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  "body": "\n## Statement\n\nLet $p \\colon E \\to X$ be a [[covering map|3XZRRE165N]], let $I = [0, 1]$, let $\\gamma \\colon I \\to X$ be a continuous path, and let $e_0 \\in E$ with $p(e_0) = \\gamma(0)$.\nThen there is a unique continuous path $\\tilde\\gamma \\colon I \\to E$ with $\\tilde\\gamma(0) = e_0$ and $p \\circ \\tilde\\gamma = \\gamma$.\n\n## Remarks\n\nFor existence, cover $X$ by evenly covered open sets.\nBy the Lebesgue number lemma, there is a subdivision $0 = s_0 \u003c s_1 \u003c \\cdots \u003c s_k = 1$ of the [[interval|0Q1D85B8G6]] with each $\\gamma([s_{j-1}, s_j])$ in one evenly covered set $U_j$.\nThe lift is built step by step: on $[s_{j-1}, s_j]$ it is $(p|_{V})^{-1} \\circ \\gamma$, where $V$ is the sheet over $U_j$ that contains the point already reached.\nUniqueness holds because the set where two lifts agree is open and closed in the connected interval $I$, and it contains $0$.\nExample: for $p(t) = e^{2\\pi i t}$ from $\\R$ to $S^1$ and $\\gamma(s) = e^{2\\pi i n s}$, the lift starting at $0$ is $\\tilde\\gamma(s) = ns$, which ends at $n$.\nThe covering hypothesis is needed: for the local homeomorphism $q = p|_{(0, 2)} \\colon (0, 2) \\to S^1$, the loop $\\gamma(s) = e^{-2\\pi i s}$ with $e_0 = 1$ has no lift.\nA lift in $(0, 2)$ is also a lift in $\\R$, so by uniqueness it would be $s \\mapsto 1 - s$, whose value at $s = 1$ is $0 \\notin (0, 2)$.\n\n## Explanations\n\n### L1 human\n\nA covering map is like a projection from many stacked floors to a ground floor.\nIf you walk along a path on the ground floor and choose a floor to start on, there is exactly one way to walk upstairs that stays above your path.\nThis lets one read off information about loops downstairs from where the lifted path ends upstairs.\n\n### L2 human\n\nThe path lifting property, together with its extension to homotopies, is the main tool of covering space theory.\nIt shows that $p_* \\colon \\pi_1(E, e_0) \\to \\pi_1(X, p(e_0))$ is injective, and it computes the [[fundamental group of the circle|6SQXPE0JPN]] through the covering $\\R \\to S^1$.\nThe endpoint $\\tilde\\gamma(1)$ of the lift of a loop defines an action of $\\pi_1(X, x_0)$ on the fibre $p^{-1}(x_0)$, which leads to the classification of covering spaces.\n",
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