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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]], let $x \\in X$ and let $r \u003e 0$ be a real number.\nThe open ball with centre $x$ and radius $r$ is the set\n$$B(x, r) = \\{ y \\in X : d(x, y) \u003c r \\}.$$\n\n## Remarks\n\nThe centre is in the ball: $x \\in B(x, r)$, because $d(x, x) = 0 \u003c r$.\nIn $\\mathbb{R}$ with the usual metric, $B(x, r)$ is the open interval $(x - r, x + r)$.\nIn the [[discrete metric|0NA4FK7B8S]], $B(x, r) = \\{x\\}$ for $r \\le 1$ and $B(x, r) = X$ for $r \u003e 1$.\nSome authors also allow $r \\le 0$. Then $B(x, r)$ is empty.\n\n## Explanations\n\n### L1 human\n\nThe open ball $B(x, r)$ is the set of points that are less than $r$ away from $x$.\nIn the plane with the Euclidean metric, it is the inside of the circle of radius $r$ about $x$, without the circle itself.\n",
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