join


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

XY:=X×[0,1]×Y/,

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