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 ${\displaystyle x_{0}}$ and ${\displaystyle y_{0}}$) the wedge sum of X and Y is the quotient space of the disjoint union of X and Y by the identification ${\displaystyle x_{0}\sim y_{0}:}$ $${\displaystyle X\vee Y=(X\amalg Y)\;/{\sim },}$$

where ${\displaystyle \,\sim \,}$ is the equivalence closure of the relation ${\displaystyle \left\{\left(x_{0},y_{0}\right)\right\}.}$ More generally, suppose ${\displaystyle \left(X_{i}\right)_{i\in I}}$ is a indexed family of pointed spaces with basepoints ${\displaystyle \left(p_{i}\right)_{i\in I}.}$ The wedge sum of the family is given by: $${\displaystyle \bigvee _{i\in I}X_{i}=\coprod _{i\in I}X_{i}\;/{\sim },}$$ where ${\displaystyle \,\sim \,}$ is the equivalence closure of the relation ${\displaystyle \left\{\left(p_{i},p_{j}\right):i,j\in I\right\}.}$ 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 ${\displaystyle \left(p_{i}\right)_{i\in I},}$ unless the spaces ${\displaystyle \left(X_{i}\right)_{i\in I}}$ are homogeneous.

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

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.