Partitions of unity.

If $M$ is a manifold a partition of unity is a collection of smooth non-negative functions $\{ \rho_\alpha\}_{\alpha \in I}$ such that every $x \in M$ has neighbourhood on which only a finite number of the $\rho$ are non-vanishing and such that $\sum_{\alpha \in I} \rho_{\alpha} = 1$.

Recall that if $f \colon M\to \mathbb{R}$ is smooth function then we define supp$(f)$ to be the closure of the set on which $f$ is non-zero. There are two basic existence results on a paracompact, Hausdorff manifold.

1.

If $\{ U_{\alpha}\}_{\alpha\in I}$ is an open cover of $M$ there is a partition of unity $\{ \rho_\alpha\}_{\alpha \in I}$ with supp$(\rho_{\alpha}) \in U_{\alpha}$. Such a partition of unity is called subordinate to the cover.

2.

If $\{ U_{\alpha}\}_{\alpha\in I}$ is an open cover of $M$ there is a partition of unity $\{ \rho_\alpha\}_{\beta \in J}$, with a possibly different indexing set $J$ with each supp$(\rho_{\beta})$ in some $U_\alpha$.