The mayer vietoris sequence is obtained by restricting forms without a partition of unity

Given: Assume countable choice. $M=\mathbb{R}$, $U=(-\mathrm{\infty },1)$, $V=(0,\mathrm{\infty })$, and the smooth overlap function $\omega (x)=1\mathrm{/}x$ on $(0,1)$.

Two open set de rham mayer vietoris cochain maps: For an open cover $M=U\cup V$, put $W=U\cap V$. The two-open-set de Rham maps are $r:{\mathrm{\Omega }}^{\bullet }(M)\to {\mathrm{\Omega }}^{\bullet }(U)\oplus {\mathrm{\Omega }}^{\bullet }(V)$, $r\omega =(\omega {\mathrm{\mid }}_U,\omega {\mathrm{\mid }}_V)$, and $s:{\mathrm{\Omega }}^{\bullet }(U)\oplus {\mathrm{\Omega }}^{\bullet }(V)\to {\mathrm{\Omega }}^{\bullet }(W)$, $s(\alpha ,\beta )=\beta {\mathrm{\mid }}_W-\alpha {\mathrm{\mid }}_W$. The complexes are def-de-rham-cochain-complex. Restrictions are pullbacks along open inclusions, so prop-pullback-is-a-morphism-of-de-rham-complexes gives $dr=rd$ and $ds=sd$. Both maps are real linear. The middle differential acts componentwise. Empty opens have zero form spaces. The order second minus first fixes the sign of every connecting map below.

The de rham mayer vietoris difference map is surjective: Assume countable choice. The difference map $s:{\mathrm{\Omega }}^k(U)\oplus {\mathrm{\Omega }}^k(V)\to {\mathrm{\Omega }}^k(U\cap V)$ is surjective in every degree.