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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 $K$ if and only if 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.

Also see