Partition of unity subordinate to an open cover

Let $M$ be a smooth manifold and let $\{U_i\}_{i\in I}$ be an open cover of $M$.

A smooth partition of unity subordinate to $\{U_i\}$ is a family of smooth functions $\{\varphi_i:M\to[0,1]\}_{i\in I}$ such that:

1. (Support condition) For each $i$, the support $\mathrm{supp}(\varphi_i)$ is contained in $U_i$.
2. (Local finiteness) The family $\{\mathrm{supp}(\varphi_i)\}$ is locally finite: every point of $M$ has a neighborhood meeting only finitely many supports.
3. (Sum to one) For all $x\in M$, $\sum_{i\in I}\varphi_i(x)=1,$ where the sum is well-defined because of local finiteness.

A fundamental theorem states that if $M$ is a paracompact manifold , then every open cover admits such a partition of unity.

Examples

1. Three-arc cover of the circle.
   Cover $S^1$ by three open arcs with pairwise overlaps. One can build smooth bump functions supported in each arc and normalize their sum to obtain a partition of unity.

2. Cover of $\mathbb R^n$ by balls.
   For an open cover of $\mathbb R^n$ by (possibly overlapping) balls, choose a locally finite refinement and bump functions supported in the refined sets; normalizing yields a subordinate partition of unity.

3. Gluing local data.
   If $\alpha_i$ are differential forms defined on $U_i$, then $\sum_i \varphi_i\,\alpha_i$ defines a global form when the $\alpha_i$ agree on overlaps in the appropriate sense; local finiteness ensures the sum is pointwise finite.