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  "body": "\n## Definition\n\nLet $L$ be a first-order language that contains a binary relation symbol $\u003c$.\nLet $M$ be an $L$-[[structure|983GZBQYT4]] in which $\u003c$ is interpreted as a [[total order|0FRXBVVR03]].\nThe structure $M$ is *o-minimal* if every subset of $M$ [[definable|CY1TXHF6KQ]] with parameters is a finite union of points and open [[intervals|0Q1D85B8G6]].\nHere the endpoints of an interval $(a, b)$ lie in $M \\cup \\set{-\\infty, +\\infty}$.\nAn $L$-[[theory|92SWXST1ZM]] $T$ is *o-minimal* if every model of $T$ is o-minimal.\n\n## Remarks\n\nExample: the ordered field $(\\R, +, \\cdot, \u003c, 0, 1)$ is o-minimal.\nBy the Tarski–Seidenberg theorem it has quantifier elimination, so a definable subset of $\\R$ is a Boolean combination of sets $\\set{x : p(x) \u003e 0}$ and $\\set{x : p(x) = 0}$ with $p$ a polynomial with real coefficients.\nEach such set is a finite union of points and open intervals, because a nonzero polynomial has finitely many roots.\n\nNon-example: the structure $(\\Z, \u003c, +)$ is not o-minimal.\nThe formula $\\exists y \\, (x = y + y)$ defines the set of even integers, which is not a finite union of points and intervals.\n\nBy a theorem of Knight, Pillay and Steinhorn, if a structure $M$ is o-minimal then its complete theory is o-minimal.\nDense linear orders without endpoints and real closed fields give o-minimal theories.\n",
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