# Definition:Homotopy Class/Path

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## Definition

Let $T = \struct {S, \tau}$ be a topological space.

Let $f: \closedint 0 1 \to S$ be a path in $T$.

The **homotopy class** of the path $f$ is the homotopy class of $f$ relative to $\set {0, 1}$.

That is, the equivalence class of $f$ under the equivalence relation defined by path-homotopy.

## Also see

## Also known as

The **homotopy class of $f$** is also sometimes called as the **path class of $f$**.

## Sources

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

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- 2018: John M. Lee:
*Introduction to Riemannian Manifolds*(2nd ed.) ... (previous) ... (next): $\S 6$: Geodesics and Distance. Closed Geodesics