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  "body": "\n## Definition\n\nLet $P$ and $Q$ be [[partially ordered sets|0QV2JS9RJ3]].\nA map $f \\colon P \\to Q$ is *Scott-continuous* if it satisfies the following.\nFor every [[directed subset|D9RY4667K6]] $D \\subseteq P$ with a [[supremum|0R3H5PANRQ]] $\\bigvee D$ in $P$, the image $f(D)$ has a supremum in $Q$ and\n$$f\\left(\\bigvee D\\right) = \\bigvee f(D).$$\n\n## Remarks\n\nA Scott-continuous map is [[monotone|0A7KV2WB7A]]: if $x \\le y$, apply the definition to $D = \\set{x, y}$, whose supremum is $y$, to get $f(x) \\le f(y)$.\nA map $f$ is Scott-continuous if and only if it is [[continuous|0X3WGX95FE]] for the [[Scott topologies|ZR8A03FV67]] on $P$ and $Q$.\nExample: on the power set of $\\N$, the map $A \\mapsto A \\cup \\set{0}$; the image of the union of a directed family $D$ is $\\bigcup_{A \\in D} (A \\cup \\set{0})$.\nNon-example: on the power set of $\\N$, let $f(A) = \\N$ if $A$ is infinite and $f(A) = \\varnothing$ if $A$ is finite.\nThis map is monotone but not Scott-continuous.\nThe finite sets $\\set{0, \\ldots, n}$ form a directed family with union $\\N$, and $f(\\N) = \\N \\ne \\varnothing = \\bigvee_n f(\\set{0, \\ldots, n})$.\n",
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