# Definition:Homotopy/Free

< Definition:Homotopy(Redirected from Definition:Free Homotopy)

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## 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**.