Short exact two open singular cochain mayer vietoris sequence

Since $i,j$ commute with boundary, precomposition by them commutes with coboundary. Their dual maps are exactly $a$ and $b$ with the displayed signs. Also ${\delta }^2\lambda =\lambda {\mathrm{\partial }}^2=0$ on small chains, so the first term is a cochain complex, including zero groups in negative degrees.

If both restrictions of $\lambda$ vanish then $\lambda$ vanishes on every small generator, so $a$ is injective. A pair $(\phi ,\psi )$ is in $\ker b$ precisely when the functions agree on overlap simplices. Define $\lambda$ on a small simplex to be $\phi$ on $U$ simplices and $\psi$ on the others. Agreement makes its restrictions the given pair. Extending by finite real sums gives a unique functional. Conversely restrictions of a single functional agree on the overlap, proving $\ker b=\mathrm{im}a$.

For $\eta \in C^k(U\cap V;\mathbb{R})$, $b(-E_{U\cap V}^U\eta ,0)=\eta$ by [F2], proving degreewise surjectivity with the required minus sign. This lift is not used as a cochain map. Empty opens or overlap give zero terms and the same formulas; if $U=V=X$, agreeing pairs are $(\lambda ,\lambda )$ and $b(\phi ,\psi )=\psi -\phi$. This includes a one-point space. The argument applies in degree zero, on repeated simplices, and on the zero groups in negative degrees, without any AC.