Given two topological spacesMathworldPlanetmath X and Y, their join, denoted by XY, is defined to be the quotient spaceMathworldPlanetmath


where the equivalence relationMathworldPlanetmath is generated by

(x,0,y1) (x,0,y2) for anyxX,y1,y2Y,and
(x1,1,y) (x2,1,y) for anyyY,x1,x2X.

Intuitively, XY is formed by taking the disjoint unionMathworldPlanetmath of the two spaces and attaching a line segment joining every point in X to every point in Y.

Some examples:

  • The join of a space X with a one-point space is called the cone of X.

  • The join of the spheres Sn and Sm is the sphere Sn+m+1.

Title join
Canonical name Join1
Date of creation 2013-03-22 13:25:40
Last modified on 2013-03-22 13:25:40
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 7
Author mathcam (2727)
Entry type Definition
Classification msc 54B99
Related topic Cone
Related topic SuspensionMathworldPlanetmath
Defines join