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  "body": "\n## Definition\n\nLet $X$ and $Y$ be topological spaces and let $A \\subseteq X$.\nLet $f, g \\colon X \\to Y$ be continuous maps with $f(a) = g(a)$ for all $a \\in A$.\nA *homotopy from $f$ to $g$ relative to $A$* is a [[homotopy|9YA6S6HD9V]] $H$ from $f$ to $g$ with $H(a, t) = f(a)$ for all $a \\in A$ and $t \\in [0, 1]$.\nThe maps $f$ and $g$ are *homotopic relative to $A$*, written $f \\simeq g \\ (\\mathrm{rel}\\ A)$, if such a homotopy exists.\n\n## Remarks\n\nFor $A = \\emptyset$ this is an ordinary homotopy.\nBeing homotopic relative to $A$ is an equivalence relation on the continuous maps $X \\to Y$ that agree with a given map on $A$.\nExample: for $X = [0, 1]$, $A = \\set{0, 1}$, $Y = \\R^2$, and two paths $f, g$ with $f(0) = g(0)$ and $f(1) = g(1)$, the map $H(s, t) = (1 - t) f(s) + t g(s)$ is a homotopy relative to $A$, since $H(0, t) = f(0)$ and $H(1, t) = f(1)$.\nNon-example: in $\\R^2 \\setminus \\set{0}$, the upper and lower half circles from $(1, 0)$ to $(-1, 0)$ are homotopic as maps, since $[0, 1]$ is contractible, but they are not homotopic relative to $\\set{0, 1}$.\n",
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