nLab
wedge sum

Definition

For pointed sets

The wedge sum $A\vee B$ of two pointed sets $A$ and $B$ is the quotient set of the disjoint union $A\uplus B$ where both copies of the basepoint (the one in $A$ and the one in $B$) are identified. The wedge sum $A\vee B$ can be identified with a subset of the cartesian product $A\times B$; if this subset is collapsed to a point, then the result is the smash product $A\wedge B$.

For general pointed objects

The wedge sum can be defined for pointed objects in any category $C$ with pushouts, and is the coproduct in the category of pointed objects in $C$. A very commonly used case is when $C=$Top is a category of topological spaces. The wedge sum also makes sense for any family of pointed objects, not just for two of them.