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  "body": "\n## Definition\n\nLet $X$ be a [[topological space|0CQHFPGJ45]] and let $(A_i)_{i \\in I}$ be a family of subsets of $X$.\nThe family is *locally finite* if every point $x \\in X$ has a [[neighbourhood|0ZCY0Z80B6]] $N$ such that the set $\\set{i \\in I : N \\cap A_i \\ne \\emptyset}$ is finite.\n\n## Remarks\n\nExample: in $\\R$, the family $([n, n + 1])_{n \\in \\Z}$ is locally finite.\nFor $x \\in \\R$, the neighbourhood $(x - 1, x + 1)$ meets $[n, n + 1]$ only when $x - 2 \u003c n \u003c x + 1$, which holds for at most three integers $n$.\n\nNon-example: in $\\R$, the family $((0, 1/n))_{n \\ge 1}$ is not locally finite.\nEvery neighbourhood of $0$ contains an interval $(-\\varepsilon, \\varepsilon)$, which meets every $(0, 1/n)$.\n\nIf $(A_i)_{i \\in I}$ is locally finite, then the [[closure|0QEAJB91GZ]] of $\\bigcup_i A_i$ equals $\\bigcup_i \\overline{A_i}$.\nIn particular, the union of a locally finite family of [[closed sets|0ZCK8DQYN1]] is closed.\n",
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