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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]] and let ${}^*X$ be its extension in a model of [[nonstandard analysis|WM62GKCBWN]] that is an enlargement.\nFor an [[open set|02BH3MCBQB]] $U \\subseteq X$, write ${}^*U \\subseteq {}^*X$ for its extension.\nThe *monad* of a point $x \\in X$ is\n$$\\mu(x) = \\bigcap \\set{{}^*U : U \\subseteq X \\text{ open}, \\ x \\in U}.$$\nA point $y \\in {}^*X$ is *infinitely close* to $x$ if $y \\in \\mu(x)$.\n\n## Remarks\n\nExample: for $X = \\R$ with the usual topology, ${}^*X$ is the field of [[hyperreal numbers|421QMTFG9T]].\nA hyperreal $y$ lies in $\\mu(x)$ exactly when $\\abs{y - x} \u003c 1/n$ for every positive integer $n$, that is, when $y - x$ is infinitesimal; so $x + \\varepsilon \\in \\mu(x)$ for every infinitesimal $\\varepsilon$.\nNon-example: for $x = 0$ and a positive infinitesimal $\\varepsilon$, the hyperreal $1 + \\varepsilon$ is not in $\\mu(0)$, since it is not in ${}^*(-1/2, 1/2)$.\nThe [[standard part|1ZBY4W4E81]] of a finite hyperreal $y$ is the unique real $x$ with $y \\in \\mu(x)$.\nStandard characterizations in an enlargement: $X$ is [[Hausdorff|0S1FJ8VEX9]] if and only if distinct points have disjoint monads.\n$X$ is [[compact|0TVJZYB6SH]] if and only if every point of ${}^*X$ lies in the monad of some point of $X$.\nA map $f \\colon X \\to Y$ is [[continuous|0X3WGX95FE]] at $x$ if and only if ${}^*f(\\mu(x)) \\subseteq \\mu(f(x))$.\n",
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