Definition:Smooth Homotopy
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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 $g$.