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  "body": "\n## Definition\n\nLet $E$ and $X$ be topological spaces and let $p \\colon E \\to X$ be a [[continuous map|0VGFHTM75Y]].\nAn open set $U \\subseteq X$ is *evenly covered* by $p$ if $p^{-1}(U)$ is a union of pairwise disjoint open sets $V_i \\subseteq E$, $i \\in J$, with one more property.\nFor each $i$, the restriction $p|_{V_i} \\colon V_i \\to U$ is a [[homeomorphism|0QT00PB5G2]], where $V_i$ and $U$ carry the [[subspace topology|0WS9P12FQR]].\nThe map $p$ is a *covering map* if every point of $X$ has an evenly covered open neighbourhood.\nThen $E$ is a *covering space* of $X$.\n\n## Remarks\n\nThe sets $V_i$ are the *sheets* over $U$.\nMany authors also require $p$ to be surjective, or $E$ to be nonempty and connected; this varies between textbooks.\nEvery covering map is a [[local homeomorphism|MDH71ZZ8XX]].\nExample: $p \\colon \\R \\to S^1$, $p(t) = e^{2\\pi i t}$, is a covering map.\nFor $z = e^{2\\pi i a}$, the arc $U = p((a - 1/2, a + 1/2))$ is evenly covered, with sheets $(a - 1/2 + n, a + 1/2 + n)$ for $n \\in \\Z$.\nNon-example: the restriction of this $p$ to $(0, 2)$ is a surjective local homeomorphism but not a covering map.\nFor every open arc $U$ around $1$, the part of $p^{-1}(U)$ near $0$ is a half-open piece $(0, \\varepsilon)$ that is not mapped onto $U$.\n",
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