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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]].\nFor $x \\in P$, write ${\\uparrow} x = \\set{y \\in P : x \\le y}$.\nThe *lower topology* on $P$ is the topology with [[subbasis|09S8DV34Y0]] the sets $P \\setminus {\\uparrow} x$ for $x \\in P$.\nThe *Lawson topology* on $P$ is the smallest [[topology|0CQHFPGJ45]] on $P$ that contains both the [[Scott topology|ZR8A03FV67]] and the lower topology.\nEquivalently, it is the topology generated by the union of these two topologies as a subbasis.\n\n## Remarks\n\nExample: on $P = [0, 1]$ with the usual order, the Scott open sets are $\\emptyset$, $[0, 1]$ and the intervals $(a, 1]$ with $0 \\le a \u003c 1$.\nThe lower open sets are generated by the intervals $P \\setminus {\\uparrow} b = [0, b)$.\nFinite intersections of these sets give all intervals $(a, b)$, $[0, b)$ and $(a, 1]$, so the Lawson topology is the usual topology of $[0, 1]$.\n\nNon-example: the Scott topology of $[0, 1]$ is not its Lawson topology.\nThe set $[0, 1/2)$ is Lawson open, but it is not an upper set, so it is not Scott open.\n\nThe Lawson topology of a [[continuous lattice|RXBD58SGXZ]] is compact and Hausdorff.\nSome authors define the Lawson topology only for directed-complete partial orders or for continuous posets, where it is best behaved.\n",
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