Symmetric product (topology)

In algebraic topology, the symmetric product of a topological space consists of unordered -tuples of distinct points in . The infinite symmetric product is the colimit of this process, and appears in the Dold–Thom theorem.
Definition.
For a topological space , the th symmetric product of is the space
that is, the orbit space given by the quotient of the "n"-fold product of by the natural action of the symmetric group defined by
Infinite symmetric product.
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", "X"2, "X"3, ... obtained by identifying points ("x"1...,"x""n") with ("x"1...,"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.
Category-theoretic definition.
The infinite symmetric product is also defined as the colimit