Definition:Homotopy/Path

Definition
Let $X$ be a topological space.

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

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

Also see

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