# Definition:Homotopy Class

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## Definition

Let $X$ and $Y$ be topological spaces.

Let $K \subseteq X$ be any subset.

Let $f : X \to Y$ be a continuous mapping.

The **homotopy class** or **$K$-homotopy class** of $f$ is the equivalence class of $f$ under the equivalence relation defined by homotopy relative to $K$.

### Homotopy class of path

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.