Join

Summary

The join $X * Y$ of two topological spaces is the space of all segments that run from a point of $X$ to a point of $Y$. Formally it is $X \times Y \times [0, 1]$ with the end $t = 0$ collapsed onto $X$ and the end $t = 1$ collapsed onto $Y$. Joins generalise cones and suspensions, and the join of two spheres is again a sphere: $S^m * S^n \cong S^{m+n+1}$.