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  "body": "\n## Definition\n\nLet $A$ and $X$ be topological spaces and let $I = [0, 1]$.\nLet $i \\colon A \\to X$ be a [[continuous map|0VGFHTM75Y]].\nThe map $i$ is a *cofibration* if the following holds for every topological space $Y$.\nFor every continuous map $f \\colon X \\to Y$ and every [[homotopy|9YA6S6HD9V]] $h \\colon A \\times I \\to Y$ with $h(a, 0) = f(i(a))$ for all $a \\in A$, there is a homotopy $H \\colon X \\times I \\to Y$ with:\n\n1. $H(x, 0) = f(x)$ for all $x \\in X$;\n2. $H(i(a), t) = h(a, t)$ for all $a \\in A$ and $t \\in I$.\n\n## Remarks\n\nFor the inclusion of a subspace $A \\subseteq X$, this says that the pair $(X, A)$ has the [[homotopy extension property|D42X0F1EG1]].\nEvery cofibration is a homeomorphism onto its image, and if $X$ is Hausdorff the image is closed in $X$.\nExample: the inclusion $S^{n-1} \\to D^n$ is a cofibration, and so is the inclusion of a subcomplex of a CW complex.\nNon-example: the inclusion $(0, 1) \\to [0, 1]$ is not a cofibration, since $[0, 1]$ is Hausdorff and $(0, 1)$ is not closed in it.\nCofibrations are stable under pushout and under composition.\n",
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