# Definition:Smooth Homotopy

## Definition

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

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

Then $f$ and $g$ are smoothly homotopic if and only if there exists a smooth 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 smooth homotopy between $f$ and $g$.