Definition:Concatenation of Paths

Definition
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}$

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