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  "body": "\n## Definition\n\nLet $P$ be a [[directed-complete partial order|7TSXP95QN6]] and let $\\ll$ be its [[way-below relation|RDM7J5J9HQ]].\nThe dcpo $P$ is a *continuous domain* if for every $x \\in P$ the set $\\set{y \\in P : y \\ll x}$ is [[directed|D9RY4667K6]] with supremum $x$.\nIt is an *algebraic domain* if for every $x \\in P$ the set of compact elements below $x$ is directed with supremum $x$.\n*Domain theory* is the study of continuous domains and related classes such as [[Scott domains|1SAAZG46DG]].\nIt studies them together with the [[Scott-continuous maps|C6GAHS1Z2P]] between them and their [[Scott topologies|ZR8A03FV67]].\n\n## Remarks\n\nExample: the interval $[0, 1]$ is a continuous domain that is not algebraic; $\\set{y : y \\ll x} = [0, x)$ for $x \u003e 0$, with supremum $x$, while the only compact element is $0$.\nExample: the partial functions from $\\N$ to $\\N$, ordered by extension, form an algebraic domain whose compact elements are the finite partial functions.\nNon-example: the lattice of closed subsets of $[0, 1]$, ordered by inclusion, is a dcpo, but only $\\varnothing$ is way below $[0, 1]$, so it is not a continuous domain.\nA central tool is the [[least fixed point|DBD64TH52D]]: a Scott-continuous map $f$ on a pointed dcpo has the least fixed point $\\bigvee_n f^n(\\bot)$.\nSuch fixed points give the meaning of recursive programs and of terms of the [[lambda calculus|BD756KR8FM]].\n",
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