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  "body": "\n## Statement\n\nLet $(X, d)$ be a [[metric space|063W5JMY85]].\nThen the set of all [[open balls|0DC3MPR7C9]] $B(x, r)$, with $x \\in X$ and real $r \u003e 0$, is a [[basis|06EX0FHBTF]] of the [[metric topology|0A6YRNZB2R]] of $X$.\nAlso, the balls $B(x, 1/n)$ with $x \\in X$ and integer $n \\ge 1$ form a basis.\n\n## Remarks\n\nProof: each ball is open, by [[open balls are open|0B2305GNAZ]]. If $U$ is open and $x \\in U$, then some ball $B(x, r)$ lies in $U$, and $B(x, 1/n) \\subseteq B(x, r)$ for $n \u003e 1/r$.\nFor each point $x$, the balls $B(x, 1/n)$ form a countable neighbourhood basis at $x$. Thus each metric space is [[first-countable|0HG11FW8MJ]].\nIn $\\mathbb{R}$, the open balls are the bounded open intervals.\n\n## Explanations\n\n### L1 human\n\nEach open set of a metric space is a union of open balls.\nSo, to check a property for all open sets, it is often enough to check it for balls.\n",
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