Π∞ preserves coproducts

Proof. Every continuous map from a topological simplex into a coproduct of spaces lands in a unique summand, since the simplex is connected. Hence the singular complex functor preserves coproducts. The claim follows because \(\Pi _{\infty }(X)\) is the colimit in \(\An \) of the simplicial set \(\Sing (X)\) and colimits commute with colimits. □