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  "body": "\n## Definition\n\nLet $D$ be a set with a [[preorder|0VABYWJWH9]] $\\le$.\nThe set $D$ is *directed* (upward directed) when:\n\n1. $D$ is nonempty.\n2. For all $a, b \\in D$ there is $c \\in D$ with $a \\le c$ and $b \\le c$, that is, every two elements have an upper bound in $D$.\n\n## Remarks\n\nExample: the set of neighbourhoods of a point $x$ in a topological space, ordered by reverse inclusion ($U \\le V$ when $V \\subseteq U$), is directed.\nCheck: $x$ has the neighbourhood consisting of the whole space, and for neighbourhoods $U, V$ the set $U \\cap V$ is a neighbourhood with $U \\le U \\cap V$ and $V \\le U \\cap V$.\n\nNon-example: the set $\\set{a, b}$ with the trivial order, where each element is only below itself, fails clause 2, since $a$ and $b$ have no common upper bound.\n\nBy induction, every finite subset of a directed set has an upper bound in it.\nA downward directed set is defined dually, with lower bounds.\nSome authors, and the Mathlib predicate `IsDirected`, drop clause 1.\n",
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