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 one-point space is called the cone of $X$.

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