Definition:Homotopy/Path

Definition
Let $X$ be a topological space.

Let $f, g: \closedint 0 1 \to X$ be paths.

We say that $f$ and $g$ are path-homotopic if they are homotopic relative to $\set {0, 1}$.

The relative homotopy $H : \closedint 0 1 \times \closedint 0 1 \to X$ is a continuous map characterized by:


 * $\forall s \in \closedint 0 1 : \map H {s, 0} = \map f s $


 * $\forall s \in \closedint 0 1 : \map H {s, 1} = \map g s $

and:


 * $\forall t \in \closedint 0 1 : \map H {0, t} = \map f 0 = \map g 0 $


 * $\forall t \in \closedint 0 1 : \map H {1, t} = \map f 1 = \map g 1 $

$H$ is called a path homotopy between $f$ and $g$.

Also see

 * Homotopic Paths have Same Endpoints
 * Relative Homotopy is Equivalence Relation