  Fix $\alpha \in (0,1/3)$. We show that, from a topological point of view,
almost all sets $A\subseteq \mathbb{N}$ have the property that, if $A^\prime=A$
for all but $o(n^{\alpha})$ elements, then $A^\prime$ is not a nontrivial
sumset $B+C$. In particular, almost all $A$ are totally irreducible. In
addition, we prove that the measure analogue holds with $\alpha=1$.
