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• CommentRowNumber1.
• CommentAuthorDavidRoberts
• CommentTimeAug 6th 2011

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?’.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeAug 6th 2011

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

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMay 12th 2017

I have spelled out the detailed proof that smooth manifolds admit smooth partitions of unity, here

• CommentRowNumber4.
• CommentAuthorDmitri Pavlov
• CommentTimeMay 12th 2017

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.

• CommentRowNumber5.
• CommentAuthorDmitri Pavlov
• CommentTimeMay 12th 2017

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?

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeMay 12th 2017

Thanks, yes, fixed now.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeMay 30th 2017

• CommentRowNumber8.
• CommentAuthorDmitri Pavlov
• CommentTimeMar 19th 2020

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.

• CommentRowNumber9.
• CommentAuthorarsmath
• CommentTimeMay 17th 2020

Add reference to Engelking for Mather result.

• CommentRowNumber10.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 16th 2021

### The case of non-Hausdorff spaces

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.

### The case of locales

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.