\begin{definition}[Definition:Homotopy/Relative]
Let $X$ and $Y$ be topological spaces.
Let $f: X \to Y$, $g: X \to Y$ be continuous mappings.
Let $K \subseteq X$ be a subset of $X$.
We say that $f$ and $g$ are '''homotopic relative to $K$''' {{iff}} there exists a free homotopy $H$ between $f$ and $g$, and:
:$(1): \quad \forall x \in K: \map f x = \map g x$
:$(2): \quad \forall x \in K, t \in \closedint 0 1: \map H {x, t} = \map f x$
Trivially, if $K = \O$, then $(1)$ and $(2)$ are vacuous truths, so '''relative homotopy''' generalises free homotopy.
\end{definition}