The de rham mayer vietoris difference map is surjective

The partition construction supplies a locally finite family $({\varphi }_i)$ with nonnegative smooth terms summing to one and each closed support contained in $U$ or $V$. Its countable-choice implementation uses all admissible coordinate-ball tuples; a fixed countable basis and countable choice select covering tuple representatives. For unions $H_r$ of their first $r$ compact closures, take least larger indices giving $H_r\subseteq \mathrm{int}{H}_{r\mathrm{\prime }}$. For the resulting exhaustion $K_m$, the compact annulus $K_m\setminus \mathrm{int}{K}_{m-1}$ has a finite covering list of nested chart pairs inside selected balls and inside $\mathrm{int}{K}_{m+1}\setminus K_{m-2}$. Countable choice selects these finite lists and their countably many bumps. The annulus separation makes their supports locally finite, and division by their positive smooth sum gives $({\varphi }_i)$. All eligible tuples are formed before selection; least-index recursion uses no dependent choice.

Assign $i$ to $U$ if $\mathrm{supp}{\varphi }_i\subseteq U$, and to $V$ otherwise. Set ${\rho }_U={\sum }_{i\text{ assigned to }U}{\varphi }_i$ and ${\rho }_V={\sum }_{i\text{ assigned to }V}{\varphi }_i$. Locally these are finite smooth sums and they sum to one. Each grouped union of closed supports is closed by local finiteness and lies in its assigned open, so $\mathrm{supp}{\rho }_U\subseteq U$ and $\mathrm{supp}{\rho }_V\subseteq V$.

Define $\alpha =-{\rho }_V\omega$ on $U\cap V$ and zero on $U\setminus \mathrm{supp}{\rho }_V$. These two open sets cover $U$, and the expressions agree where they overlap; hence $\alpha$ is a smooth form on $U$. Similarly $\beta ={\rho }_U\omega$ on the overlap and zero on $V\setminus \mathrm{supp}{\rho }_U$ is smooth on $V$. On the overlap, $\beta -\alpha =({\rho }_U+{\rho }_V)\omega =\omega$, proving surjectivity. Empty overlap or out-of-range degree has only $\omega =0$, lifted by $(0,0)$; empty $M$ uses the empty family.