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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]].\nAn element $c \\in P$ is *compact* if for every [[directed subset|D9RY4667K6]] $D \\subseteq P$ that has a [[supremum|0R3H5PANRQ]] $\\bigvee D$ in $P$ with $c \\le \\bigvee D$, there is $d \\in D$ with $c \\le d$.\n\n## Remarks\n\nIn a [[complete lattice|D7A0ZGZT5S]] $L$, an element $c$ is compact if and only if for every $S \\subseteq L$ with $c \\le \\bigvee S$ there is a finite $F \\subseteq S$ with $c \\le \\bigvee F$.\nCompact elements are also called finite elements, and in domain theory $c$ is compact exactly when $c \\ll c$ for the way-below relation.\nExample: in the power set of a set $X$, a finite set $c$ is compact.\nIf $c \\subseteq \\bigcup_{d \\in D} d$ for a directed family $D$, each of the finitely many points of $c$ lies in some member of $D$.\nDirectedness then gives one member containing all of them.\nNon-example: in the power set of $\\N$, the set $\\N$ is not compact.\nIt is the union of the directed family of sets $\\set{0, \\ldots, n}$ and lies in none of them.\nThe least element, when it exists, is always compact.\n",
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