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  "body": "\n## Definition\n\nLet $E$, $B$ and $X$ be [[topological spaces|0CQHFPGJ45]], let $p \\colon E \\to B$ be a [[continuous map|0X3WGX95FE]], and let $I = [0, 1]$.\nThe map $p$ has the *homotopy lifting property* with respect to $X$ if the following holds.\nFor every continuous map $\\tilde f \\colon X \\to E$ and every [[homotopy|9YA6S6HD9V]] $H \\colon X \\times I \\to B$ with $H(x, 0) = p(\\tilde f(x))$ for all $x \\in X$, there is a homotopy $\\tilde H \\colon X \\times I \\to E$ such that:\n\n1. $p \\circ \\tilde H = H$.\n2. $\\tilde H(x, 0) = \\tilde f(x)$ for all $x \\in X$.\n\n## Remarks\n\nExample: every [[covering map|3XZRRE165N]] has the homotopy lifting property with respect to every space, and the lift is then unique.\nFor $X$ a point this is the path lifting property: for $p \\colon \\R \\to S^1$, $p(t) = e^{2\\pi i t}$, the loop $H(t) = e^{2\\pi i t}$ with starting point $0$ lifts to $\\tilde H(t) = t$.\n\nNon-example: let $E = \\set{0, 1}$ with the discrete topology, $B = [0, 1]$ and $p$ the inclusion.\nFor $X$ a point, $\\tilde f = 0$ and $H(t) = t$, clauses 1 and 2 would force $\\tilde H$ to be a path in $E$ from $0$ to $1$, but $E$ is not path-connected.\n\nIn diagram form: every commutative square with $X \\to E$ on top, $X \\times \\set{0} \\hookrightarrow X \\times I$ on the left and $p$ on the right has a diagonal $X \\times I \\to E$ making both triangles commute.\nA map with this property for all spaces $X$ is a [[Hurewicz fibration|Y773B7EQA7]]; for all cubes $I^n$, a [[Serre fibration|PB9JXQ2KC3]].\n",
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