Partition of unity

@lee2013smooth page 57.

In a manifold $M$, a partition of unity subordinate to an open cover $\mathcal{U}=\{U_{\alpha}\}$ is a collection of smooth functions $\varphi_{\alpha}:M\to \mathbb{R}$ such that:

• $0\leq \varphi_{\alpha}\leq 1$

• The support of $\varphi_{\alpha}$ is contained in $U_{\alpha}$.

• The set of supports $\{supp(\varphi_{\alpha})\}$ is locally finite (every point $x\in M$ belongs only to a finite number of $supp(\varphi_{\alpha})$)

• $\sum_{\alpha} \varphi_{\alpha}(x)=1$ for $x\in M$. The sum is always well defined, because of the previous condition.

Theorem. If $M$ is a smooth manifold and $\mathcal{U}=\{U_{\alpha}\}$ is any open cover of $M$ then there exist a partition of unity subordinate to $\mathcal{U}$.$\blacksquare$

Keep an eye: in the proof it is used that a manifold is always paracompact. @lee2013smooth page 37.

Theorem. A topological space $X$ is paracompact Hausdorff if and only if every open cover admits a subordinate partition of unity.$\blacksquare$ See here.

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es