# Definition:Homotopy/Free

(Redirected from Definition:Free 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 (freely) homotopic if and only if 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 $g$.

## Also known as

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