What is "wedge sum"

n. (context topology English) A one-point union of a family of topological spaces.

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 x and y) the wedge sum of X and Y is the quotient space of the disjoint union of X and Y by the identification x ∼ y:

X ∨ Y = (X ∐ Y) / ∼ ,  

where ∼ is the equivalence closure of the relation {(x,y)}. More generally, suppose (X) is a family of pointed spaces with basepoints {p}. The wedge sum of the family is given by:

⋁X = ∐X / ∼ ,  

where ∼ is the equivalence relation {(p, p) | 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}, unless the spaces {X} 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.