# Definition:Homotopy/Free

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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:

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

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

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

and:

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

$H$ 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

## Sources

- 2011: John M. Lee:
*Introduction to Topological Manifolds*(2nd ed.) ... (next): $\S 7$: Homotopy and the Fundamental Group. Homotopy