Definition:Homotopy Class

This page is about Homotopy Class in the context of Topology. For other uses, see 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 $K$.

Homotopy class of path

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