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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]].\nA *pseudogroup* on $X$ is a set $\\Gamma$ of [[homeomorphisms|0QT00PB5G2]] $f \\colon U \\to V$ between open subsets $U, V$ of $X$ with these properties.\n\n1. The identity map of $X$ is in $\\Gamma$.\n2. If $f \\colon U \\to V$ is in $\\Gamma$, then $f^{-1} \\colon V \\to U$ is in $\\Gamma$.\n3. If $f \\colon U \\to V$ and $h \\colon U' \\to V'$ are in $\\Gamma$, then the composite $h \\circ f$, defined on the open set $f^{-1}(V \\cap U')$, is in $\\Gamma$.\n4. If $f \\colon U \\to V$ is in $\\Gamma$ and $W \\subseteq U$ is open, then the restriction $f|_W \\colon W \\to f(W)$ is in $\\Gamma$.\n5. Let $f \\colon U \\to V$ be a homeomorphism between open subsets of $X$, and let $U = \\bigcup_i U_i$ be a cover by open sets. If each restriction $f|_{U_i}$ is in $\\Gamma$, then $f$ is in $\\Gamma$.\n\n## Remarks\n\nExample: the set of all $C^\\infty$ diffeomorphisms between open subsets of $\\R^n$ is a pseudogroup on $\\R^n$.\nEach clause holds because being a diffeomorphism is preserved by inversion, composition and restriction, and it is a local property, which gives clause 5.\nThe same holds for real-analytic diffeomorphisms, for biholomorphic maps between open subsets of $\\C^n$ and for symplectomorphisms between open subsets of $\\R^{2n}$.\n\nNon-example: on $\\R$, let $\\Gamma$ be the set of restrictions to open sets of the bijective affine maps $x \\mapsto ax + b$, $a \\ne 0$.\nIt satisfies clauses 1 to 4 but fails clause 5.\nThe map on $(0, 1) \\cup (2, 3)$ that is the identity on $(0, 1)$ and $x \\mapsto x + 10$ on $(2, 3)$ is affine on each piece but is not the restriction of a single affine map.\n\nA pseudogroup generated by a set of local homeomorphisms is the smallest pseudogroup containing them.\nA $\\Gamma$-structure on a space, such as a smooth structure, is an atlas whose transition maps lie in $\\Gamma$.\n\n## Explanations\n\n### L1 human\n\nA pseudogroup is a set of local symmetries that can be composed, inverted, shrunk and patched together.\n",
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