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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 if and only if there exists a continuous mapping:

$H: X \times \closedint 0 1 \to Y$

such that, for all $x \in X$:

$\map H {x, 0} = \map f x$


$\map H {x, 1} = \map g x$

$H$ is called a (free) homotopy between $f$ and $g$.

Also known as

When relative homotopy is not under consideration, free homotopy is usually referred to as simply homotopy.

Also see