I fixed (I hope) the previous proof of the existence of partitions of unity by smooth functions on smooth manifolds.

(The previous proof claimed to pertain to the general case but really only worked in the compact case, for which case however it was overly complicated.)

I have split it up now into a lemma which asserts existence of locally finite refinemens of open covers by closed subsets diffeomorphic to closed balls, and then a direct corollary by applying bump functions.

(My battery is dying right now. Not double-checked yet.)

]]>added to partition of unity a paragraph on how to build Cech coboundaries using partitions of unity (but have been lazy about getting the relative signs right).

It would be good (for me) if we could add some more about smooth partitions of unity, too, eventually.

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