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  "body": "\n## Summary\n\nA Priestley space is a compact topological space with a partial order in which any two points $x \\not\\le y$ are separated by a clopen upper set.\nPriestley spaces are the duals of bounded [[distributive lattices|0HM0HQCK6N]] in [[Priestley duality|SA22SEC75K]].\nForgetting the order, every Priestley space is a [[Stone space|ZV38HA5PAQ]], and a Stone space with the equality order is a Priestley space.\nSo Priestley spaces extend Stone duality from Boolean algebras to distributive lattices.\n\n## Explanations\n\n### L1 human\n\nA Priestley space carries two structures at once: a compact, very disconnected topology and an order.\nOne rule ties them together: whenever $x$ is not below $y$, some set that is open, closed and closed upward contains $x$ but not $y$.\nA finite poset with the discrete topology is the simplest example.\nThe clopen upper sets of a Priestley space form a distributive lattice, and every bounded distributive lattice arises in this way.\n",
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