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

$(a,\; b\_1,\; 0)\; sim\; (a,\; b\_2,\; 0)\; quadmbox\{for\; all\; \}\; a\; in\; A\; mbox\{\; and\; \}\; b\_1,b\_2\; in\; B,$

$(a\_1,\; b,\; 1)\; sim\; (a\_2,\; b,\; 1)\; quadmbox\{for\; all\; \}\; a\_1,a\_2\; in\; A\; mbox\{\; and\; \}\; b\; in\; B.$

In effect, one is collapsing $Atimes\; Btimes\; \{0\}$ to $A$ and $Atimes\; Btimes\; \{1\}$ 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 $S^\{n+m+1\}$.

See also

• Cone (topology)
• Suspension (topology)

References

• Hatcher, Allen, Algebraic topology. Cambridge University Press, Cambridge, 2002. xii+544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0