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  "body": "\n## Definition\n\nLet $X$ be a [[path-connected|0WB6DH499T]] topological space.\nA *universal cover* of $X$ is a [[covering map|3XZRRE165N]] $p \\colon \\tilde X \\to X$ such that $\\tilde X$ is simply connected.\nHere $\\tilde X$ is simply connected if it is path-connected and its [[fundamental group|S1XK3HXHFX]] $\\pi_1(\\tilde X, \\tilde x)$ is trivial for one, hence every, point $\\tilde x \\in \\tilde X$.\n\n## Remarks\n\nExample: $p \\colon \\R \\to S^1$, $p(t) = (\\cos 2\\pi t, \\sin 2\\pi t)$, is a covering map.\nIts total space $\\R$ is path-connected and contractible, so $\\pi_1(\\R) = 1$, and $p$ is a universal cover of the circle.\nFor $n \\ge 2$, the quotient map $S^n \\to \\R P^n$ is a universal cover, because $S^n$ is simply connected.\n\nNon-example: the map $S^1 \\to S^1$, $z \\mapsto z^2$, on the unit circle in $\\C$ is a covering map.\nIts total space $S^1$ has fundamental group $\\Z$, so it is not a universal cover.\n\nIf $X$ is connected, locally path-connected and [[semilocally simply connected|R3SHZX0NZR]], then a universal cover exists.\nIf $X$ is locally path-connected, two universal covers of $X$ are isomorphic over $X$.\nA universal cover is often written $\\tilde X \\to X$.\n",
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