Corollary: Continuous candidate families

The same criterion holds when $\mathcal{A}$ is a specified vector space of continuous corrections and the candidate maps are continuous. Vanishing means membership in the actual image $\partial \mathcal{A}$, not merely its closure.

Proof. The proof uses the exact equation $d_0+\partial a=0$ in the specified correction space. It does not replace the image of $\partial$ by a closure or completion.