Construction 4.3.4.1. Let $X$ and $Y$ be topological spaces, and let $[0,1] = | \Delta ^1 |$ denote the unit interval. We let $X \star Y$ denote the topological space given by the iterated pushout
\[ X \coprod _{ (X \times \{ 0\} \times Y)} (X \times [0,1] \times Y) \coprod _{ (X \times \{ 1\} \times Y) } Y. \]
We will refer to $X \star Y$ as the join of $X$ and $Y$.