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  "body": "\n## Statement\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]], let $x \\in X$ and let $r \u003e 0$.\nThen the [[open ball|0DC3MPR7C9]] $B(x, r)$ is an [[open set|01ZQQDCERP]] of $X$.\n\n## Remarks\n\nProof: let $y \\in B(x, r)$ and let $\\varepsilon = r - d(x, y) \u003e 0$. If $d(y, z) \u003c \\varepsilon$, then $d(x, z) \\le d(x, y) + d(y, z) \u003c r$ by the [[triangle inequality|01M6CW9VM6]]. Thus $B(y, \\varepsilon) \\subseteq B(x, r)$.\nThe open sets of $X$ are exactly the unions of open balls.\n\n## Explanations\n\n### L1 human\n\nA point inside an open ball is at some distance from the edge of the ball.\nA smaller ball about that point, with radius equal to this distance, stays inside the big ball.\n",
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