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  "body": "\n## Statement\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]] and let $U \\subseteq X$.\nThen $U$ is [[open|01ZQQDCERP]] in the [[metric topology|0A6YRNZB2R]] if and only if for each $x \\in U$ there is a real $\\varepsilon \u003e 0$ with $B(x, \\varepsilon) \\subseteq U$.\nHere $B(x, \\varepsilon)$ is the [[open ball|0DC3MPR7C9]] with centre $x$ and radius $\\varepsilon$.\n\n## Remarks\n\nThe number $\\varepsilon$ can depend on $x$.\nIn $\\mathbb{R}$, the set $(0, \\infty)$ is open: for $x \u003e 0$, take $\\varepsilon = x$.\nThe set $\\mathbb{Q}$ is not open in $\\mathbb{R}$, because each ball about a rational number contains an irrational number.\n\n## Explanations\n\n### L1 human\n\nThis is the test for open sets that you use in practice.\nYou pick any point of the set, and you find a small ball about it inside the set.\n",
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