Suppose that $X$ is a topological space, covered by open sets $A$ and $B$. When deriving the Mayer-Vietoris sequence for simplicial homology, the standard argument is to argue for the exactness of the sequence of singular chains $$ 0 \rightarrow C_p(A\cap B) \rightarrow C_p(A)\oplus C_p(B) \rightarrow C_p(A+B)\rightarrow 0$$ which tends to be fairly straightforward.
What is less clear is whether or not this sequence will give a description of the homology of $X$. This can be resolved by using Proposition 2.21 of Hatcher's Algebraic Topology book, where he argues that the inclusion map $\iota:C_p(A+B)\rightarrow C_p(X)$ is chain-homotopic to the identity map. This in turn gives us isomorphisms $H_p(A+B) \cong H_p(X)$, and we can then derive a Mayer-Vietoris sequence for $X$ using the above short exact sequence.
I am curious about the reliance of Hatcher's argument on continuity vs. smoothness. Suppose now that $X$ is a smooth manifold, and we instead consider the smooth singular chains for $X$, $A$, $B$ and $A\cap B$. In this case, we still have a short exact sequence in the smooth case: $$ 0 \rightarrow C^\infty_p(A\cap B) \rightarrow C^\infty_p(A)\oplus C^\infty_p(B) \rightarrow C^\infty_p(A+B)\rightarrow 0$$ however, is it not clear to me whether there is a "smooth chain homotopy" between the spaces $C^\infty_p(A+B)$ and $C^\infty(X)$. Is it possible to simply repeat the argument of Hatcher and perform "smooth barycentric subdivisions" everywhere in order to obtain the smooth equivalence $H_p^\infty(A+B)\cong H^\infty_p(X)$?