Definition:Concatenation of Paths

Definition
Let $X$ be a topological space.

Let $f, g: \left[{0 \,.\,.\, 1}\right] \to X$ be paths.

Suppose that $f \left({1}\right) = g \left({0}\right)$.

The composition of $f$ and $g$ is the mapping $f * g: \left[{0 \,.\,.\, 1}\right] \to X$ defined by:


 * $\displaystyle (f * g) \left({s}\right)= \begin{cases}

f \left({2 s}\right) & 0 \le s \le \dfrac 1 2 \\ g \left({2 s - 1}\right) & \dfrac 1 2 \le s \le 1 \end{cases}$

Also known as
The composition of paths is also called concatenation or product.

Also denoted as
The composition of $f$ and $g$ can also be denoted by $fg$.

Also see

 * Composition of Paths is Path
 * Definition:Multiplication of Homotopy Classes of Paths