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  "body": "\n## Summary\n\nLet $X$ be a topological space and ${}^*X$ its extension in [[nonstandard analysis|WM62GKCBWN]].\nThe monad of a point $x$ of $X$ is the set of points of ${}^*X$ that are infinitely close to $x$.\nIt is the intersection of the nonstandard extensions of all open neighbourhoods of $x$.\nMonads turn topological notions into simple statements about infinitely close points.\nFor example, a space is Hausdorff exactly when distinct points have disjoint monads.\n\n## Explanations\n\n### L1 human\n\nIn the [[hyperreal numbers|421QMTFG9T]] there are infinitesimals, numbers smaller in size than every positive real number but not zero.\nThe monad of a real number $x$ is the cloud of hyperreal numbers $x + \\varepsilon$ with $\\varepsilon$ infinitesimal.\nIt is a picture of \"all points that are infinitely near $x$\".\nFor a general space, the open sets around $x$ take the place of distances, and the monad is what all of them have in common after the nonstandard extension.\nContinuity then reads: points infinitely near $x$ go to points infinitely near $f(x)$.\n",
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