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

	$\displaystyle(x,0,y_{1})$	$\displaystyle\sim(x,0,y_{2})$	$\displaystyle\text{for any}\,x\in X,\,y_{1},y_{2}\in Y,\,\text{and}$
	$\displaystyle(x_{1},1,y)$	$\displaystyle\sim(x_{2},1,y)$	$\displaystyle\text{for any}\,y\in Y,\,x_{1},x_{2}\in X.$

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 one-point 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	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	Suspension
Defines	join