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  "body": "\n## Definition\n\nLet $E$ and $B$ be [[topological spaces|0CQHFPGJ45]] and let $p \\colon E \\to B$ be a [[continuous map|0VGFHTM75Y]].\nWrite $I = [0, 1]$.\nThe map $p$ is a *Hurewicz fibration* if it has the [[homotopy lifting property|G4QDM01P6J]] with respect to every topological space $X$.\nThis means the following.\nLet $H \\colon X \\times I \\to B$ be a [[homotopy|9YA6S6HD9V]] and let $\\tilde{h}_0 \\colon X \\to E$ be continuous with $p \\circ \\tilde{h}_0 = H(-, 0)$.\nThen there is a homotopy $\\tilde{H} \\colon X \\times I \\to E$ with\n$$p \\circ \\tilde{H} = H \\quad \\text{and} \\quad \\tilde{H}(-, 0) = \\tilde{h}_0.$$\n\n## Remarks\n\nExample: for spaces $B$ and $F$, the projection $p \\colon B \\times F \\to B$ is a Hurewicz fibration.\nGiven $H$ and $\\tilde{h}_0 = (H(-, 0), g)$, the lift $\\tilde{H}(x, t) = (H(x, t), g(x))$ works.\n\nNon-example: the inclusion $p \\colon \\set{0} \\to I$ is not a fibration.\nTake $X$ a point, $H(t) = t$ and $\\tilde{h}_0 = 0$: a lift would satisfy $p(\\tilde{H}(1)) = 1$, but $1$ is not in the image of $p$.\n\nCovering maps and fibre bundles over paracompact Hausdorff bases are Hurewicz fibrations.\nIf $p$ is a Hurewicz fibration and $B$ is path-connected, all fibres $p^{-1}(b)$ are homotopy equivalent.\nEvery Hurewicz fibration is a [[Serre fibration|PB9JXQ2KC3]], where the lifting property is only required for $X$ a cube $I^n$ or a CW complex, but not conversely.\n",
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