Path Product

Definition

Consider a Topological Space $(X,\mathcal{T})$ and paths on it $p_1,p_2$ such that $p_1(1)=p_2(0)$. The path product of $p_1$ and $p_2$ is a path defined as $p_1\ast p_2(t)=\{\begin{array}{ll}{p}_1(2t)& 0\le t\le \frac{1}{2}\\ p_2(2t-1)& \frac{1}{2}\le t\le 1\end{array}$

Facts

Path product is a Continuous Function.