Smooth partition of unity on a manifold with boundary

Let $M$ be a smooth manifold with boundary in the sense of Smooth charts, atlases, and structures with boundary, and let $(U_j{)}_{j\in J}$ be an indexed open cover of $M$. A smooth partition of unity subordinate to this cover is a family $({\varphi }_j{)}_{j\in J}$ of functions ${\varphi }_j:M\to [0,1]$ such that:

1. every ${\varphi }_j$ is smooth in boundary charts, using the local-extension convention of Smooth functions on relatively open half-space sets;
2. the family $(\mathrm{supp}{\varphi }_j{)}_{j\in J}$ is locally finite (Refinements, locally finite families, point-finite families, and star refinements);
3. $\mathrm{supp}{\varphi }_j\subseteq U_j$ for every $j$; and
4. ${\sum }_{j\in J}{\varphi }_j(p)=1$ for every $p\in M$.

Here $\mathrm{supp}{\varphi }_j$ is the closure in $M$ of $\{p:{\varphi }_j(p)\mathrm{\ne }0\}$. Local finiteness makes the pointwise sum locally finite. The definition includes $M=\mathrm{\varnothing }$, when the zero family indexed by an empty cover satisfies the conditions.