The cone CX is the union of all line segments joining points of X to an external vertex, and similarly the suspension SX is the union of all line segments joining points of X to two external vertices. More generally, given X and a second space Y , one can define the space of all line segments joining points in X to points in Y . This is the join X ∗ Y , the quotient space of X × Y × I under the identifications (x, y 1 , 0) ∼ (x, y 2 , 0) and (x 1 , y, 1) ∼ (x 2 , y, 1). Points in the join can be written as formal convex combinations, and can be extended to joins of several spaces.

In general, the join of n points is a convex polyhedron of dimension n − 1 called a Simplex.