\begin{definition}[Definition:Homotopy/Free]
Let $X$ and $Y$ be topological spaces.
Let $f: X \to Y$, $g: X \to Y$ be continuous mappings.
Then $f$ and $g$ are '''(freely) homotopic''' {{iff}} there exists a continuous mapping:
: $H: X \times \left[{0 \,.\,.\, 1}\right] \to Y$
such that, for all $x \in X$:
: $H \left({x, 0}\right) = f \left({x}\right)$
and:
: $H \left({x, 1}\right) = g \left({x}\right)$
$H$ is called a '''(free) homotopy between $f$ and $g$'''.
\end{definition}