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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]] and let $X$ be the set of [[prime ideals|00MHHZ9SNE]] of $R$.\nFor a subset $E \\subseteq R$, let\n$$V(E) = \\set{\\mathfrak p \\in X : E \\subseteq \\mathfrak p}.$$\nThe *Zariski topology* on $X$ is the [[topology|0CQHFPGJ45]] whose [[closed sets|0ZCK8DQYN1]] are exactly the sets $V(E)$ for $E \\subseteq R$.\n\n## Remarks\n\nThese sets are the closed sets of a topology:\n\n1. $V(\\set{0}) = X$ and $V(\\set{1}) = \\emptyset$;\n2. $\\bigcap_i V(E_i) = V\\left(\\bigcup_i E_i\\right)$ for any family of subsets $E_i$;\n3. $V(I) \\cup V(J) = V(IJ)$ for ideals $I, J$, because a prime ideal contains $IJ$ only if it contains $I$ or $J$.\n\nAlso $V(E) = V(I)$ for the ideal $I$ generated by $E$, and $V(I) = V(\\sqrt I)$.\nThe closure of a point $\\mathfrak p$ is $V(\\mathfrak p)$, so $\\set{\\mathfrak p}$ is closed exactly when $\\mathfrak p$ is a [[maximal ideal|0PV74TR5FP]].\n\nExample: in $\\operatorname{Spec} \\Z$, the closed sets are $\\operatorname{Spec} \\Z$ itself and the finite sets of nonzero prime ideals.\nIndeed $V(n\\Z)$ for $n \\ne 0$ is the finite set of $p\\Z$ with $p \\mid n$.\n\nNon-example: $\\operatorname{Spec} \\Z$ with this topology is not [[Hausdorff|0S1FJ8VEX9]].\nThe point $(0)$ is not closed: its closure is $V(0)$, the whole space.\nAny nonempty open set contains $(0)$, so no two nonempty open sets are disjoint.\n\nThe open sets $D(f) = X \\setminus V(\\set{f})$ are the [[basic open sets|N6JX9BCKQP]] and form a basis of the topology.\n",
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