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  "body": "\n## Statement\n\nLet $p \\colon E \\to B$ be a [[covering map|3XZRRE165N]], and let $X$ be a [[topological space|0CQHFPGJ45]].\nLet $H \\colon X \\times [0, 1] \\to B$ be a [[homotopy|9YA6S6HD9V]], and let $\\tilde f \\colon X \\to E$ be a continuous map with $p \\circ \\tilde f = H(\\cdot, 0)$.\nThen there is a unique homotopy $\\tilde H \\colon X \\times [0, 1] \\to E$ such that $p \\circ \\tilde H = H$ and $\\tilde H(\\cdot, 0) = \\tilde f$.\n\n## Remarks\n\nFor each $x \\in X$, cover $\\{x\\} \\times [0, 1]$ by sets of the form $N \\times [t_i, t_{i+1}]$ that $H$ maps into evenly covered open sets of $B$.\nOn each such piece the lift is forced: it is the composite with the inverse of $p$ on the sheet that contains the lift already built.\nUniqueness of lifts on the connected sets $\\{x\\} \\times [0, 1]$ makes these local lifts agree, so they glue to a continuous $\\tilde H$.\nWhen $X$ is a point, the statement is the path lifting property: each path in $B$ lifts uniquely once the start point of the lift is chosen.\nApplied to $X = [0, 1]$ and a homotopy of loops rel endpoints, it shows that $p_* \\colon \\pi_1(E, e_0) \\to \\pi_1(B, p(e_0))$ is injective, where $\\pi_1$ is the [[fundamental group|S1XK3HXHFX]].\nFor example, take $p \\colon \\R \\to S^1$, $p(t) = (\\cos 2\\pi t, \\sin 2\\pi t)$, and a loop at $(1, 0)$ whose lift runs from $0$ to $n \\ne 0$.\nThis loop is not null-homotopic, because a null-homotopy would lift to a homotopy rel endpoints in $\\R$.\n\n## Explanations\n\n### L1 human\n\nA covering map looks locally like several stacked copies of the base.\nIf a movie of maps into the base is given, and the first frame is already lifted to the cover, then the whole movie lifts, and in only one way.\nThe picture is a helix over a circle: a motion on the circle can be followed upstairs on the helix without any choice once the starting level is fixed.\n\n### L2 human\n\nThis property is the engine of covering space theory.\nIt gives the injectivity of $p_*$ on fundamental groups, the computation $\\pi_1(S^1) \\cong \\Z$, and the lifting criterion behind the classification of covering spaces.\nIt also says that a covering map is a fibration with discrete fibers, so the long exact sequence of homotopy groups applies and gives $\\pi_n(E) \\cong \\pi_n(B)$ for $n \\ge 2$.\n",
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