I have seen the following fact used somewhere (for example to show that $\mathbb{RP}^3$ is not homotopy equivalent to $S^3\vee\mathbb{RP}^2$):
Let $X,Y$ be two path connected pointed spaces such that the base points each have a contractible neighborhood. Then: $$H^\bullet(X\vee Y)\cong H^\bullet(X)\oplus H^\bullet(Y)$$
I have two questions:
- In what category do we have to take the direct sum? Intuitively, I would say the category of $R$-algebras. Is it correct, or should we do it in the category of rings, or something else?
- How can I show this? It is pretty easy to show something similar, namely that the reduced cohomology of the wedge of such spaces is isomorphic to the product of the reduced cohomologies as $R$-modules (and this is true for arbitrary wedges, not only finite ones). However, i don't know how to proceed for the statement above. Should I try to show that the universal property holds?