In other words, the wedge sum is the joining of several spaces at a single point.

The wedge sum of the family is given by:

A rose is a wedge sum of circles.

The wedge sum of two circles is homeomorphic to a figure-eight space.

The wedge sum can be understood as the coproduct in the category of pointed spaces.

Some coproducts, such as Direct sum and Wedge sum, are named to evoke their connection with addition.

In topology, the wedge sum is a "one-point union" of a family of topological spaces.

If A is a space with one point then the adjunction is the wedge sum of X and Y.

The smash product of two pointed spaces is essentially the quotient of the direct product and the wedge sum.

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

Formality is preserved under wedge sums and direct products; it is also preserved under connected sums for manifolds.

They are complex oriented (at least after being periodified by taking the wedge sum of (p 1) shifted copies), and the formal group they define has height n.

It is homotopy equivalent to the wedge sum of two circles, and thus has fundamental group isomorphic to the free group on two generators (one generator for each circle).

A special case of the above is the wedge sum or one-point union; here we take X and Y to be pointed spaces and Z the one-point space.

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.

The Hawaiian earring looks very similar to the wedge sum of countably infinitely many circles; that is, the rose with infinitely many petals, but those two spaces are not homeomorphic.

The wedge sum of n circles is often called a bouquet of circles, while a wedge product of arbitrary spheres is often called a bouquet of spheres.

In the category of pointed spaces, fundamental in homotopy theory, the coproduct is the wedge sum (which amounts to joining a collection of spaces with base points at a common base point).

Note that the above defined the wedge sum of two functions, which was possible because , which was the point that is equivalenced in the wedge sum of the underlying spaces.

In particular, the fundamental group of the wedge sum of two spaces (i.e. the space obtained by joining two spaces together at a single point) is simply the free product of the fundamental groups of the spaces.

This space is known to be the wedge sum of 2n circles (also called a bouquet of circles), which further is known to have fundamental group isomorphic to the free group with 2n generators, which in this case can be represented by the edges themselves: .

It is also seen in the fact that the wedge sum is not compact: the complement of the distinguished point is a union of open intervals; to those add a small open neighborhood of the distinguished point to get an open cover with no finite subcover.

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 of the disjoint union of X and Y by the identification x y:

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.