 Small-chain locality theorem for singular homology

For an open cover $\mathcal{U}$ of a space $X$, let $C_{\ast }^{\mathcal{U}}(X)$ be the subgroup generated by singular simplices whose images lie in a single member of the cover. It is a chain complex, and its inclusion into the full singular chain complex induces an isomorphism on homology.

For a finite chain, enough iterations of barycentric subdivision make every simplex small: pull the cover back to each compact standard simplex and use its Lebesgue number. The subdivision operator commutes with the boundary and is related to the identity by the standard subdivision chain homotopy, which keeps each produced simplex inside its original image. Therefore any cycle becomes homologous to a small cycle, proving surjectivity.

If a small cycle $z$ bounds a full chain $b$, choose $N$ so that ${\mathrm{Sd}}^{N}b$ is small. Write ${\mathrm{Sd}}^N-1=\partial Q+Q\partial$. Since the homotopy preserves simplex images, $Qz$ is also small. Then $z=\partial ({\mathrm{Sd}}^{N}b-Qz)$ is a small boundary, proving injectivity. This is the chain-level locality input in the proof of the Mayer–Vietoris sequence.