Wedge Sum

In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if X and Y are pointed spaces (i.e. topological spaces with distinguished basepoints x₀ and y₀) the wedge sum of X and Y is the quotient of the disjoint union of X and Y by the identification x₀ ∼ y₀:

where ∼ is the equivalence closure of the relation {(x₀,y₀)}. More generally, suppose (X_i )_i∈I is a family of pointed spaces with basepoints {p_i }. The wedge sum of the family is given by:

where ∼ is the equivalence relation {(p_i, p_j ) | i,j ∈ I }. In other words, the wedge sum is the joining of several spaces at a single point. This definition is sensitive to the choice of the basepoints {p_i}, unless the spaces {X_i } are homogeneous.

The wedge sum is again a pointed space, and the binary operation is associative and commutative (up to isomorphism).

Sometimes the wedge sum is called the wedge product, but this is not the same concept as the exterior product, which is also often called the wedge product.