I have a quastion about product spaces in singular cohomology. I only know a formula for sinugular homology for product spaces from lecture, the universal coefficient theorem. Let $R$ be a commutative unital ring, $(X, A), (X',A')$ pairs of topological spaces, such that $$H^*(X,A;R)\cong H^*(X',A';R).$$ Let $Y$ be a topological space. Under which conditions (or is it always correct?) holds $$H^*(X\times Y ,A\times Y;R)\cong H^*(X'\times Y,A'\times Y;R)?$$
There is an universal coefficient theorem for cohomology, such that we have a few certain conditions under which $H^*(X\times Y ,A\times Y;R)\cong H^*(X'\times Y,A'\times Y;R)$ holds, if $H^*(X,A;R)\cong H^*(X',A';R).$
The background of my question is Why is $H^*(S^1\times X,\{\text{pt}\}\times X;R)\cong H^*(D^1\times X,\partial D^1\times X;R)$? becuase I only was able to prove $H^*(S^1,\{\text{pt}\};R)\cong H^*(D^1 ,\partial D^1;R)$ with excision and with the long exact sequence in cohomology for the pair $(S^1,\{\text{pt}\})$. It would be great if there is an general argument to conclude that the cohomology of the product spaces are isomorphic.
Best.
