join
Given two topological spaces^{} $X$ and $Y$, their join, denoted by $X\star Y,$ is defined to be the quotient space^{}
$$X\star Y:=X\times [0,1]\times Y/\sim ,$$ 
where the equivalence relation^{} $\sim $ is generated by
$(x,0,{y}_{1})$  $\sim (x,0,{y}_{2})$  $\text{for any}x\in X,{y}_{1},{y}_{2}\in Y,\text{and}$  
$({x}_{1},1,y)$  $\sim ({x}_{2},1,y)$  $\text{for any}y\in Y,{x}_{1},{x}_{2}\in X.$ 
Intuitively, $X\star Y$ is formed by taking the disjoint union^{} 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 onepoint space is called the cone of $X$.

•
The join of the spheres ${S}^{n}$ and ${S}^{m}$ is the sphere ${S}^{n+m+1}$.
Title  join 

Canonical name  Join1 
Date of creation  20130322 13:25:40 
Last modified on  20130322 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  Suspension^{} 
Defines  join 