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  "body": "\n## Statement\n\nLet $X$ be a [[path-connected|0WB6DH499T]], [[locally path-connected|1ZVR8BRK65]] and [[semilocally simply connected|R3SHZX0NZR]] space, and let $x_0 \\in X$.\nTwo pointed covering maps $p_1 \\colon (\\tilde X_1, \\tilde x_1) \\to (X, x_0)$ and $p_2 \\colon (\\tilde X_2, \\tilde x_2) \\to (X, x_0)$ are *isomorphic* if there is a homeomorphism $h \\colon \\tilde X_1 \\to \\tilde X_2$ with $p_2 \\circ h = p_1$ and $h(\\tilde x_1) = \\tilde x_2$.\nThen:\n\n1. Consider pointed [[covering maps|3XZRRE165N]] $p \\colon (\\tilde X, \\tilde x_0) \\to (X, x_0)$ with $\\tilde X$ path-connected. The rule $p \\mapsto p_*\\bigl(\\pi_1(\\tilde X, \\tilde x_0)\\bigr)$ is a bijection from their isomorphism classes onto the subgroups of the [[fundamental group|S1XK3HXHFX]] $\\pi_1(X, x_0)$.\n2. Consider isomorphisms that need not preserve base points. Then the same rule gives a bijection from isomorphism classes of covering maps with path-connected total space onto the [[conjugacy classes|0H11ZPX4ND]] of subgroups of $\\pi_1(X, x_0)$.\n\n## Remarks\n\nThe covering that belongs to a subgroup $H$ is built like the universal cover from homotopy classes of paths that start at $x_0$.\nTwo such classes are identified when the paths end at the same point and differ by a loop whose class lies in $H$.\nInjectivity uses the lifting criterion: a map from a path-connected, locally path-connected space lifts through a covering exactly when its image on $\\pi_1$ lies in the image of the covering.\nChanging the base point inside the fibre replaces the subgroup by a conjugate, which explains the second clause.\nFor $X = S^1$, the subgroups of $\\pi_1(S^1) = \\Z$ are $n\\Z$ with $n \\ge 0$; they belong to the coverings $z \\mapsto z^n$ for $n \\ge 1$ and to the universal cover $\\R \\to S^1$ for $n = 0$.\nWithout the semilocal condition the trivial subgroup may have no covering: the [[Hawaiian earring|73EBKQPP8Q]] has no simply connected covering space.\n\n## Explanations\n\n### L1 human\n\nFor a reasonable space, the connected coverings are described completely by the subgroups of its fundamental group.\nThe whole group belongs to the space itself, the trivial subgroup to the universal cover, and the other subgroups to the coverings in between.\nSo a topological question, which coverings exist, becomes an algebraic one, which subgroups exist.\n\n### L2 human\n\nThe theorem is the topological analogue of the fundamental theorem of Galois theory, with the universal cover in the role of an algebraic closure.\nIt is used to build coverings with given properties.\nFor example, each subgroup of a free group gives a covering of a graph, and this yields a quick proof that subgroups of free groups are free.\nFurther properties of the correspondence, about deck transformations and normal subgroups, are collected in the [[Galois correspondence for coverings|MTWNAJMH1D]].\n",
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