Example 4.3.4.5 (Cones). Let $\ast $ denote the topological space consisting of a single point. For any topological space $X$, we write $X^{\triangleleft }$ for the join $\ast \star X$, and $X^{\triangleright }$ for the join $X \star \ast $, given more concretely by the formulae

\[ X^{\triangleleft } = \ast \coprod _{ (\{ 0\} \times X) }( [0,1] \times X) \quad \quad X^{\triangleright } = (X \times [0,1]) \coprod _{ (X \times \{ 1\} ) } \ast . \]

We will refer to both $X^{\triangleleft }$ and $X^{\triangleright }$ as the *cone on $X$* (note that they are canonically homeomorphic, by virtue of Remark 4.3.4.4).