Small singular chains for an open cover ($C_{\ast }^{\mathcal{U}}(X)\hookrightarrow C_{\ast }(X)$)

For an open cover, take the subcomplex generated by singular simplices whose images lie in single cover members. Repeated barycentric subdivision makes each finite singular chain small, and the standard subdivision chain homotopy shows that inclusion induces homology isomorphisms. For a two-set cover, the short exact sequence of these small chains gives the Mayer–Vietoris sequence; excising the complement of one open set from the pair with the other set gives its equivalent relative-homology formulation.