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  "body": "\n## Definition\n\nLet $X$ and $Y$ be [[topological spaces|0CQHFPGJ45]] and let $f \\colon X \\to Y$ be a map.\nThe map $f$ is a *local homeomorphism* if every $x \\in X$ has an [[open|02BH3MCBQB]] neighbourhood $U$ with the following two properties.\nThe set $f(U)$ is open in $Y$.\nThe restriction $f|_U \\colon U \\to f(U)$ is a [[homeomorphism|0QT00PB5G2]], where $U$ and $f(U)$ carry the [[subspace topology|0WS9P12FQR]].\n\n## Remarks\n\nA local homeomorphism is [[continuous|0VGFHTM75Y]] and [[open|0YCG648KDD]].\nEvery homeomorphism is a local homeomorphism, and so is the inclusion of an open subset.\n\nExample: $p \\colon \\R \\to S^1$, $p(t) = (\\cos 2\\pi t, \\sin 2\\pi t)$.\nFor $x \\in \\R$, take $U = (x - 1/2, x + 1/2)$.\nThen $p(U)$ is the circle minus one point, which is open, and $p|_U$ is a continuous bijection onto it with a continuous inverse given by a branch of the angle.\n\nNon-example: $f \\colon \\C \\to \\C$, $f(z) = z^2$.\nEvery neighbourhood $U$ of $0$ contains some $\\varepsilon \\ne 0$ and $-\\varepsilon$, with $f(\\varepsilon) = f(-\\varepsilon)$, so $f|_U$ is not injective.\nAt every $z \\ne 0$, however, $f$ is a local homeomorphism.\n",
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