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.
Question: Are there compelling reasons to define total cohomology as $\bigoplus_i H^i$ or as $\prod_i H^i$?
Addendum:
The question itself is quite concrete, but there are other reasons I am contemplating this, so perhaps I should list them.
If I think of $H^*\mathbb{C}P^\infty$ as somehow Koszul dual to a circle, this question might be closer to whether one should think of (this kind of) Koszul duality as always happening in a filtered/pro setting. If there are strong views/philosophies on viewing infinite projective space as an instance of Koszul duality, or on whether Koszul duality should always ask for filtration (e.g., adic-near-a-point) structures, do share.
One can think of $\mathbb{C}P^\infty$ as a space in its own right, or as a filtered diagram of spaces. This changes, for example, what kind of condensed set I think of $\mathbb{C}P^\infty$ as. Accordingly, the cohomology of the condensed set obtained as an ind-object of $\mathbb{C}P^n$ should look more pro-y (and hence look more like a power series), while the cohomology of the condensed set called "what does $\mathbb{C}P^\infty$ represent as a space" feels more like a polynomial ring.