Homotopic Paths have Same Endpoints

Definition

Let $X$ be a topological space.

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

Let $f$ and $g$ be homotopic.

Then $f$ and $g$ have the same endpoints.

That is:

$\map f 0 = \map g 0$ and $\map f 1 = \map g 1$.

Proof

By Definition of Path-Homotopic, $f$ and $g$ are homotopic relative to $\set {0, 1}$.

By Definition of Relative Homotopy:

$\forall x \in \set {0, 1}: \map f x = \map g x$

The result follows.

$\blacksquare$