Definition:Homotopy Class

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 $K$.

Also see

 * Relative Homotopy is Equivalence Relation