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  "body": "\n## Definition\n\nLet $(X, \u003c)$ be a [[totally ordered set|0FRXBVVR03]].\nThe *order topology* on $X$ is the [[topology|0CQHFPGJ45]] on $X$ with [[subbasis|09S8DV34Y0]] the open rays\n$$(a, \\to) = \\set{x \\in X : a \u003c x} \\quad\\text{and}\\quad (\\gets, b) = \\set{x \\in X : x \u003c b}, \\qquad a, b \\in X.$$\n\n## Remarks\n\nWhen $X$ has more than one point, the open intervals $(a, b)$ together with the open rays form a basis of the order topology.\n\nExample: on $\\R$ with its usual order, the open rays generate the open intervals, so the order topology is the usual topology of $\\R$.\n\nNon-example: let $Y = \\set{-1} \\cup \\set{1/n : n \\ge 1}$ as a subset of $\\R$.\nWith the subspace topology from $\\R$, the point $-1$ is isolated.\nIn the order topology of $Y$, every open set containing $-1$ contains a ray $(\\gets, 1/n)$, which contains $1/m$ for all $m \u003e n$.\nSo the subspace topology of $Y$ is not its order topology, although the order of $Y$ is the restriction of the order of $\\R$.\n\nSome authors also use the name for the topology generated by the same rays on an arbitrary partially ordered set.\n",
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