Definition:Homotopy Class

Definition
Let $X$ be a topological space and $x_0 \in X$.

Let $\mathbb S^n \subseteq \R^{n+1}$ be the $n$-sphere, and $K \subseteq \mathbb S^n$.

Let $c : \mathbb S^n \to X$ be a continuous mapping.

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