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 ${\phi }_{\alpha }:M\to \mathbb{R}$ such that:

• $0\le {\phi }_{\alpha }\le 1$
• The support of ${\phi }_{\alpha }$ is contained in $U_{\alpha }$.
• The set of supports $\{supp({\phi }_{\alpha })\}$ is locally finite (every point $x\in M$ belongs only to a finite number of $supp({\phi }_{\alpha })$)
• $\sum _{\alpha }{\phi }_{\alpha }(x)=1$ 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 $\mathcal{U}$.$◼$

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.$◼$ See here.