Products of co-sober spaces need not be co-sober

We answer negatively the question whether the product of two co-sober spaces must be co-sober. For a nonempty closed subset $A$ of a $T_1$-space $(X,τ)$, we consider the topology consisting of the co-finite open sets together with the $τ$-open sets disjoint from $A$. We determine its $k$-irreducible