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 $\prod_i H^i$?