Given two topological spaces $X$ and $Y$ their join, denoted by $X\star Y,$ is defined to be the quotient space $$ X\star Y := X\cross[0,1]\cross Y/\sim, $$ where the equivalence relation $\sim$ is generated by \begin{eqnarray*} (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. \end{eqnarray*} 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 $S^{n+m+1}$