= Small singular chains for an open cover
{title2=$C_*^{\mathcal U}(X)\hookrightarrow C_*(X)$}
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.
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