where the cylinder is attached to the original spaces and along the natural projections of the faces of the cylinder:
Intuitively, is formed by taking the disjoint union of the two spaces and attaching line segments joining every point in to every point in .
Note that usually it is implicitly assumed that and are non-empty, in which case the definition is often phrased a bit differently: instead of attaching the faces of the cylinder to the spaces and , these faces are simply collapsed in a way suggested by the attachment projections :
The join of a space with a one-point space is called the cone of .
The join of a space with (the 0-dimensional sphere, or, the discrete space with two points) is called the suspension of .
The join of the spheres and is the sphere .
The join of two pairs of isolated points is a square (without interior). The join of a square with a third pair of isolated points is an octahedron (again, without interior). In general, the join of pairs of isolated points is an -dimensional octahedral sphere.
The join of two abstract simplicial complexes and on disjoint vertex sets is the abstract simplicial complex . I.e., any simplex in the join is the union of a simplex from and a simplex from . For example, if each of and contain two isolated points, and , then , a "square" graph.
Properties[]
The join of two spaces is homeomorphic to a sum of cartesian products of cones over the spaces and the spaces themselves, where the sum is taken over the cartesian product of the spaces:
Given basepointed CW complexes and , the "reduced join"