In Tom Dieck p.$9.7.2$ there is a proof of a the relative version of the Eilenberg-Zilber Theorem which I'm going to attach here
What I don't get here is how $S_\bullet((X,A)\times (Y,B))$ is defined since I'd like to prove that if $$A \times Y \cup X \times B = \text{int}(A \times Y) \cup \text{int}(X \times B)$$
Then we have an isomorphism (using $9.7.2$) between the homologies of $S_\bullet(X,A) \otimes S(Y,B)$ and $S_\bullet((X,A)\times (Y,B))$ but I'm unable to do it since I don't recognize the spaces involved.
Maybe this isomorphism is not crucial, but I think it could be useful further on when cup and cap product are involed.
Best Answer
Based on context, we have already defined $$S(X,A) = \frac{S(X)}{S(A)}$$ and $$(X,A) \times (Y,B) = (X\times Y, X\times B \cup A\times Y).$$ Altogether, we have $$S((X,A) \times (Y,B)) = \frac{S(X \times Y)}{S(X \times B \cup A \times Y)}$$