Definition:Homotopy Class/Path

Definition
Let $X$ be a topological space.

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

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

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

Also see

 * Relative Homotopy is Equivalence Relation