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  "body": "\n## Definition\n\nLet $L$ be a [[signature|WDVVEYD6AT]], let $T$ be an $L$-[[theory|92SWXST1ZM]], and let $n \\ge 1$.\nAn *$n$-type* of $T$ is a set $p(x_1, \\dots, x_n)$ of $L$-[[formulas|HYJ10AF6RT]] whose free variables are among $x_1, \\dots, x_n$, such that $T \\cup p$ is satisfiable.\nThis means that some [[model|MVCH6N9SNW]] $M$ of $T$ has a tuple $(a_1, \\dots, a_n)$ that satisfies every formula of $p$.\nThe type $p$ is *complete* if for every $L$-formula $\\varphi(x_1, \\dots, x_n)$, either $\\varphi \\in p$ or $\\neg \\varphi \\in p$.\nA model $M$ of $T$ *realizes* $p$ if some tuple of $M$ satisfies every formula of $p$, and *omits* $p$ otherwise.\nThe set of complete $n$-types of $T$ is written $S_n(T)$.\n\nLet $M$ be a model and $A$ a set of elements of $M$.\nLet $L_A$ be $L$ extended by a new constant for each element of $A$.\nA type over $A$ is a type of the $L_A$-theory of $M$.\nThe set of complete $n$-types over $A$ is written $S_n^M(A)$.\n\n## Remarks\n\nFor a tuple $\\bar{a}$ of a model $M$, the set $\\operatorname{tp}(\\bar{a})$ of all formulas satisfied by $\\bar{a}$ in $M$ is a complete type of the theory of $M$.\n\nExample: let $T$ be the theory of $(\\N, 0, 1, +, \\cdot, \u003c)$ and let $p(x) = \\set{x \u003e 1 + \\cdots + 1 \\ (k \\text{ terms}) : k \\ge 1}$.\nEach finite part of $p$ is satisfied in $\\N$ by a large number, so by compactness $T \\cup p$ has a model, and $p$ is a $1$-type.\nThe standard model $\\N$ omits $p$, while every nonstandard model of $T$ realizes it.\n\nNon-example: in the theory of fields, $\\set{x = 0, \\; x \\cdot x = 1}$ is not a type, since $x = 0$ gives $x \\cdot x = 0 \\ne 1$ in every field.\n\nThe set $S_n(T)$ carries a natural topology, with basic open sets $\\set{p : \\varphi \\in p}$; it is a compact, totally disconnected Hausdorff space by the compactness theorem.\n",
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