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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]], let $\\mathcal F$ be a presheaf or a [[sheaf|TB6TM6R4MD]] of sets on $X$, and let $x \\in X$.\nConsider pairs $(U, s)$ where $U$ is an [[open set|02BH3MCBQB]] containing $x$ and $s \\in \\mathcal F(U)$.\nTwo pairs $(U, s)$ and $(V, t)$ are equivalent if there is an open set $W$ with $x \\in W \\subseteq U \\cap V$ and $s|_W = t|_W$.\nThe *stalk* of $\\mathcal F$ at $x$ is the set $\\mathcal F_x$ of equivalence classes.\nThe class of $(U, s)$ is the *germ* of $s$ at $x$, written $s_x$.\n\n## Remarks\n\nThe relation is an [[equivalence relation|0G8VEZ3WTK]]: for transitivity, intersect the two open sets $W$.\nIn the language of categories, $\\mathcal F_x$ is the direct limit of the sets $\\mathcal F(U)$ over the open neighbourhoods $U$ of $x$, ordered by reverse inclusion.\nIf $\\mathcal F$ is a sheaf of abelian groups or of rings, then each stalk is an abelian group or a ring, with $s_x + t_x = (s|_W + t|_W)_x$ for $W$ a common neighbourhood.\n\nExample: for the sheaf of continuous real functions on $\\R$, the stalk at $0$ is the ring of [[germs|VZXF7RNFKS]] of continuous functions at $0$.\nIt is a [[local ring|MZ02BKPHJA]]: the germs with $f(0) \\ne 0$ are units, since $1/f$ is continuous near $0$, and the other germs form the unique maximal ideal.\n\nExample: for the sheaf of holomorphic functions on $\\C$, the stalk at $0$ is the ring of power series $\\sum a_n z^n$ with positive radius of convergence.\n\nNon-example: a germ is not a function value.\nThe functions $0$ and $x \\mapsto \\max(x, 0)$ have the same value at $0$, but they have different germs at $0$, since they differ on every interval around $0$.\n\nFor a sheaf, a section $s \\in \\mathcal F(U)$ is determined by its germs $s_x$ for $x \\in U$, by the locality condition.\n",
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