Infinite symmetric product

In algebraic topology, the infinite symmetric product SP(X) of a topological space X with given basepoint e is the quotient of the disjoint union of all powers X, X², X³, ... obtained by identifying points (x₁,...,x_n) with (x₁,...,x_n,e) and identifying any point with any other point given by permuting its coordinates. In other words its underlying set is the free commutative monoid generated by X (with unit e), and is the abelianization of the James reduced product.

The infinite symmetric product appears in the Dold–Thom theorem.