User:Anghel/Sandbox

Theorem
Let $X$ be a topological space.

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

Then $f$ and $g$ are path-homotopic, there exists a path homotopy between $f$ and $g$.

Necessary Condition
By, there exists a homotopy $H : \closedint 0 1 \times \closedint 0 1 \to X$ such that:


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


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

Homotopic Paths have Same Endpoints shows that $\map f 0 = \map g 0$, and $\map f 1 = \map g 1$.

As $f$ and $g$ are homotopic relative to $\set{ 0, 1 }$, we have:

$\forall s' \in \set{ 0, 1 } , t \in \closedint 0 1 : \map H { s' , t } = \map f { s' } = \map g { s' }$

By, it follows that $H$ is a path homotopy.

Sufficient Condition
Let $H: \closedint 0 1 \times \closedint 0 1 \to Y$ be a path homotopy between $f$ and $g$.

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