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  "body": "\n## Definition\n\nLet $X$ be a topological space, let $A \\subseteq X$ carry the [[subspace topology|0WS9P12FQR]], let $Y$ be a topological space, and let $I = [0, 1]$.\nLet $f \\colon X \\to Y$ be continuous, and let $h \\colon A \\times I \\to Y$ be a [[homotopy|9YA6S6HD9V]] with $h(a, 0) = f(a)$ for all $a \\in A$.\nAn *extension* of $h$ along $f$ is a homotopy $H \\colon X \\times I \\to Y$ such that:\n\n1. $H(x, 0) = f(x)$ for all $x \\in X$;\n2. $H(a, t) = h(a, t)$ for all $a \\in A$ and $t \\in I$.\n\nThe pair $(X, A)$ has the *homotopy extension property with respect to $Y$* if every such $h$ has an extension along every such $f$.\nThe pair $(X, A)$ has the *homotopy extension property* if it has it with respect to every topological space $Y$.\n\n## Remarks\n\nIf $A$ is closed in $X$, then $(X, A)$ has the homotopy extension property if and only if $X \\times \\set{0} \\cup A \\times I$ is a retract of $X \\times I$.\nExample: $(D^n, S^{n-1})$ has it, because the radial projection from the point $(0, \\ldots, 0, 2)$ of $\\R^{n+1}$ gives a retraction of $D^n \\times I$ onto $D^n \\times \\set{0} \\cup S^{n-1} \\times I$.\nNon-example: let $X = \\set{0} \\cup \\set{1/n : n \\ge 1}$ and $A = \\set{0}$.\nA retraction $r$ of $X \\times I$ onto $X \\times \\set{0} \\cup A \\times I$ would fix $(0, 1)$, and would send each connected set $\\set{1/n} \\times I$ to $(1/n, 0)$.\nThen $r(1/n, 1) = (1/n, 0) \\to (0, 0) \\neq r(0, 1)$, so $r$ is not continuous and the pair fails the property.\nEvery CW pair has the homotopy extension property.\n",
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