Infinite symmetric products of rational algebras and spaces

We show that the infinite symmetric product of a connected graded-commutative algebra over ℚ is naturally isomorphic to free on positive degree subspace original algebra. In particular, commutative (in usual sense) graded polynomial Applied topology, we obtain quick proof Dold–Thom theorem in rational homotopy theory for spaces finite type. also products certain simple algebras are free; this allows us determine minimal Sullivan models complex projective spaces.