# 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 Join1 2013-03-22 13:25:40 2013-03-22 13:25:40 mathcam (2727) mathcam (2727) 7 mathcam (2727) Definition msc 54B99 Cone Suspension join