Definition:Homotopy

Definition
Let $X$ and $Y$ be topological spaces.

Let $f: X \to Y$, $g: X \to Y$ be continuous mappings.

Then $f$ and $g$ are homotopic iff there exists a continuous mapping:
 * $H: X \times \left[{0 \,.\,.\, 1}\right] \to Y$

such that:
 * $H \left({x, 0}\right) = f \left({x}\right)$

and:
 * $H \left({x, 1}\right) = g \left({x}\right)$

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

Also known as
Homotopy is sometimes known as free homotopy so as to distinguish it from relative homotopy.

Also see

 * Homotopy Class
 * Smooth Homotopy