Definitions

# Join (topology)

In topology, a field of mathematics, the join of two topological spaces A and B, often denoted by $Astar B$, is defined to be the quotient space
$A times B times I / R, ,$
where I is the interval [0, 1] and R is the relation defined by
$\left(a, b_1, 0\right) sim \left(a, b_2, 0\right) quadmbox\left\{for all \right\} a in A mbox\left\{ and \right\} b_1,b_2 in B,$
$\left(a_1, b, 1\right) sim \left(a_2, b, 1\right) quadmbox\left\{for all \right\} a_1,a_2 in A mbox\left\{ and \right\} b in B.$
In effect, one is collapsing $Atimes Btimes \left\{0\right\}$ to $A$ and $Atimes Btimes \left\{1\right\}$ to $B$.

Intuitively, $Astar B$ 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.

## Examples

• 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 $Lambda X$ of X.
• The join of a space X with $S^0$ (the 0-dimensional sphere, or, the discrete space with two points) is called the suspension $SX$ of X.
• The join of the spheres $S^n$ and $S^m$ is the sphere $S^\left\{n+m+1\right\}$.