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