Some people define total cohomology of a space $X$ to be $\bigoplus_{i \geq 0} H^i(X)$, which would make $H^*(\mathbb{C} P^\infty)$ a polynomial ring in one generator of degree 2.

However, it seems like thinking of $H^*(\mathbb{C} P^\infty)$ as a power series ring is more natural for several reasons. For one thing, if cohomology is like the dual of homology, then the dual to an infinite direct sum is a direct product. Many algebraic formulas are also simplified if one allows for the entire power series ring rather than the polynomial ring.

Are there compelling reasons to define total cohomology as $\bigoplus_i H^i$ or as $\prod_i H^i$?