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  "body": "\n## Definition\n\nLet $X$ be a set with a [[topology|0CQHFPGJ45]] $\\tau$ and a [[partial order|0QV2JS9RJ3]] $\\le$.\nThe triple $(X, \\tau, \\le)$ is a *Priestley space* if:\n\n1. $(X, \\tau)$ is [[compact|0TVJZYB6SH]];\n2. for all $x, y \\in X$ with $x \\not\\le y$, some $U \\subseteq X$ has $x \\in U$ and $y \\notin U$, is open and closed in $\\tau$, and is an [[upper set|ETCE287E2R]].\n\nClause 2 is called the Priestley separation axiom.\n\n## Remarks\n\nA Priestley space is Hausdorff and totally disconnected, so its underlying space is a [[Stone space|ZV38HA5PAQ]].\nIndeed, for $x \\ne y$, say $x \\not\\le y$, the sets $U$ and $X \\setminus U$ from clause 2 separate $x$ and $y$.\n\nExample: a finite poset with the discrete topology.\nEvery subset is clopen, and for $x \\not\\le y$ the upper set ${\\uparrow} x = \\set{z : x \\le z}$ contains $x$ but not $y$.\n\nNon-example: the interval $[0, 1]$ with its usual topology and usual order.\nIt is compact, but its only clopen sets are $\\emptyset$ and $[0, 1]$, so no clopen set separates $1 \\not\\le 0$, and clause 2 fails.\n\nThe morphisms of Priestley spaces are the continuous order-preserving maps.\n",
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