Definition:Homotopy Class/Path

Definition
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.

Also see

 * Relative Homotopy is Equivalence Relation

Also known as
The homotopy class of $f$ is also sometimes called as the path class of $f$.