notes, Cambridge (1964).
{#Mather66} Michael R. Mather, Products in spectral sequences and other topics, PhD dissertation, Cambridge (1966).
Hm, on my system I don’t see the problem you describe. Could you check again, maybe reload the page?
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