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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|011C29Q1S1]] and let $U$ be a [[subset|0FE8Q5S42Y]] of $P$.\nThe set $U$ is an *upper set* when, for all $x, y \\in P$, if $x \\in U$ and $x \\le y$, then $y \\in U$.\nDually, a subset $L$ of $P$ is a *lower set* when, for all $x, y \\in P$, if $y \\in L$ and $x \\le y$, then $x \\in L$.\n\n## Remarks\n\nIn $(\\N, \\le)$, the set $\\set{n \\in \\N : n \\ge 5}$ is an upper set: if $n \\ge 5$ and $n \\le m$, then $m \\ge 5$.\nThe set of even numbers is not an upper set of $(\\N, \\le)$, since $2$ is in it, $2 \\le 3$, and $3$ is not.\nA subset is an upper set if and only if its complement is a lower set.\nUnions and intersections of any family of upper sets are upper sets.\nOther names are up-set and upward closed set for upper sets, and down-set, decreasing set and order ideal for lower sets.\nThe smallest lower set containing a subset $S$ is its [[down-closure|TGW5NTV3AA]].\n\n## Explanations\n\n### L1 human\n\nAn upper set never stops partway up: whatever lies above a member is also a member.\nA lower set is the same thing turned upside down.\n",
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