Join (topology)

In topology, a field of mathematics, the join of two topological spaces "A" and "B", often denoted by formula_1 or formula_2, is defined to be the quotient space
where "I" is the interval [0, 1] and "R" is the equivalence relation generated by
At the endpoints, this collapses formula_6 to formula_7 and formula_8 to formula_9.
Intuitively, formula_2 is formed by taking the disjoint union of the two spaces and attaching line segments joining every point in "A" to every point in "B".
Properties.
is homeomorphic to the reduced suspension
of the smash product. Consequently, since formula_20 is contractible, there is a homotopy equivalence