Let $X,Y$ be locally compact (including Hausdorff) topological spaces. Iversen in "Cohomology of Sheaves" defines the cohomological dimension of $X$ as the smallest integer $n$ such that $$ H^n_c(X,\mathcal{F})=0 $$ for all sheaves $\mathcal{F}$ on $X$.
My question is if $X\times Y$ has to be finite dimensional if $X$ and $Y$ are finite dimensional. Proving something about all sheaves on a product space seems like a difficult thing to do, but neither could I find a counterexample.