Definition:Homotopy/Path

Definition

Let $X$ be a topological space.

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

Then:

$f$ and $g$ are path-homotopic

if and only if:

$f$ and $g$ are homotopic relative to $\set {0, 1}$.

Path Homotopy

Let $H : \closedint 0 1 \times \closedint 0 1 \to X$ be a continuous map such that:

$\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 $

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

Also see

• Path-Homotopic Paths have Path Homotopy
• Homotopic Paths have Same Endpoints
• Relative Homotopy is Equivalence Relation

Sources

• 2000: James R. Munkres: Topology (2nd ed.): $\S 51$
• 2011: John M. Lee: Introduction to Topological Manifolds (2nd ed.) ... (previous) ... (next): $\S 7$: Homotopy and the Fundamental Group. Homotopy