Definition:Homotopy/Relative

Definition
Let $X$ and $Y$ be topological spaces.

Let $K \subseteq X$ be a subset of $X$.

Let $f: X \to Y$, $g: X \to Y$ be continuous mappings.

We say that $f$ and $g$ a homotopic relative to $K$ if there exists a homotopy $H$ between $f$ and $g$, and:
 * $(1): \quad \forall x \in K: f \left({x}\right) = g \left({x}\right)$
 * $(2): \quad \forall x \in K, t \in \left[{0 \,.\,.\, 1}\right]: H\left({x, t}\right)) = f \left({x}\right)$

Trivially, if $K = \varnothing$, then $(1)$ and $(2)$ are vacuous truths, so relative homotopy generalises free homotopy.

Also see

 * Homotopy Class
 * Smooth Homotopy