\begin{definition}[Definition:Concatenation of Paths]
Let $X$ be a topological space.
Let $f, g: \closedint 0 1 \to X$ be paths.
Let $\map f 1 = \map g 0$.
The '''concatenation''' of $f$ and $g$ is the mapping $f * g: \closedint 0 1 \to X$ defined by:
:$\ds \map {\paren {f * g} } s = \begin {cases}
\map f {2 s} & : 0 \le s \le \dfrac 1 2 \\
\map g {2 s - 1} & : \dfrac 1 2 \le s \le 1
\end {cases}$
\end{definition}