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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]].\nA subset $U \\subseteq P$ is *Scott-open* if:\n\n1. $U$ is an [[upper set|ETCE287E2R]]: if $x \\in U$ and $x \\le y$, then $y \\in U$;\n2. $U$ is inaccessible by directed suprema: for every [[directed subset|D9RY4667K6]] $D \\subseteq P$ that has a [[supremum|0R3H5PANRQ]] with $\\bigvee D \\in U$, the set $D \\cap U$ is nonempty.\n\nThe Scott-open sets form a topology on $P$, the *Scott topology*.\n\n## Remarks\n\nThe Scott topology makes $P$ a [[topological space|0CQHFPGJ45]] that is $T_0$, and its [[specialization preorder|CMYVK9CEKZ]] is the order $\\le$; the closure of $\\set{x}$ is $\\set{y : y \\le x}$.\nIt is $T_1$ only when the order is trivial.\nExample: in $\\N \\cup \\set{\\infty}$, the set $U = \\set{n \\in \\N : n \\ge 3} \\cup \\set{\\infty}$ is Scott-open.\nIt is an upper set, and a directed $D$ with $\\bigvee D \\in U$ either contains its supremum, or is infinite and so contains some $n \\ge 3$.\nNon-example: in the same poset, $\\set{\\infty}$ is an upper set but is not Scott-open, since $D = \\N$ has supremum $\\infty$ and $D \\cap \\set{\\infty} = \\varnothing$.\nClosed sets of the Scott topology are the down-sets closed under suprema of directed subsets.\n",
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