In Hatcher on page 241, he says that there is a general relative version of the cap product $H_k(X; R) \otimes H^l(X, A; R) \to H_{k-l}(X,B;R)$. I've been trying to prove this, but I'm not entirely sure about my argument. Since the cap product $ C_k(X;R) \otimes C^l(X;R) \to C_{k-l}(X;R) $ restricts to zero on the module $ C_k(A+B;R) \otimes C^l(X,A;R)$, doesn't this pretty much directly imply that there is an induced map $ C_k(X, A+B;R) \otimes C^l(X,A;R) \to C_{k-l}(X, B)$? By the formula for $\partial(\sigma \frown \phi)$, this map passes to (co)homology so we acquire a map $H_k(X, A+B;R) \otimes H^l(X,A;R) \to H_{k-l}(X, B)$, and since $A$ and $B$ are open there is an isomorphism $H_k(X, A+B;R) \cong H_k(X, A \cup B;R)$, and we are done.
Something feels off about my solution, but I'm not quite sure where it goes south.