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


 * $\displaystyle \left({f * g}\right) \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 concatenation of paths is also called composition or product.

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

Also see

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