, a field of mathematics
, the join
of two topological spaces A
, often denoted by
, is defined to be the quotient space
is the interval
[0, 1] and R
is the relation defined by
In effect, one is collapsing
Intuitively, is formed by taking the disjoint union of the two spaces and attaching a line segment joining every point in A to every point in B.
- The join of A and B, regarded as subsets of n-dimensional Euclidean space is homotopy equivalent to the space of paths in n-dimensional Euclidean space, beginning in A and ending in B.
- The join of a space X with a one-point space is called the cone 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 .