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

Linguistic Note

The word concatenation derives from the Latin com- for with/together and the Latin word catena for chain.

However, the end result of such an operation is not to be confused with a (set theoretical) chain.

Sources

• 2000: James R. Munkres: Topology (2nd ed.) ... (next): $9$: The Fundamental Group: $\S 52$: The Fundamental Group
• 2011: John M. Lee: Introduction to Topological Manifolds (2nd ed.) ... (previous) ... (next): $\S 7$: Homotopy and the Fundamental Group. Homotopy