Definition. I say a topological space $Y$ has finite-type if $Y$ is a finite union of open sets $Y=U_1\cup \dotsb \cup U_N$ such that each possible intersection of the $U_i$ is either empty or is contractible to a point.

If $Y$ is finite-type, I am trying to figure out if it is possible to prove the Künneth decomposition formula for $X\times Y$ (where $X$ is an arbitrary topological space) using induction on $N$ above and Mayer–Vietoris.

To do so, the natural path is to consider the MV long-exact sequence of $Y=(U_1\cup \dotsb \cup U_{N-1})\cup U_N$, tensor it with the cohomology of $X$, and then use the Five-Lemma.

The issue is that tensoring with $H^i(X)$ could destroy the exactness. (That's, of course, not an issue if we work over a field.) Is there any way to fix this argument and make the desired conclusion?