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Let $U= \{ (-n,n): n=1,2,3 . . . \}$ be an open cover for $\mathbb{R}$. I need to construct a smooth partitions of unity subordinate to the open cover $U$ .

Now, I know from the definition that I need to find a collection of functions $\{\phi_{\alpha} :M \rightarrow \mathbb{R}\}_{\alpha \in A}$ ($M$ is a smooth manifold) which satisfies the following properties -

1) $0 \leq \phi_{\alpha}(x) \leq 1 $ for every $\alpha \in A , x \in M$

2) support $\phi_{\alpha} \subset U_{\alpha}$

3) $\{ $Support$ \space \phi_{\alpha}\}$ is locally finite

4) $\sum_{\alpha \in A} \phi_{\alpha}(x) = 1$ for every $x \in M$

But I am unable to think of any such kind of set of functions. Any kind of help is appreciated!

Dark_Knight
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  • https://en.wikipedia.org/wiki/Bump_function#Properties_and_uses Bump functions are a typical choice. B-splines are often another possibility. https://en.wikipedia.org/wiki/B-spline, http://math.stackexchange.com/questions/758284/is-partition-of-unity-a-property-of-b-spline-bases – Chill2Macht Sep 07 '16 at 16:04
  • Well I'm familiar with the concept of Bump functions but not of B-splines. – Dark_Knight Sep 07 '16 at 16:06

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