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  "body": "\n## Summary\n\nThe down-closure of a subset $S$ of a [[partially ordered set|011C29Q1S1]] is the set of all elements that lie below some element of $S$.\nIt is the smallest [[lower set|BWYFKCWRW5]] that contains $S$.\nThe down-closure of a single element $x$ is the principal down-set $\\downarrow x$ of all elements below $x$.\n\n## Explanations\n\n### L1 human\n\nTo take the down-closure of a set, add everything that lies below one of its elements.\nIn the real numbers, the down-closure of $\\set{1, 5}$ is the half-line $(-\\infty, 5]$.\nIn the positive integers ordered by divisibility, the down-closure of $\\set{12}$ is the set of divisors of $12$.\nThe result is closed downwards, and it is the least such set that contains the original one.\n",
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