Definition:Homotopy Class
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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 $K$-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 known as
A $K$-homotopy class is also known just as a homotopy class if $K \subseteq X$ is understood.
Also see
- Results about homotopy classes can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): homotopy group
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): homotopy group