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  "body": "\n## Summary\n\nThe Lawson topology on a partially ordered set is the coarsest topology that contains both the [[Scott topology|ZR8A03FV67]] and the lower topology.\nThe Scott topology sees only the \"upward\" behaviour of the order, so it is rarely Hausdorff.\nAdding the lower topology repairs this: on a [[continuous lattice|RXBD58SGXZ]] the Lawson topology is compact and Hausdorff.\nIt is the main tool for treating domains and continuous lattices as genuine compact spaces.\n\n## Explanations\n\n### L1 human\n\nThe Scott topology has only open sets that are closed upward, so a point cannot be separated from the points above it.\nThe lower topology has the opposite bias: its basic open sets are the points that are not above some fixed elements.\nThe Lawson topology lets both kinds of sets be open, so it can separate points in both directions.\nOn the unit interval $[0, 1]$ with its usual order, the proper nonempty Scott open sets are the intervals $(a, 1]$.\nThe lower topology is generated by the intervals $[0, b)$.\nTogether they give the usual topology of $[0, 1]$.\n",
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