Definition:Homotopy Class/Path

Definition
Let $X$ be a topological space.

Let $f: \left[{0 \,.\,.\, 1}\right] \to X$ be a path.

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

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

Also see

 * Relative Homotopy is Equivalence Relation