# Join (topology)

In topology, a field of mathematics, the **join** of two topological spaces *A* and *B*, often denoted by or , 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 to and to .

Intuitively, 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*.

## Examples

- The join of a space
*X*with a one-point space is called the cone*CX*of*X*. - The join of a space
*X*with (the 0-dimensional sphere, or, the discrete space with two points) is called the suspension of*X*. - The join of the spheres and is the sphere .

## 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 (
*A*,*a*_{0}) and (*B*,*b*_{0}), the "reduced join"

is homeomorphic to the reduced suspension

of the smash product. Consequently, since is contractible, there is a homotopy equivalence

## See also

## References

- Hatcher, Allen,
*Algebraic topology.*Cambridge University Press, Cambridge, 2002. xii+544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0 *This article incorporates material from Join on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*- Brown, Ronald,
*Topology and Groupoids*Section 5.7 Joins.

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