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:

$X\backslash vee\; Y\; =\; (X\backslash sqcup\; Y)\backslash ;/\; \backslash ;\backslash \{x\_0\; \backslash sim\; y\_0\backslash \}$

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:

$\backslash bigvee\_i\; X\_i\; :=\; \backslash coprod\_i\; X\_i\backslash ;/\; \backslash ;\backslash \{p\_i\backslash sim\; p\_j\; \backslash mid\; i,j\; \backslash in\; I\backslash \}$

In other words, the wedge sum is the joining of several spaces at a single point. This definition of course depends on the choice of {p_i} unless the spaces {X_i} are homogeneous.

The wedge sum can be understood as the coproduct in the category of pointed spaces. Alternatively, the wedge sum can be seen as the pushout of the diagram X ← {•} → Y in the category of topological spaces (where {•} is any one point space).

For example, the wedge product of two circles is homeomorphic to a figure-eight space. The wedge product of n-circles is often called a bouquet of circles, while a wedge product of arbitrary spheres is often called a "bouquet of spheres".

Van Kampen's theorem gives certain conditions (which are usually fulfilled for well-behaved spaces, such as CW complexes) under which the fundamental group of the wedge sum of two spaces X and Y is the free product of the fundamental groups of X and Y.

See also

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• Smash product

Topology | Binary operations

Bouquet (topologia)