Short exact two open cover small singular chain sequence

Regard chains in $U$, $V$, and $U\cap V$ as chains in $X$ by their basis inclusions: a map with image in a subspace has a unique corestriction, continuous by the subspace topology. A face of a small simplex remains small. Hence these spaces and the span $C^{U,V}$ are subcomplexes, and both displayed arrows commute with boundary. The sign choice equals [F2] precomposed by $-\mathrm{id}$ on the overlap.

The map $i$ is injective since its second component is $c$. To split a finite small chain, assign a simplex to the $U$ component whenever its image lies in $U$, and otherwise to $V$. This formula gives a preimage under $j$, proving surjectivity without choosing from an arbitrary family.

If $a+b=0$, coefficient comparison in the supplied $X$ basis forces coefficients of $a$ outside the overlap to vanish, and likewise for $b$. Thus $b=-a$ is an overlap chain, and $(a,b)=i(b)$. Conversely $ji(c)=-c+c=0$. These prove equality of kernel and image. If an open set or overlap is empty the same argument gives the appropriate zero term; when $U=V=X$ it is the diagonal signed sequence. In negative degrees all terms are zero, and degree zero uses the same basis argument with zero differential. Constant simplices remain generators, and no AC is used.