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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]] and let $x, y \\in P$.\nThe element $x$ is *way below* $y$, written $x \\ll y$, if the following holds.\nFor every [[directed subset|D9RY4667K6]] $D \\subseteq P$ with a [[supremum|0R3H5PANRQ]] such that $y \\le \\bigvee D$, there is $d \\in D$ with $x \\le d$.\n\n## Remarks\n\nIf $x \\ll y$, then $x \\le y$, by taking $D = \\set{y}$.\nIf $w \\le x \\ll y \\le z$, then $w \\ll z$.\nAn element $c$ is [[compact|9F7TYES4KE]] if and only if $c \\ll c$.\nExample: in $[0, 1]$ with the usual order, $x \\ll y$ holds if and only if $x = 0$ or $x \u003c y$; for $x \u003c y \\le \\bigvee D$, some $d \\in D$ exceeds $x$ by the definition of the supremum.\nNon-example: in $[0, 1]$, $1/2 \\not\\ll 1/2$, because $D = \\set{1/2 - 1/n : n \\ge 3}$ is directed with supremum $1/2$ and no member is at least $1/2$.\n",
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