Join (topology)

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 imes B imes 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 A imes B imes {0} to A and A imes B imes {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}.

ee also

*Cone (topology)
*Suspension (topology)

References

*Hatcher, Allen, [http://www.math.cornell.edu/~hatcher/AT/ATpage.html "Algebraic topology."] Cambridge University Press, Cambridge, 2002. xii+544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0
*planetmath|id=3985|title=Join


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